36:I[651451,["/_next/static/chunks/2msykuvi-vydm.js?dpl=nogit-1790362613","/_next/static/chunks/2mu6-wo6ccpi3.js?dpl=nogit-1790362613","/_next/static/chunks/2epfhfe5im7r6.js?dpl=nogit-1790362613","/_next/static/chunks/1ew75_i487l-5.js?dpl=nogit-1790362613","/_next/static/chunks/0p4rexedanx97.js?dpl=nogit-1790362613","/_next/static/chunks/0u008gt6051u7.js?dpl=nogit-1790362613","/_next/static/chunks/1odwrcl12kfnl.js?dpl=nogit-1790362613","/_next/static/chunks/0zkcxwtbsqliz.js?dpl=nogit-1790362613","/_next/static/chunks/3lf9kdvvybemj.js?dpl=nogit-1790362613","/_next/static/chunks/2hyw6fyf3iqeh.js?dpl=nogit-1790362613","/_next/static/chunks/03q735zs9dbpu.js?dpl=nogit-1790362613","/_next/static/chunks/3r5hh4t0jf4es.js?dpl=nogit-1790362613","/_next/static/chunks/3ml3vooxhiupb.js?dpl=nogit-1790362613","/_next/static/chunks/1h4wccj42vygd.js?dpl=nogit-1790362613","/_next/static/chunks/12xoy6h_w_le6.js?dpl=nogit-1790362613","/_next/static/chunks/2os7nxouq91eq.js?dpl=nogit-1790362613","/_next/static/chunks/3ht89vb0p09bm.js?dpl=nogit-1790362613","/_next/static/chunks/3ocoqwe4aht_u.js?dpl=nogit-1790362613","/_next/static/chunks/32qmkt01ta8b2.js?dpl=nogit-1790362613","/_next/static/chunks/3jpl-42mbad5n.js?dpl=nogit-1790362613","/_next/static/chunks/3ao1re0xmmur_.js?dpl=nogit-1790362613","/_next/static/chunks/3xa2z7fbv2riw.js?dpl=nogit-1790362613","/_next/static/chunks/14kf-an-d2n-e.js?dpl=nogit-1790362613","/_next/static/chunks/381o-754ev9ia.js?dpl=nogit-1790362613","/_next/static/chunks/2tr2cll18vqv3.js?dpl=nogit-1790362613","/_next/static/chunks/0v4byuex1x38h.js?dpl=nogit-1790362613","/_next/static/chunks/3kom3ps_bn2q6.js?dpl=nogit-1790362613","/_next/static/chunks/1x_yxljsdq0vl.js?dpl=nogit-1790362613","/_next/static/chunks/1xkpwfqqdrdio.js?dpl=nogit-1790362613","/_next/static/chunks/04zr-wqafvwoa.js?dpl=nogit-1790362613","/_next/static/chunks/2dx-vzwyw9a3g.js?dpl=nogit-1790362613","/_next/static/chunks/2mv9l1xl_aa4j.js?dpl=nogit-1790362613","/_next/static/chunks/1ku-vdy4ggo41.js?dpl=nogit-1790362613","/_next/static/chunks/0k-y_4mcfub-p.js?dpl=nogit-1790362613","/_next/static/chunks/440m38krgocdi.js?dpl=nogit-1790362613","/_next/static/chunks/2p6gbbs6zvfpy.js?dpl=nogit-1790362613","/_next/static/chunks/172wsar4r7zzx.js?dpl=nogit-1790362613","/_next/static/chunks/0llqhg6-bxn_b.js?dpl=nogit-1790362613","/_next/static/chunks/1rbw_-uux6mpq.js?dpl=nogit-1790362613","/_next/static/chunks/1s6j_7_p0ugo9.js?dpl=nogit-1790362613","/_next/static/chunks/34ux0fo6f-rm8.js?dpl=nogit-1790362613","/_next/static/chunks/1pp_uf_jx5j9r.js?dpl=nogit-1790362613","/_next/static/chunks/2b6r02yb2zswo.js?dpl=nogit-1790362613","/_next/static/chunks/422ba61wql3d7.js?dpl=nogit-1790362613","/_next/static/chunks/2vg15qyptj-oc.js?dpl=nogit-1790362613","/_next/static/chunks/0d2u5k628p_lw.js?dpl=nogit-1790362613","/_next/static/chunks/3jzaxe43n3e1e.js?dpl=nogit-1790362613","/_next/static/chunks/1lsxkfxy262n-.js?dpl=nogit-1790362613","/_next/static/chunks/02e_f03j5ij1l.js?dpl=nogit-1790362613","/_next/static/chunks/12mv6p8rwcl5o.js?dpl=nogit-1790362613","/_next/static/chunks/0v5zhouho0zlq.js?dpl=nogit-1790362613","/_next/static/chunks/41x40gd5jiq0g.js?dpl=nogit-1790362613","/_next/static/chunks/2nnmvgmq4euk9.js?dpl=nogit-1790362613","/_next/static/chunks/1ksyn9m68446d.js?dpl=nogit-1790362613","/_next/static/chunks/3wqp-h5xnebw9.js?dpl=nogit-1790362613","/_next/static/chunks/30dopbgd1dkf9.js?dpl=nogit-1790362613","/_next/static/chunks/22z8pa0ys3w4l.js?dpl=nogit-1790362613","/_next/static/chunks/0zg1om_k0m5zv.js?dpl=nogit-1790362613","/_next/static/chunks/2ywf8cxod8dkv.js?dpl=nogit-1790362613","/_next/static/chunks/43ajf4ofaeu5y.js?dpl=nogit-1790362613","/_next/static/chunks/1__39-dgg7p8-.js?dpl=nogit-1790362613","/_next/static/chunks/1zt4k83kx3bkw.js?dpl=nogit-1790362613","/_next/static/chunks/0rt2_go1it43r.js?dpl=nogit-1790362613","/_next/static/chunks/200x0be4hbjpj.js?dpl=nogit-1790362613","/_next/static/chunks/35bn8r0ao86_s.js?dpl=nogit-1790362613","/_next/static/chunks/3xskuweujaw05.js?dpl=nogit-1790362613","/_next/static/chunks/2wlqtkfegn9hx.js?dpl=nogit-1790362613","/_next/static/chunks/2y341-r8ncizo.js?dpl=nogit-1790362613","/_next/static/chunks/03ai9i_h50rnc.js?dpl=nogit-1790362613","/_next/static/chunks/0pdzmybpnr872.js?dpl=nogit-1790362613"],"UserProgressProvider"] 37:I[196793,["/_next/static/chunks/2msykuvi-vydm.js?dpl=nogit-1790362613","/_next/static/chunks/2mu6-wo6ccpi3.js?dpl=nogit-1790362613","/_next/static/chunks/2epfhfe5im7r6.js?dpl=nogit-1790362613","/_next/static/chunks/1ew75_i487l-5.js?dpl=nogit-1790362613","/_next/static/chunks/0p4rexedanx97.js?dpl=nogit-1790362613","/_next/static/chunks/0u008gt6051u7.js?dpl=nogit-1790362613","/_next/static/chunks/1odwrcl12kfnl.js?dpl=nogit-1790362613","/_next/static/chunks/0zkcxwtbsqliz.js?dpl=nogit-1790362613","/_next/static/chunks/3lf9kdvvybemj.js?dpl=nogit-1790362613","/_next/static/chunks/2hyw6fyf3iqeh.js?dpl=nogit-1790362613","/_next/static/chunks/03q735zs9dbpu.js?dpl=nogit-1790362613","/_next/static/chunks/3r5hh4t0jf4es.js?dpl=nogit-1790362613","/_next/static/chunks/3ml3vooxhiupb.js?dpl=nogit-1790362613","/_next/static/chunks/1h4wccj42vygd.js?dpl=nogit-1790362613","/_next/static/chunks/12xoy6h_w_le6.js?dpl=nogit-1790362613","/_next/static/chunks/2os7nxouq91eq.js?dpl=nogit-1790362613","/_next/static/chunks/3ht89vb0p09bm.js?dpl=nogit-1790362613","/_next/static/chunks/3ocoqwe4aht_u.js?dpl=nogit-1790362613","/_next/static/chunks/32qmkt01ta8b2.js?dpl=nogit-1790362613","/_next/static/chunks/3jpl-42mbad5n.js?dpl=nogit-1790362613","/_next/static/chunks/3ao1re0xmmur_.js?dpl=nogit-1790362613","/_next/static/chunks/3xa2z7fbv2riw.js?dpl=nogit-1790362613","/_next/static/chunks/14kf-an-d2n-e.js?dpl=nogit-1790362613","/_next/static/chunks/381o-754ev9ia.js?dpl=nogit-1790362613","/_next/static/chunks/2tr2cll18vqv3.js?dpl=nogit-1790362613","/_next/static/chunks/0v4byuex1x38h.js?dpl=nogit-1790362613","/_next/static/chunks/3kom3ps_bn2q6.js?dpl=nogit-1790362613","/_next/static/chunks/1x_yxljsdq0vl.js?dpl=nogit-1790362613","/_next/static/chunks/1xkpwfqqdrdio.js?dpl=nogit-1790362613","/_next/static/chunks/04zr-wqafvwoa.js?dpl=nogit-1790362613","/_next/static/chunks/2dx-vzwyw9a3g.js?dpl=nogit-1790362613","/_next/static/chunks/2mv9l1xl_aa4j.js?dpl=nogit-1790362613","/_next/static/chunks/1ku-vdy4ggo41.js?dpl=nogit-1790362613","/_next/static/chunks/0k-y_4mcfub-p.js?dpl=nogit-1790362613","/_next/static/chunks/440m38krgocdi.js?dpl=nogit-1790362613","/_next/static/chunks/2p6gbbs6zvfpy.js?dpl=nogit-1790362613","/_next/static/chunks/172wsar4r7zzx.js?dpl=nogit-1790362613","/_next/static/chunks/0llqhg6-bxn_b.js?dpl=nogit-1790362613","/_next/static/chunks/1rbw_-uux6mpq.js?dpl=nogit-1790362613","/_next/static/chunks/1s6j_7_p0ugo9.js?dpl=nogit-1790362613","/_next/static/chunks/34ux0fo6f-rm8.js?dpl=nogit-1790362613","/_next/static/chunks/1pp_uf_jx5j9r.js?dpl=nogit-1790362613","/_next/static/chunks/2b6r02yb2zswo.js?dpl=nogit-1790362613","/_next/static/chunks/422ba61wql3d7.js?dpl=nogit-1790362613","/_next/static/chunks/2vg15qyptj-oc.js?dpl=nogit-1790362613","/_next/static/chunks/0d2u5k628p_lw.js?dpl=nogit-1790362613","/_next/static/chunks/3jzaxe43n3e1e.js?dpl=nogit-1790362613","/_next/static/chunks/1lsxkfxy262n-.js?dpl=nogit-1790362613","/_next/static/chunks/02e_f03j5ij1l.js?dpl=nogit-1790362613","/_next/static/chunks/12mv6p8rwcl5o.js?dpl=nogit-1790362613","/_next/static/chunks/0v5zhouho0zlq.js?dpl=nogit-1790362613","/_next/static/chunks/41x40gd5jiq0g.js?dpl=nogit-1790362613","/_next/static/chunks/2nnmvgmq4euk9.js?dpl=nogit-1790362613","/_next/static/chunks/1ksyn9m68446d.js?dpl=nogit-1790362613","/_next/static/chunks/3wqp-h5xnebw9.js?dpl=nogit-1790362613","/_next/static/chunks/30dopbgd1dkf9.js?dpl=nogit-1790362613","/_next/static/chunks/22z8pa0ys3w4l.js?dpl=nogit-1790362613","/_next/static/chunks/0zg1om_k0m5zv.js?dpl=nogit-1790362613","/_next/static/chunks/2ywf8cxod8dkv.js?dpl=nogit-1790362613","/_next/static/chunks/43ajf4ofaeu5y.js?dpl=nogit-1790362613","/_next/static/chunks/1__39-dgg7p8-.js?dpl=nogit-1790362613","/_next/static/chunks/1zt4k83kx3bkw.js?dpl=nogit-1790362613","/_next/static/chunks/0rt2_go1it43r.js?dpl=nogit-1790362613","/_next/static/chunks/200x0be4hbjpj.js?dpl=nogit-1790362613","/_next/static/chunks/35bn8r0ao86_s.js?dpl=nogit-1790362613","/_next/static/chunks/3xskuweujaw05.js?dpl=nogit-1790362613","/_next/static/chunks/2wlqtkfegn9hx.js?dpl=nogit-1790362613","/_next/static/chunks/2y341-r8ncizo.js?dpl=nogit-1790362613","/_next/static/chunks/03ai9i_h50rnc.js?dpl=nogit-1790362613","/_next/static/chunks/0pdzmybpnr872.js?dpl=nogit-1790362613"],"Gingerbread"] 39:I[716297,["/_next/static/chunks/2msykuvi-vydm.js?dpl=nogit-1790362613","/_next/static/chunks/2mu6-wo6ccpi3.js?dpl=nogit-1790362613","/_next/static/chunks/2epfhfe5im7r6.js?dpl=nogit-1790362613","/_next/static/chunks/1ew75_i487l-5.js?dpl=nogit-1790362613","/_next/static/chunks/0p4rexedanx97.js?dpl=nogit-1790362613","/_next/static/chunks/0u008gt6051u7.js?dpl=nogit-1790362613","/_next/static/chunks/1odwrcl12kfnl.js?dpl=nogit-1790362613","/_next/static/chunks/0zkcxwtbsqliz.js?dpl=nogit-1790362613","/_next/static/chunks/3lf9kdvvybemj.js?dpl=nogit-1790362613","/_next/static/chunks/2hyw6fyf3iqeh.js?dpl=nogit-1790362613","/_next/static/chunks/03q735zs9dbpu.js?dpl=nogit-1790362613","/_next/static/chunks/3r5hh4t0jf4es.js?dpl=nogit-1790362613","/_next/static/chunks/3ml3vooxhiupb.js?dpl=nogit-1790362613","/_next/static/chunks/1h4wccj42vygd.js?dpl=nogit-1790362613","/_next/static/chunks/12xoy6h_w_le6.js?dpl=nogit-1790362613","/_next/static/chunks/2os7nxouq91eq.js?dpl=nogit-1790362613","/_next/static/chunks/3ht89vb0p09bm.js?dpl=nogit-1790362613","/_next/static/chunks/3ocoqwe4aht_u.js?dpl=nogit-1790362613","/_next/static/chunks/32qmkt01ta8b2.js?dpl=nogit-1790362613","/_next/static/chunks/3jpl-42mbad5n.js?dpl=nogit-1790362613","/_next/static/chunks/3ao1re0xmmur_.js?dpl=nogit-1790362613","/_next/static/chunks/3xa2z7fbv2riw.js?dpl=nogit-1790362613","/_next/static/chunks/14kf-an-d2n-e.js?dpl=nogit-1790362613","/_next/static/chunks/381o-754ev9ia.js?dpl=nogit-1790362613","/_next/static/chunks/2tr2cll18vqv3.js?dpl=nogit-1790362613","/_next/static/chunks/0v4byuex1x38h.js?dpl=nogit-1790362613","/_next/static/chunks/3kom3ps_bn2q6.js?dpl=nogit-1790362613","/_next/static/chunks/1x_yxljsdq0vl.js?dpl=nogit-1790362613","/_next/static/chunks/1xkpwfqqdrdio.js?dpl=nogit-1790362613","/_next/static/chunks/04zr-wqafvwoa.js?dpl=nogit-1790362613","/_next/static/chunks/2dx-vzwyw9a3g.js?dpl=nogit-1790362613","/_next/static/chunks/2mv9l1xl_aa4j.js?dpl=nogit-1790362613","/_next/static/chunks/1ku-vdy4ggo41.js?dpl=nogit-1790362613","/_next/static/chunks/0k-y_4mcfub-p.js?dpl=nogit-1790362613","/_next/static/chunks/440m38krgocdi.js?dpl=nogit-1790362613","/_next/static/chunks/2p6gbbs6zvfpy.js?dpl=nogit-1790362613","/_next/static/chunks/172wsar4r7zzx.js?dpl=nogit-1790362613","/_next/static/chunks/0llqhg6-bxn_b.js?dpl=nogit-1790362613","/_next/static/chunks/1rbw_-uux6mpq.js?dpl=nogit-1790362613","/_next/static/chunks/1s6j_7_p0ugo9.js?dpl=nogit-1790362613","/_next/static/chunks/34ux0fo6f-rm8.js?dpl=nogit-1790362613","/_next/static/chunks/1pp_uf_jx5j9r.js?dpl=nogit-1790362613","/_next/static/chunks/2b6r02yb2zswo.js?dpl=nogit-1790362613","/_next/static/chunks/422ba61wql3d7.js?dpl=nogit-1790362613","/_next/static/chunks/2vg15qyptj-oc.js?dpl=nogit-1790362613","/_next/static/chunks/0d2u5k628p_lw.js?dpl=nogit-1790362613","/_next/static/chunks/3jzaxe43n3e1e.js?dpl=nogit-1790362613","/_next/static/chunks/1lsxkfxy262n-.js?dpl=nogit-1790362613","/_next/static/chunks/02e_f03j5ij1l.js?dpl=nogit-1790362613","/_next/static/chunks/12mv6p8rwcl5o.js?dpl=nogit-1790362613","/_next/static/chunks/0v5zhouho0zlq.js?dpl=nogit-1790362613","/_next/static/chunks/41x40gd5jiq0g.js?dpl=nogit-1790362613","/_next/static/chunks/2nnmvgmq4euk9.js?dpl=nogit-1790362613","/_next/static/chunks/1ksyn9m68446d.js?dpl=nogit-1790362613","/_next/static/chunks/3wqp-h5xnebw9.js?dpl=nogit-1790362613","/_next/static/chunks/30dopbgd1dkf9.js?dpl=nogit-1790362613","/_next/static/chunks/22z8pa0ys3w4l.js?dpl=nogit-1790362613","/_next/static/chunks/0zg1om_k0m5zv.js?dpl=nogit-1790362613","/_next/static/chunks/2ywf8cxod8dkv.js?dpl=nogit-1790362613","/_next/static/chunks/43ajf4ofaeu5y.js?dpl=nogit-1790362613","/_next/static/chunks/1__39-dgg7p8-.js?dpl=nogit-1790362613","/_next/static/chunks/1zt4k83kx3bkw.js?dpl=nogit-1790362613","/_next/static/chunks/0rt2_go1it43r.js?dpl=nogit-1790362613","/_next/static/chunks/200x0be4hbjpj.js?dpl=nogit-1790362613","/_next/static/chunks/35bn8r0ao86_s.js?dpl=nogit-1790362613","/_next/static/chunks/3xskuweujaw05.js?dpl=nogit-1790362613","/_next/static/chunks/2wlqtkfegn9hx.js?dpl=nogit-1790362613","/_next/static/chunks/2y341-r8ncizo.js?dpl=nogit-1790362613","/_next/static/chunks/03ai9i_h50rnc.js?dpl=nogit-1790362613","/_next/static/chunks/0pdzmybpnr872.js?dpl=nogit-1790362613"],"TopicRecentlyViewed"] 3a:I[647935,["/_next/static/chunks/2msykuvi-vydm.js?dpl=nogit-1790362613","/_next/static/chunks/2mu6-wo6ccpi3.js?dpl=nogit-1790362613","/_next/static/chunks/2epfhfe5im7r6.js?dpl=nogit-1790362613","/_next/static/chunks/1ew75_i487l-5.js?dpl=nogit-1790362613","/_next/static/chunks/0p4rexedanx97.js?dpl=nogit-1790362613","/_next/static/chunks/0u008gt6051u7.js?dpl=nogit-1790362613","/_next/static/chunks/1odwrcl12kfnl.js?dpl=nogit-1790362613","/_next/static/chunks/0zkcxwtbsqliz.js?dpl=nogit-1790362613","/_next/static/chunks/3lf9kdvvybemj.js?dpl=nogit-1790362613","/_next/static/chunks/2hyw6fyf3iqeh.js?dpl=nogit-1790362613","/_next/static/chunks/03q735zs9dbpu.js?dpl=nogit-1790362613","/_next/static/chunks/3r5hh4t0jf4es.js?dpl=nogit-1790362613","/_next/static/chunks/3ml3vooxhiupb.js?dpl=nogit-1790362613","/_next/static/chunks/1h4wccj42vygd.js?dpl=nogit-1790362613","/_next/static/chunks/12xoy6h_w_le6.js?dpl=nogit-1790362613","/_next/static/chunks/2os7nxouq91eq.js?dpl=nogit-1790362613","/_next/static/chunks/3ht89vb0p09bm.js?dpl=nogit-1790362613","/_next/static/chunks/3ocoqwe4aht_u.js?dpl=nogit-1790362613","/_next/static/chunks/32qmkt01ta8b2.js?dpl=nogit-1790362613","/_next/static/chunks/3jpl-42mbad5n.js?dpl=nogit-1790362613","/_next/static/chunks/3ao1re0xmmur_.js?dpl=nogit-1790362613","/_next/static/chunks/3xa2z7fbv2riw.js?dpl=nogit-1790362613","/_next/static/chunks/14kf-an-d2n-e.js?dpl=nogit-1790362613","/_next/static/chunks/381o-754ev9ia.js?dpl=nogit-1790362613","/_next/static/chunks/2tr2cll18vqv3.js?dpl=nogit-1790362613","/_next/static/chunks/0v4byuex1x38h.js?dpl=nogit-1790362613","/_next/static/chunks/3kom3ps_bn2q6.js?dpl=nogit-1790362613","/_next/static/chunks/1x_yxljsdq0vl.js?dpl=nogit-1790362613","/_next/static/chunks/1xkpwfqqdrdio.js?dpl=nogit-1790362613","/_next/static/chunks/04zr-wqafvwoa.js?dpl=nogit-1790362613","/_next/static/chunks/2dx-vzwyw9a3g.js?dpl=nogit-1790362613","/_next/static/chunks/2mv9l1xl_aa4j.js?dpl=nogit-1790362613","/_next/static/chunks/1ku-vdy4ggo41.js?dpl=nogit-1790362613","/_next/static/chunks/0k-y_4mcfub-p.js?dpl=nogit-1790362613","/_next/static/chunks/440m38krgocdi.js?dpl=nogit-1790362613","/_next/static/chunks/2p6gbbs6zvfpy.js?dpl=nogit-1790362613","/_next/static/chunks/172wsar4r7zzx.js?dpl=nogit-1790362613","/_next/static/chunks/0llqhg6-bxn_b.js?dpl=nogit-1790362613","/_next/static/chunks/1rbw_-uux6mpq.js?dpl=nogit-1790362613","/_next/static/chunks/1s6j_7_p0ugo9.js?dpl=nogit-1790362613","/_next/static/chunks/34ux0fo6f-rm8.js?dpl=nogit-1790362613","/_next/static/chunks/1pp_uf_jx5j9r.js?dpl=nogit-1790362613","/_next/static/chunks/2b6r02yb2zswo.js?dpl=nogit-1790362613","/_next/static/chunks/422ba61wql3d7.js?dpl=nogit-1790362613","/_next/static/chunks/2vg15qyptj-oc.js?dpl=nogit-1790362613","/_next/static/chunks/0d2u5k628p_lw.js?dpl=nogit-1790362613","/_next/static/chunks/3jzaxe43n3e1e.js?dpl=nogit-1790362613","/_next/static/chunks/1lsxkfxy262n-.js?dpl=nogit-1790362613","/_next/static/chunks/02e_f03j5ij1l.js?dpl=nogit-1790362613","/_next/static/chunks/12mv6p8rwcl5o.js?dpl=nogit-1790362613","/_next/static/chunks/0v5zhouho0zlq.js?dpl=nogit-1790362613","/_next/static/chunks/41x40gd5jiq0g.js?dpl=nogit-1790362613","/_next/static/chunks/2nnmvgmq4euk9.js?dpl=nogit-1790362613","/_next/static/chunks/1ksyn9m68446d.js?dpl=nogit-1790362613","/_next/static/chunks/3wqp-h5xnebw9.js?dpl=nogit-1790362613","/_next/static/chunks/30dopbgd1dkf9.js?dpl=nogit-1790362613","/_next/static/chunks/22z8pa0ys3w4l.js?dpl=nogit-1790362613","/_next/static/chunks/0zg1om_k0m5zv.js?dpl=nogit-1790362613","/_next/static/chunks/2ywf8cxod8dkv.js?dpl=nogit-1790362613","/_next/static/chunks/43ajf4ofaeu5y.js?dpl=nogit-1790362613","/_next/static/chunks/1__39-dgg7p8-.js?dpl=nogit-1790362613","/_next/static/chunks/1zt4k83kx3bkw.js?dpl=nogit-1790362613","/_next/static/chunks/0rt2_go1it43r.js?dpl=nogit-1790362613","/_next/static/chunks/200x0be4hbjpj.js?dpl=nogit-1790362613","/_next/static/chunks/35bn8r0ao86_s.js?dpl=nogit-1790362613","/_next/static/chunks/3xskuweujaw05.js?dpl=nogit-1790362613","/_next/static/chunks/2wlqtkfegn9hx.js?dpl=nogit-1790362613","/_next/static/chunks/2y341-r8ncizo.js?dpl=nogit-1790362613","/_next/static/chunks/03ai9i_h50rnc.js?dpl=nogit-1790362613","/_next/static/chunks/0pdzmybpnr872.js?dpl=nogit-1790362613"],"RecordResourceResume"] 3b:I[426249,["/_next/static/chunks/2msykuvi-vydm.js?dpl=nogit-1790362613","/_next/static/chunks/2mu6-wo6ccpi3.js?dpl=nogit-1790362613","/_next/static/chunks/2epfhfe5im7r6.js?dpl=nogit-1790362613","/_next/static/chunks/1ew75_i487l-5.js?dpl=nogit-1790362613","/_next/static/chunks/0p4rexedanx97.js?dpl=nogit-1790362613","/_next/static/chunks/0u008gt6051u7.js?dpl=nogit-1790362613","/_next/static/chunks/1odwrcl12kfnl.js?dpl=nogit-1790362613","/_next/static/chunks/0zkcxwtbsqliz.js?dpl=nogit-1790362613","/_next/static/chunks/3lf9kdvvybemj.js?dpl=nogit-1790362613","/_next/static/chunks/2hyw6fyf3iqeh.js?dpl=nogit-1790362613","/_next/static/chunks/03q735zs9dbpu.js?dpl=nogit-1790362613","/_next/static/chunks/3r5hh4t0jf4es.js?dpl=nogit-1790362613","/_next/static/chunks/3ml3vooxhiupb.js?dpl=nogit-1790362613","/_next/static/chunks/1h4wccj42vygd.js?dpl=nogit-1790362613","/_next/static/chunks/12xoy6h_w_le6.js?dpl=nogit-1790362613","/_next/static/chunks/2os7nxouq91eq.js?dpl=nogit-1790362613","/_next/static/chunks/3ht89vb0p09bm.js?dpl=nogit-1790362613","/_next/static/chunks/3ocoqwe4aht_u.js?dpl=nogit-1790362613","/_next/static/chunks/32qmkt01ta8b2.js?dpl=nogit-1790362613","/_next/static/chunks/3jpl-42mbad5n.js?dpl=nogit-1790362613","/_next/static/chunks/3ao1re0xmmur_.js?dpl=nogit-1790362613","/_next/static/chunks/3xa2z7fbv2riw.js?dpl=nogit-1790362613","/_next/static/chunks/14kf-an-d2n-e.js?dpl=nogit-1790362613","/_next/static/chunks/381o-754ev9ia.js?dpl=nogit-1790362613","/_next/static/chunks/2tr2cll18vqv3.js?dpl=nogit-1790362613","/_next/static/chunks/0v4byuex1x38h.js?dpl=nogit-1790362613","/_next/static/chunks/3kom3ps_bn2q6.js?dpl=nogit-1790362613","/_next/static/chunks/1x_yxljsdq0vl.js?dpl=nogit-1790362613","/_next/static/chunks/1xkpwfqqdrdio.js?dpl=nogit-1790362613","/_next/static/chunks/04zr-wqafvwoa.js?dpl=nogit-1790362613","/_next/static/chunks/2dx-vzwyw9a3g.js?dpl=nogit-1790362613","/_next/static/chunks/2mv9l1xl_aa4j.js?dpl=nogit-1790362613","/_next/static/chunks/1ku-vdy4ggo41.js?dpl=nogit-1790362613","/_next/static/chunks/0k-y_4mcfub-p.js?dpl=nogit-1790362613","/_next/static/chunks/440m38krgocdi.js?dpl=nogit-1790362613","/_next/static/chunks/2p6gbbs6zvfpy.js?dpl=nogit-1790362613","/_next/static/chunks/172wsar4r7zzx.js?dpl=nogit-1790362613","/_next/static/chunks/0llqhg6-bxn_b.js?dpl=nogit-1790362613","/_next/static/chunks/1rbw_-uux6mpq.js?dpl=nogit-1790362613","/_next/static/chunks/1s6j_7_p0ugo9.js?dpl=nogit-1790362613","/_next/static/chunks/34ux0fo6f-rm8.js?dpl=nogit-1790362613","/_next/static/chunks/1pp_uf_jx5j9r.js?dpl=nogit-1790362613","/_next/static/chunks/2b6r02yb2zswo.js?dpl=nogit-1790362613","/_next/static/chunks/422ba61wql3d7.js?dpl=nogit-1790362613","/_next/static/chunks/2vg15qyptj-oc.js?dpl=nogit-1790362613","/_next/static/chunks/0d2u5k628p_lw.js?dpl=nogit-1790362613","/_next/static/chunks/3jzaxe43n3e1e.js?dpl=nogit-1790362613","/_next/static/chunks/1lsxkfxy262n-.js?dpl=nogit-1790362613","/_next/static/chunks/02e_f03j5ij1l.js?dpl=nogit-1790362613","/_next/static/chunks/12mv6p8rwcl5o.js?dpl=nogit-1790362613","/_next/static/chunks/0v5zhouho0zlq.js?dpl=nogit-1790362613","/_next/static/chunks/41x40gd5jiq0g.js?dpl=nogit-1790362613","/_next/static/chunks/2nnmvgmq4euk9.js?dpl=nogit-1790362613","/_next/static/chunks/1ksyn9m68446d.js?dpl=nogit-1790362613","/_next/static/chunks/3wqp-h5xnebw9.js?dpl=nogit-1790362613","/_next/static/chunks/30dopbgd1dkf9.js?dpl=nogit-1790362613","/_next/static/chunks/22z8pa0ys3w4l.js?dpl=nogit-1790362613","/_next/static/chunks/0zg1om_k0m5zv.js?dpl=nogit-1790362613","/_next/static/chunks/2ywf8cxod8dkv.js?dpl=nogit-1790362613","/_next/static/chunks/43ajf4ofaeu5y.js?dpl=nogit-1790362613","/_next/static/chunks/1__39-dgg7p8-.js?dpl=nogit-1790362613","/_next/static/chunks/1zt4k83kx3bkw.js?dpl=nogit-1790362613","/_next/static/chunks/0rt2_go1it43r.js?dpl=nogit-1790362613","/_next/static/chunks/200x0be4hbjpj.js?dpl=nogit-1790362613","/_next/static/chunks/35bn8r0ao86_s.js?dpl=nogit-1790362613","/_next/static/chunks/3xskuweujaw05.js?dpl=nogit-1790362613","/_next/static/chunks/2wlqtkfegn9hx.js?dpl=nogit-1790362613","/_next/static/chunks/2y341-r8ncizo.js?dpl=nogit-1790362613","/_next/static/chunks/03ai9i_h50rnc.js?dpl=nogit-1790362613","/_next/static/chunks/0pdzmybpnr872.js?dpl=nogit-1790362613"],"TopicQuestionBank"] 24:["$","$L36",null,{"bundle":{"topicIds":["f9661294-95d4-42ef-9ff4-0bf0c7ac3086"],"paperIds":[],"stageIds":[],"stageTopicIds":[],"stageAssessmentIds":[],"topicNoteIds":[]},"children":[["$","$L37",null,{"feature":"question-bank"}],"$L38",["$","$L39",null,{"subjectCode":"a_level_maths__aqa","subjectName":"Maths","topicSlug":"the-normal-distribution","topicTitle":"2.4.2 The Normal distribution (A-level only)","sectionTitle":null,"activity":"worksheet","resourceTitle":"2.4.2 The Normal distribution (A-level only) Questions","resourceHref":"/a-level/maths/aqa/the-normal-distribution/questions/q1","saveTrail":false}],["$","$L3a",null,{"href":"/a-level/maths/aqa/the-normal-distribution/questions/q1","label":"2.4.2 The Normal distribution (A-level only)","subjectCode":"a_level_maths__aqa","surface":"questions"}],["$","script",null,{"type":"application/ld+json","dangerouslySetInnerHTML":{"__html":"{\"@context\":\"https://schema.org\",\"@type\":\"LearningResource\",\"name\":\"Question 69 – 2.4.2 The Normal distribution (A-level only) - Questions\",\"description\":\"Practise 2.4.2 The Normal distribution (A-level only) A Level Maths exam-style questions with worked solutions on MathsGenie.\",\"learningResourceType\":\"Practice questions\",\"educationalLevel\":\"A Level\",\"about\":{\"@type\":\"Thing\",\"name\":\"2.4.2 The Normal distribution (A-level only)\"},\"url\":\"https://mathsgenie.co.uk/a-level/maths/aqa/the-normal-distribution/questions/q69\",\"inLanguage\":\"en-GB\",\"provider\":{\"@id\":\"https://mathsgenie.co.uk/#organization\"}}"}}],["$","div",null,{"className":"min-h-0 w-full flex-1 overflow-hidden bg-background","children":["$","$L3b",null,{"backHref":"/a-level/maths/aqa/questions","basePath":"/a-level/maths/aqa/the-normal-distribution/questions","formulaUrl":"https://filestore.aqa.org.uk/resources/mathematics/AQA-AS-A-MATHS-FORMULAE.PDF","initialIndex":68,"initialQuestionWindow":[{"id":"c38d64e5-443c-4967-91b2-6203ba0fdbb9","revisionId":"2729cbf1-b9d0-406a-89d2-05f3457a7d26","questionNumber":null,"questionType":"LA","specification":"The lifespan of a certain brand of light bulb is normally distributed with a mean of 1200 hours and a standard deviation of 80 hours.","difficulty":1,"difficultyBand":"medium","tier":null,"totalMarks":6,"sourceTitle":null,"calculatorUse":null,"isCalculation":true,"parts":[{"id":"eefea5a1-2a0f-4f3b-9631-fc6adccbae51","label":"a","content":"Find the probability that a randomly selected light bulb lasts for less than 1050 hours.","markscheme":"- $P(X < 1050) = 0.0304$ (3sf) M1 A1","marks":2,"order":0,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Attempt to find $P(X < 1050)$ using normal distribution parameters.","code":"M"},{"id":"p2","marks":1,"description":"Correct probability of 0.0304 (3sf).","code":"A"}],"finalAnswers":["0.0304"],"generatedAt":"2026-05-06T14:04:51.913Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"direct_calculator","label":"Direct Calculator Method","steps":[{"type":"text","order":0,"title":"Identify the distribution parameters","content":"First, we define our random variable and its distribution using the information given in the question.\n\nLet $X$ be the lifespan of a randomly selected light bulb in hours. The question states that $X$ is normally distributed with:\n* Mean, $\\mu = 1200$\n* Standard deviation, $\\sigma = 80$\n\nWe write this distribution mathematically as:\n$$X \\sim \\text{N}(1200, 80^2)$$\n\nWe need to find the probability that a bulb lasts less than 1050 hours, which is written as:\n$$P(X < 1050)$$","highlights":[{"text":"mean of 1200 hours","location":"specification"},{"text":"standard deviation of 80 hours","location":"specification"},{"text":"less than 1050 hours","location":"specification"}]},{"type":"text","order":1,"title":"Use your GDC directly","content":"Rather than calculating a $Z$-score and using tables, we can find this probability directly using the cumulative normal distribution function on a graphing calculator.\n\nTo find $P(X < 1050)$, we define our lower and upper bounds:\n* **Lower Bound**: a very large negative number, such as $-99999$ or $-1 \\times 10^{99}$\n* **Upper Bound**: $1050$\n* **Standard Deviation ($\\sigma$)**: $80$\n* **Mean ($\\mu$)**: $1200$","calculator":[{"gdc":{"expression":"NormCD(-99999, 1050, 80, 1200)","instructions":"1. Go to **Menu** > **Probability** > **Distributions** > **Normal Cdf** (or **Main Menu** > **Statistics** > **Dist** > **NORM** > **Ncd**).\n2. Set the fields:\n - **Lower**: -99999\n - **Upper**: 1050\n - **σ**: 80\n - **μ**: 1200\n3. Execute the calculation."},"ti84":{"expression":"normalcdf(-1E99, 1050, 1200, 80)","instructions":"1. Press [2nd] [VARS] to access the **DISTR** menu.\n2. Select **2: normalcdf**.\n3. Enter the following parameters:\n - **lower**: -1E99\n - **upper**: 1050\n - **μ**: 1200\n - **σ**: 80\n4. Highlight **Paste** and press [ENTER] twice."},"desmos":{"expression":"normaldist(1200, 80).cdf(-\\infty, 1050)","instructions":"In a fresh cell, define the normal distribution and cumulative probability:\n1. Type `normaldist(1200, 80)`.\n2. Check the **Find Cumulative Probability (CDF)** box.\n3. Set the **Min** to `-infinity` (leave blank) and the **Max** to `1050`."},"description":"Calculate the normal cumulative probability using your graphing calculator."}]},{"type":"text","order":2,"title":"State the final probability","content":"The calculator output gives us the exact cumulative probability:\n$$P(X < 1050) \\approx 0.030396...$$\n\nRounding this value to $3$ significant figures gives:\n$$P(X < 1050) = 0.0304$$"}]},{"id":"standardization_tables","label":"Standardization and Z-score Method","steps":[{"type":"text","order":0,"title":"Define the normal distribution","content":"Let the random variable $X$ represent the lifespan of a light bulb in hours.\n\nFrom the question, $X$ is normally distributed with a mean of $\\mu = 1200$ and a standard deviation of $\\sigma = 80$:\n\n$$X \\sim N(1200, 80^2)$$\n\nWe need to find the probability that a bulb lasts less than 1050 hours, which is written as $P(X < 1050)$.","highlights":[{"text":"mean of 1200 hours","location":"specification"},{"text":"standard deviation of 80 hours","location":"specification"},{"text":"less than 1050 hours","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$$Z = \\frac{\\htmlData{anchor=x-val}{\\color{@sky}{X}} - \\htmlData{anchor=mean}{\\color{@amber}{\\mu}}}{\\htmlData{anchor=sd}{\\color{@purple}{\\sigma}}}$$"},{"latex":"$$$Z = \\frac{\\htmlData{anchor=x-sub}{\\color{@sky}{1050}} - \\htmlData{anchor=mean-sub}{\\color{@amber}{1200}}}{\\htmlData{anchor=sd-sub}{\\color{@purple}{80}}}$$"},{"latex":"$$$Z = -1.875$$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"x-val"},{"color":"amber","style":"text-color","anchor":"mean"},{"color":"purple","style":"text-color","anchor":"sd"}],"connections":[{"to":"x-sub","from":"x-val","color":"sky","label":"Boundary value","style":"solid"},{"to":"mean-sub","from":"mean","color":"amber","label":"Mean","style":"solid"},{"to":"sd-sub","from":"sd","color":"purple","label":"Standard deviation","style":"solid"}]},"order":1,"title":"Calculate the z-score","description":"To standardize our value of 1050, we use the standard normal distribution formula to find its $z$-score."},{"type":"text","order":2,"title":"Find the probability","content":"Now, we find the probability of being less than this $z$-score:\n\n$$P(X < 1050) = P(Z < -1.875)$$\n\nUsing the symmetry of the standard normal distribution:\n\n$$P(Z < -1.875) = 1 - \\Phi(1.875)$$\n\nFrom the standard normal distribution tables (or using a calculator):\n\n$$\\Phi(1.875) \\approx 0.9696$$\n\nTherefore:\n\n$$P(Z < -1.875) = 1 - 0.9696 = 0.0304$$\n\nThus, the probability that a light bulb lasts less than 1050 hours is **$0.0304$** (to 3 significant figures).","calculator":[{"gdc":{"expression":"NormCD(-9999, -1.875, 1, 0)","instructions":"Go to Statistics -> Distribution -> Normal -> Ncd. Set Lower: -9999, Upper: -1.875, σ: 1, μ: 0."},"ti84":{"expression":"normalcdf(-9999, -1.875, 0, 1)","instructions":"Press [2nd] [VARS] (DISTR), select 2:normalcdf(. Enter lower: -9999, upper: -1.875, μ: 0, σ: 1, and select Paste."},"desmos":{"expression":"normaldist(0,1).cdf(-\\infty, -1.875)","instructions":"Type 'normaldist(0, 1).cdf(-infinity, -1.875)' into the input bar."},"description":"Calculate the cumulative probability using standard normal distribution functions."}]}]}],"generatedAt":"2026-06-09T09:24:23.102Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]},{"id":"6b4a912c-b5f5-453b-b9fd-e9f27043b5a9","label":"b","content":"An engineer claims that the manufacturing process has changed, causing the mean lifespan to decrease.\n\nState the engineer's null and alternative hypotheses.","markscheme":"- $H_0: \\mu = 1200$ and $H_1: \\mu < 1200$ B1","marks":1,"order":1,"markingRubric":{"totalMarks":1,"points":[{"id":"p1","marks":1,"description":"Correct hypotheses with mean 1200.","code":"B"}],"finalAnswers":["H_0: \\mu = 1200, H_1: \\mu < 1200"],"generatedAt":"2026-05-06T14:04:51.913Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"symbolic_hypotheses","label":"Symbolic Hypothesis Formulation","steps":[{"type":"text","order":0,"title":"Define the population parameter","content":"To formulate our hypotheses, we first need to define the population parameter we are investigating.\n\nHere, we are interested in the mean lifespan of the light bulbs. Let's define:\n\n* $\\mu$: the population mean lifespan of the light bulbs (in hours)."},{"type":"text","order":1,"title":"Formulate the null hypothesis","content":"The null hypothesis, denoted by $H_0$, represents the status quo or the assumption that nothing has changed.\n\nOriginally, the light bulbs had a mean lifespan of $1200$ hours. Therefore, our null hypothesis states that the mean lifespan remains at this value:\n\n$$H_0: \\mu = 1200$$","highlights":[{"text":"mean of 1200 hours","location":"specification"}]},{"type":"text","order":2,"title":"Formulate the alternative hypothesis","content":"The alternative hypothesis, denoted by $H_1$, represents the engineer's claim that we want to test.\n\nThe engineer claims that the manufacturing change has caused the mean lifespan to **decrease** (less than $1200$ hours). Since we are only looking for a change in one direction, this is a one-tailed test:\n\n$$H_1: \\mu < 1200$$","highlights":[{"text":"causing the mean lifespan to decrease","location":"specification"}]},{"type":"text","order":3,"title":"State the final hypotheses","content":"Combining both parts, the final hypotheses are:\n\n$$H_0: \\mu = 1200$$\n$$H_1: \\mu < 1200$$"}]}],"generatedAt":"2026-06-09T09:24:50.035Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]},{"id":"f8461670-a156-45b8-9d2f-eab558f183eb","label":"c","content":"A sample of 30 light bulbs is tested and the mean lifespan is found to be 1175 hours.\n\nCarry out a test at the 10% significance level to determine if there is evidence that the mean lifespan is less than 1200 hours.","markscheme":"- Identifies distribution of sample mean: $\\bar{X} \\sim N(1200, \\frac{80^2}{30})$ M1\n- Calculates $P(\\bar{X} < 1175) = 0.0435$ (3sf) A1\n- Compares $0.0435 < 0.10$, rejects $H_0$ and concludes there is evidence to suggest the mean lifespan is less than 1200 hours. A1","marks":3,"order":2,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"Distribution $\\bar{X} \\sim N(1200, \\frac{80^2}{30})$ or standard error $\\frac{80}{\\sqrt{30}}$ identified.","code":"M"},{"id":"p2","marks":1,"description":"Correct p-value of 0.0435.","code":"A"},{"id":"p3","marks":1,"description":"Comparison to 0.10 and rejection of $H_0$ with conclusion.","code":"A"}],"finalAnswers":["0.0435","Reject H_0"],"generatedAt":"2026-05-06T14:04:51.913Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"p_value_approach","label":"p-value Approach","steps":[{"type":"text","order":0,"title":"Define the hypotheses","content":"First, we state our null hypothesis ($H_0$) and alternative hypothesis ($H_1$). The null hypothesis represents the status quo, which is that the mean lifespan is 1200 hours. The alternative hypothesis represents what we are testing for—whether the mean lifespan is less than 1200 hours: $$H_0: \\mu = 1200$$ $$H_1: \\mu < 1200$$ Where $\\mu$ is the population mean lifespan of the light bulbs.","highlights":[{"text":"mean lifespan is less than 1200 hours","location":"specification"}]},{"type":"text","order":1,"title":"Distribution of sample mean","content":"Under the null hypothesis $H_0$, the individual bulb lifespans are distributed as $X \\sim N(1200, 80^2)$. For a sample of size $n = 30$, the sample mean $\\bar{X}$ is normally distributed with the same mean but a reduced variance: $$\\bar{X} \\sim N\\left(1200, \\frac{80^2}{30}\\right)$$ This distribution has a standard error of: $$\\text{SE} = \\frac{80}{\\sqrt{30}} \\approx 14.61\\text{ hours}$$ Note: Identifying this correct distribution for $\\bar{X}$ earns you the first mark.","highlights":[{"text":"normally distributed with a mean of 1200 hours and a standard deviation of 80 hours","location":"specification"},{"text":"sample of 30 light bulbs","location":"specification"}]},{"type":"text","order":2,"title":"Calculate the p-value","content":"Our observed sample mean is $\\bar{x} = 1175$ hours. Since we are conducting a one-tailed test to see if the mean is less than 1200, the p-value is the probability of obtaining a sample mean as low as or lower than 1175: $$p\\text{-value} = P(\\bar{X} \\le 1175)$$ Using a graphing calculator to find this normal cumulative probability gives: $$P(\\bar{X} \\le 1175) \\approx 0.0435\\text{ (3 sf)}$$ Note: Calculating this probability correctly earns the second mark.","highlights":[{"text":"mean lifespan is found to be 1175 hours","location":"specification"}]},{"type":"text","order":3,"title":"Make the decision","content":"Now we compare our calculated $p$-value to the significance level $\\alpha = 0.10$: $$0.0435 < 0.10$$ Because the $p$-value ($0.0435$) is less than the significance level ($0.10$), we reject the null hypothesis $H_0$. We conclude that there is sufficient evidence at the 10% significance level to suggest that the mean lifespan of the light bulbs is less than 1200 hours. Note: Correctly comparing the p-value to the significance level and writing this conclusion in context earns the final mark.","highlights":[{"text":"10% significance level","location":"specification"}]}]},{"id":"critical_value_approach","label":"Critical Value Approach","steps":[{"type":"text","order":0,"title":"Define hypotheses","content":"First, we establish our null hypothesis ($H_0$) and alternative hypothesis ($H_1$). Since we want to test if there is evidence that the mean lifespan is less than 1200 hours: $$H_0: \\mu = 1200$$ $$H_1: \\mu < 1200$$ This is a one-tailed test with a significance level of $\\alpha = 0.10$.","highlights":[{"text":"mean lifespan is less than 1200 hours","location":"specification"},{"text":"10% significance level","location":"specification"}]},{"type":"text","order":1,"title":"State sample mean distribution","content":"Under the null hypothesis $H_0$, the sample mean $\\bar{X}$ of $n = 30$ light bulbs is normally distributed. Using the formula for the standard error, $\\sigma_{\\bar{x}} = \\frac{\\sigma}{\\sqrt{n}}$: $$\\bar{X} \\sim N\\left(1200, \\frac{80^2}{30}\\right)$$ Award 1 mark for identifying the correct distribution of the sample mean.","highlights":[{"text":"A sample of 30 light bulbs is tested","location":"specification"},{"text":"mean of 1200 hours and a standard deviation of 80 hours","location":"specification"}]},{"type":"text","order":2,"title":"Calculate critical value","content":"Using the Critical Value Approach, we find the critical value of the sample mean, $\\bar{x}_{crit}$, that corresponds to the boundary of our 10% rejection region in the lower tail. For a standard normal distribution, the critical $z$-value for a one-tailed test at the 10% level is: $$z_{crit} = -1.2816$$ Now, we convert this critical $z$-value back to the sample mean scale using: $$\\bar{x}_{crit} = \\mu + z_{crit} \\times \\frac{\\sigma}{\\sqrt{n}}$$ $$\\bar{x}_{crit} = 1200 + (-1.2816) \\times \\frac{80}{\\sqrt{30}} \\approx 1181.28\\text{ hours}$$"},{"type":"text","order":3,"title":"Compare and conclude","content":"Our observed sample mean is $\\bar{x} = 1175$ hours. Since our observed sample mean lies in the rejection region: $$1175 < 1181.28$$ Alternatively, we can compute the $z$-statistic for our sample mean: $$z = \\frac{1175 - 1200}{\\frac{80}{\\sqrt{30}}} \\approx -1.712$$ Since $z \\approx -1.712 < -1.2816$, we reject $H_0$. Conclusion: There is sufficient evidence at the 10% significance level to suggest that the mean lifespan of the light bulbs is less than 1200 hours.","highlights":[{"text":"mean lifespan is found to be 1175 hours","location":"specification"}]}]}],"generatedAt":"2026-06-09T09:25:54.214Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"eefea5a1-2a0f-4f3b-9631-fc6adccbae51:direct_calculator","label":"(a) Direct Calculator Method","steps":[{"type":"text","order":0,"title":"Identify the distribution parameters","content":"First, we define our random variable and its distribution using the information given in the question.\n\nLet $X$ be the lifespan of a randomly selected light bulb in hours. The question states that $X$ is normally distributed with:\n* Mean, $\\mu = 1200$\n* Standard deviation, $\\sigma = 80$\n\nWe write this distribution mathematically as:\n$$X \\sim \\text{N}(1200, 80^2)$$\n\nWe need to find the probability that a bulb lasts less than 1050 hours, which is written as:\n$$P(X < 1050)$$","highlights":[{"text":"mean of 1200 hours","location":"specification"},{"text":"standard deviation of 80 hours","location":"specification"},{"text":"less than 1050 hours","location":"specification"}]},{"type":"text","order":1,"title":"Use your GDC directly","content":"Rather than calculating a $Z$-score and using tables, we can find this probability directly using the cumulative normal distribution function on a graphing calculator.\n\nTo find $P(X < 1050)$, we define our lower and upper bounds:\n* **Lower Bound**: a very large negative number, such as $-99999$ or $-1 \\times 10^{99}$\n* **Upper Bound**: $1050$\n* **Standard Deviation ($\\sigma$)**: $80$\n* **Mean ($\\mu$)**: $1200$","calculator":[{"gdc":{"expression":"NormCD(-99999, 1050, 80, 1200)","instructions":"1. Go to **Menu** > **Probability** > **Distributions** > **Normal Cdf** (or **Main Menu** > **Statistics** > **Dist** > **NORM** > **Ncd**).\n2. Set the fields:\n - **Lower**: -99999\n - **Upper**: 1050\n - **σ**: 80\n - **μ**: 1200\n3. Execute the calculation."},"ti84":{"expression":"normalcdf(-1E99, 1050, 1200, 80)","instructions":"1. Press [2nd] [VARS] to access the **DISTR** menu.\n2. Select **2: normalcdf**.\n3. Enter the following parameters:\n - **lower**: -1E99\n - **upper**: 1050\n - **μ**: 1200\n - **σ**: 80\n4. Highlight **Paste** and press [ENTER] twice."},"desmos":{"expression":"normaldist(1200, 80).cdf(-\\infty, 1050)","instructions":"In a fresh cell, define the normal distribution and cumulative probability:\n1. Type `normaldist(1200, 80)`.\n2. Check the **Find Cumulative Probability (CDF)** box.\n3. Set the **Min** to `-infinity` (leave blank) and the **Max** to `1050`."},"description":"Calculate the normal cumulative probability using your graphing calculator."}]},{"type":"text","order":2,"title":"State the final probability","content":"The calculator output gives us the exact cumulative probability:\n$$P(X < 1050) \\approx 0.030396...$$\n\nRounding this value to $3$ significant figures gives:\n$$P(X < 1050) = 0.0304$$"}]},{"id":"eefea5a1-2a0f-4f3b-9631-fc6adccbae51:standardization_tables","label":"(a) Standardization and Z-score Method","steps":[{"type":"text","order":0,"title":"Define the normal distribution","content":"Let the random variable $X$ represent the lifespan of a light bulb in hours.\n\nFrom the question, $X$ is normally distributed with a mean of $\\mu = 1200$ and a standard deviation of $\\sigma = 80$:\n\n$$X \\sim N(1200, 80^2)$$\n\nWe need to find the probability that a bulb lasts less than 1050 hours, which is written as $P(X < 1050)$.","highlights":[{"text":"mean of 1200 hours","location":"specification"},{"text":"standard deviation of 80 hours","location":"specification"},{"text":"less than 1050 hours","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$$Z = \\frac{\\htmlData{anchor=x-val}{\\color{@sky}{X}} - \\htmlData{anchor=mean}{\\color{@amber}{\\mu}}}{\\htmlData{anchor=sd}{\\color{@purple}{\\sigma}}}$$"},{"latex":"$$$Z = \\frac{\\htmlData{anchor=x-sub}{\\color{@sky}{1050}} - \\htmlData{anchor=mean-sub}{\\color{@amber}{1200}}}{\\htmlData{anchor=sd-sub}{\\color{@purple}{80}}}$$"},{"latex":"$$$Z = -1.875$$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"x-val"},{"color":"amber","style":"text-color","anchor":"mean"},{"color":"purple","style":"text-color","anchor":"sd"}],"connections":[{"to":"x-sub","from":"x-val","color":"sky","label":"Boundary value","style":"solid"},{"to":"mean-sub","from":"mean","color":"amber","label":"Mean","style":"solid"},{"to":"sd-sub","from":"sd","color":"purple","label":"Standard deviation","style":"solid"}]},"order":1,"title":"Calculate the z-score","description":"To standardize our value of 1050, we use the standard normal distribution formula to find its $z$-score."},{"type":"text","order":2,"title":"Find the probability","content":"Now, we find the probability of being less than this $z$-score:\n\n$$P(X < 1050) = P(Z < -1.875)$$\n\nUsing the symmetry of the standard normal distribution:\n\n$$P(Z < -1.875) = 1 - \\Phi(1.875)$$\n\nFrom the standard normal distribution tables (or using a calculator):\n\n$$\\Phi(1.875) \\approx 0.9696$$\n\nTherefore:\n\n$$P(Z < -1.875) = 1 - 0.9696 = 0.0304$$\n\nThus, the probability that a light bulb lasts less than 1050 hours is **$0.0304$** (to 3 significant figures).","calculator":[{"gdc":{"expression":"NormCD(-9999, -1.875, 1, 0)","instructions":"Go to Statistics -> Distribution -> Normal -> Ncd. Set Lower: -9999, Upper: -1.875, σ: 1, μ: 0."},"ti84":{"expression":"normalcdf(-9999, -1.875, 0, 1)","instructions":"Press [2nd] [VARS] (DISTR), select 2:normalcdf(. Enter lower: -9999, upper: -1.875, μ: 0, σ: 1, and select Paste."},"desmos":{"expression":"normaldist(0,1).cdf(-\\infty, -1.875)","instructions":"Type 'normaldist(0, 1).cdf(-infinity, -1.875)' into the input bar."},"description":"Calculate the cumulative probability using standard normal distribution functions."}]}]},{"id":"6b4a912c-b5f5-453b-b9fd-e9f27043b5a9:symbolic_hypotheses","label":"(b) Symbolic Hypothesis Formulation","steps":[{"type":"text","order":0,"title":"Define the population parameter","content":"To formulate our hypotheses, we first need to define the population parameter we are investigating.\n\nHere, we are interested in the mean lifespan of the light bulbs. Let's define:\n\n* $\\mu$: the population mean lifespan of the light bulbs (in hours)."},{"type":"text","order":1,"title":"Formulate the null hypothesis","content":"The null hypothesis, denoted by $H_0$, represents the status quo or the assumption that nothing has changed.\n\nOriginally, the light bulbs had a mean lifespan of $1200$ hours. Therefore, our null hypothesis states that the mean lifespan remains at this value:\n\n$$H_0: \\mu = 1200$$","highlights":[{"text":"mean of 1200 hours","location":"specification"}]},{"type":"text","order":2,"title":"Formulate the alternative hypothesis","content":"The alternative hypothesis, denoted by $H_1$, represents the engineer's claim that we want to test.\n\nThe engineer claims that the manufacturing change has caused the mean lifespan to **decrease** (less than $1200$ hours). Since we are only looking for a change in one direction, this is a one-tailed test:\n\n$$H_1: \\mu < 1200$$","highlights":[{"text":"causing the mean lifespan to decrease","location":"specification"}]},{"type":"text","order":3,"title":"State the final hypotheses","content":"Combining both parts, the final hypotheses are:\n\n$$H_0: \\mu = 1200$$\n$$H_1: \\mu < 1200$$"}]},{"id":"f8461670-a156-45b8-9d2f-eab558f183eb:p_value_approach","label":"(c) p-value Approach","steps":[{"type":"text","order":0,"title":"Define the hypotheses","content":"First, we state our null hypothesis ($H_0$) and alternative hypothesis ($H_1$). The null hypothesis represents the status quo, which is that the mean lifespan is 1200 hours. The alternative hypothesis represents what we are testing for—whether the mean lifespan is less than 1200 hours: $$H_0: \\mu = 1200$$ $$H_1: \\mu < 1200$$ Where $\\mu$ is the population mean lifespan of the light bulbs.","highlights":[{"text":"mean lifespan is less than 1200 hours","location":"specification"}]},{"type":"text","order":1,"title":"Distribution of sample mean","content":"Under the null hypothesis $H_0$, the individual bulb lifespans are distributed as $X \\sim N(1200, 80^2)$. For a sample of size $n = 30$, the sample mean $\\bar{X}$ is normally distributed with the same mean but a reduced variance: $$\\bar{X} \\sim N\\left(1200, \\frac{80^2}{30}\\right)$$ This distribution has a standard error of: $$\\text{SE} = \\frac{80}{\\sqrt{30}} \\approx 14.61\\text{ hours}$$ Note: Identifying this correct distribution for $\\bar{X}$ earns you the first mark.","highlights":[{"text":"normally distributed with a mean of 1200 hours and a standard deviation of 80 hours","location":"specification"},{"text":"sample of 30 light bulbs","location":"specification"}]},{"type":"text","order":2,"title":"Calculate the p-value","content":"Our observed sample mean is $\\bar{x} = 1175$ hours. Since we are conducting a one-tailed test to see if the mean is less than 1200, the p-value is the probability of obtaining a sample mean as low as or lower than 1175: $$p\\text{-value} = P(\\bar{X} \\le 1175)$$ Using a graphing calculator to find this normal cumulative probability gives: $$P(\\bar{X} \\le 1175) \\approx 0.0435\\text{ (3 sf)}$$ Note: Calculating this probability correctly earns the second mark.","highlights":[{"text":"mean lifespan is found to be 1175 hours","location":"specification"}]},{"type":"text","order":3,"title":"Make the decision","content":"Now we compare our calculated $p$-value to the significance level $\\alpha = 0.10$: $$0.0435 < 0.10$$ Because the $p$-value ($0.0435$) is less than the significance level ($0.10$), we reject the null hypothesis $H_0$. We conclude that there is sufficient evidence at the 10% significance level to suggest that the mean lifespan of the light bulbs is less than 1200 hours. Note: Correctly comparing the p-value to the significance level and writing this conclusion in context earns the final mark.","highlights":[{"text":"10% significance level","location":"specification"}]}]},{"id":"f8461670-a156-45b8-9d2f-eab558f183eb:critical_value_approach","label":"(c) Critical Value Approach","steps":[{"type":"text","order":0,"title":"Define hypotheses","content":"First, we establish our null hypothesis ($H_0$) and alternative hypothesis ($H_1$). Since we want to test if there is evidence that the mean lifespan is less than 1200 hours: $$H_0: \\mu = 1200$$ $$H_1: \\mu < 1200$$ This is a one-tailed test with a significance level of $\\alpha = 0.10$.","highlights":[{"text":"mean lifespan is less than 1200 hours","location":"specification"},{"text":"10% significance level","location":"specification"}]},{"type":"text","order":1,"title":"State sample mean distribution","content":"Under the null hypothesis $H_0$, the sample mean $\\bar{X}$ of $n = 30$ light bulbs is normally distributed. Using the formula for the standard error, $\\sigma_{\\bar{x}} = \\frac{\\sigma}{\\sqrt{n}}$: $$\\bar{X} \\sim N\\left(1200, \\frac{80^2}{30}\\right)$$ Award 1 mark for identifying the correct distribution of the sample mean.","highlights":[{"text":"A sample of 30 light bulbs is tested","location":"specification"},{"text":"mean of 1200 hours and a standard deviation of 80 hours","location":"specification"}]},{"type":"text","order":2,"title":"Calculate critical value","content":"Using the Critical Value Approach, we find the critical value of the sample mean, $\\bar{x}_{crit}$, that corresponds to the boundary of our 10% rejection region in the lower tail. For a standard normal distribution, the critical $z$-value for a one-tailed test at the 10% level is: $$z_{crit} = -1.2816$$ Now, we convert this critical $z$-value back to the sample mean scale using: $$\\bar{x}_{crit} = \\mu + z_{crit} \\times \\frac{\\sigma}{\\sqrt{n}}$$ $$\\bar{x}_{crit} = 1200 + (-1.2816) \\times \\frac{80}{\\sqrt{30}} \\approx 1181.28\\text{ hours}$$"},{"type":"text","order":3,"title":"Compare and conclude","content":"Our observed sample mean is $\\bar{x} = 1175$ hours. Since our observed sample mean lies in the rejection region: $$1175 < 1181.28$$ Alternatively, we can compute the $z$-statistic for our sample mean: $$z = \\frac{1175 - 1200}{\\frac{80}{\\sqrt{30}}} \\approx -1.712$$ Since $z \\approx -1.712 < -1.2816$, we reject $H_0$. Conclusion: There is sufficient evidence at the 10% significance level to suggest that the mean lifespan of the light bulbs is less than 1200 hours.","highlights":[{"text":"mean lifespan is found to be 1175 hours","location":"specification"}]}]}],"generatedAt":"2026-06-09T09:26:05.652Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"d26a8b1f-153d-4d10-bf28-ad8593b4701d","revisionId":"ab19680e-264a-46a8-b4c8-1e5d2f3e84e1","questionNumber":null,"questionType":"LA","specification":"","difficulty":2,"difficultyBand":"medium","tier":null,"totalMarks":4,"sourceTitle":null,"calculatorUse":null,"isCalculation":true,"parts":[{"id":"838e7fa7-1f7a-4e42-b729-39352ef32e07","label":"a","content":"The annual rainfall in a specific region is normally distributed with a mean of 850 mm. Records show that in 20% of years, the rainfall is more than 1000 mm.\n\nFind the standard deviation of the annual rainfall.","markscheme":"- Use standard normal distribution $z = \\frac{x - \\mu}{\\sigma}$ M1\n- From inverse normal: $z = 0.84162...$ B1\n- Set up equation: $0.84162... = \\frac{1000 - 850}{\\sigma}$ M1\n- $\\sigma = 178$ (3sf) mm A1","marks":4,"order":0,"markingRubric":{"totalMarks":4,"points":[{"id":"p1","marks":1,"description":"Evidence of using the standard normal distribution formula $z = \\frac{x - \\mu}{\\sigma}$.","code":"M"},{"id":"p2","marks":1,"description":"Correct $z$-value from inverse normal for 20% upper tail.","code":"B"},{"id":"p3","marks":1,"description":"Setting up the equation with the mean and observed value substituted correctly.","code":"M"},{"id":"p4","marks":1,"description":"Final value for standard deviation rounded to 3 significant figures.","code":"A"}],"finalAnswers":["178"],"generatedAt":"2026-05-06T14:04:37.752Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"z_score_method","label":"Z-Score Transformation","steps":[{"type":"text","order":0,"title":"Define the normal variable","content":"Let the annual rainfall be represented by the random variable $X$. \n\nWe are given that $X$ is normally distributed with a mean $\\mu = 850$ mm and an unknown standard deviation $\\sigma$:\n\n$$X \\sim N(850, \\sigma^2)$$ \n\nThe question states that the rainfall is more than $1000\\text{ mm}$ in $20\\%$ of years, which we can write in probability notation as:\n\n$$P(X > 1000) = 0.20$$","highlights":[{"text":"mean of 850 mm","location":"specification"},{"text":"in 20% of years, the rainfall is more than 1000 mm","location":"specification"}]},{"type":"text","order":1,"title":"Convert to a percentile","content":"To use the standard normal distribution, it is easiest to work with the cumulative probability (the area to the left). \n\nIf $20\\%$ of the distribution lies above $1000\\text{ mm}$, then the remaining $80\\%$ of the distribution lies below or equal to $1000\\text{ mm}$:\n\n$$P(X \\le 1000) = 1 - 0.20 = 0.80$$"},{"type":"text","order":2,"title":"Find the z-score","content":"Now we standardise the variable by transforming it to the standard normal distribution, $Z \\sim N(0, 1)$, where:\n\n$$Z = \\frac{X - \\mu}{\\sigma}$$\n\nUsing a calculator or standard normal tables, we find the $z$-score corresponding to the 80th percentile (where the area to the left is $0.80$):\n\n$$P(Z \\le z) = 0.80 \\implies z \\approx 0.84162$$","calculator":[{"gdc":{"expression":"invNorm(0.80, 0, 1)","instructions":"Go to Statistics -> Distributions -> Normal -> Inverse Normal. Set Area/Probability to 0.80, $\\mu = 0$, and $\\sigma = 1$."},"ti84":{"expression":"invNorm(0.80, 0, 1)","instructions":"Press [2nd] [VARS] (DISTR), select 3:invNorm(. Enter Area: 0.80, $\\mu$: 0, $\\sigma$: 1. Highlight Paste and press [ENTER]."},"desmos":{"expression":"inverseNormal(normaldist(0,1), 0.80)","instructions":"Type `inverseNormal(normaldist(0,1), 0.80)` into the input bar."},"description":"Find the z-score corresponding to a left-tail area of 0.80."}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$$\\htmlData{anchor=z-val}{\\color{@sky}{0.84162}} = \\frac{\\htmlData{anchor=x-val}{\\color{@amber}{1000}} - \\htmlData{anchor=mu-val}{\\color{@purple}{850}}}{\\htmlData{anchor=sigma-var}{\\color{@green}{\\sigma}}}$$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"z-val"},{"color":"amber","style":"text-color","anchor":"x-val"},{"color":"purple","style":"text-color","anchor":"mu-val"},{"color":"green","style":"text-color","anchor":"sigma-var"}],"connections":[{"to":"sigma-var","from":"z-val","color":"sky","label":"z-score","style":"dashed"}]},"order":3,"title":"Set up the equation","description":"Substitute the values of the $z$-score, $x = 1000$, and $\\mu = 850$ into the standardisation formula:"},{"type":"text","order":4,"title":"Solve for standard deviation","content":"Rearrange the equation to solve for the standard deviation, $\\sigma$:\n\n$$\\begin{aligned} 0.84162 &= \\frac{150}{\\sigma} \\\\ \\sigma &= \\frac{150}{0.84162} \\\\ \\sigma &\\approx 178.22 \\text{ mm} \\end{aligned}$$\n\nRounding to 3 significant figures, we get:\n\n$$\\sigma \\approx 178 \\text{ mm}$$"}]},{"id":"gdc_numerical_solver","label":"Calculator Numerical Solver","steps":[{"type":"text","order":0,"title":"Define the variable and parameters","content":"Let $X$ represent the annual rainfall in mm.\n\nWe are given that $X$ follows a normal distribution:\n$$X \\sim N(\\mu, \\sigma^2)$$\n\nWhere:\n- The mean is $\\mu = 850$ mm\n- The standard deviation $\\sigma$ is unknown.\n\nWe are also told that in 20% of years, the rainfall is more than 1000 mm. This can be written as a probability:\n$$P(X > 1000) = 0.20$$","highlights":[{"text":"mean of 850 mm","location":"specification"},{"text":"in 20% of years, the rainfall is more than 1000 mm","location":"specification"}]},{"type":"text","order":1,"title":"Rewrite the probability equation","content":"To solve this using our calculator's numerical solver, we should express the equation in terms of the standard cumulative distribution function, which calculates the area to the left (less than a value).\n\nSince total probability is $1$:\n$$P(X \\le 1000) = 1 - P(X > 1000)$$\n$$P(X \\le 1000) = 1 - 0.20 = 0.80$$"},{"type":"text","order":2,"title":"Set up the calculator solver","content":"We can set up the numerical solver on a graphing calculator to find the unknown standard deviation $\\sigma$ (which we will enter as the variable $x$).\n\nThe probability equation we want to solve is:\n$$\\text{NormalCDF}(-\\infty, 1000, 850, x) = 0.80$$","calculator":[{"gdc":{"expression":"NormCD(-9e99, 1000, x, 850) = 0.80","instructions":"Go to the Equation/Solver menu, select 'Num Solve', and enter the equation: NormCD(-∞, 1000, x, 850) = 0.80 (Note: Casio format is NormCD(Lower, Upper, σ, μ)). Enter an initial estimate for x (e.g., 100) and press [SOLVE]."},"ti84":{"expression":"normalcdf(-1E99, 1000, 850, x) - 0.80 = 0","instructions":"Press [MATH] and select 'Solver...' or 'Numeric Solver...'. Enter the equation: E1: normalcdf(-1E99, 1000, 850, x) - 0.80 = 0. Set a reasonable guess for the standard deviation (e.g., x = 100) and press [ALPHA][SOLVE]."},"desmos":{"expression":"normaldist(850, s).cdf(1000) = 0.80","instructions":"Plot the cumulative normal distribution with mean 850 and standard deviation s as a slider: y = normaldist(850, s).cdf(1000). Set y = 0.80 and find the value of s where they intersect, or type the equation directly to solve."},"description":"Solve for the unknown standard deviation $x$ using a numerical solver or graph intersection."}]},{"type":"text","order":3,"title":"Solve and state the final answer","content":"Running the numerical solver on your graphing calculator yields the solution:\n$$\\sigma \\approx 178.225...$$\n\nRounding to 3 significant figures, we get:\n$$\\sigma = 178\\text{ mm}$$"}]}],"generatedAt":"2026-06-09T01:58:24.913Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"838e7fa7-1f7a-4e42-b729-39352ef32e07:z_score_method","label":"Z-Score Transformation","steps":[{"type":"text","order":0,"title":"Define the normal variable","content":"Let the annual rainfall be represented by the random variable $X$. \n\nWe are given that $X$ is normally distributed with a mean $\\mu = 850$ mm and an unknown standard deviation $\\sigma$:\n\n$$X \\sim N(850, \\sigma^2)$$ \n\nThe question states that the rainfall is more than $1000\\text{ mm}$ in $20\\%$ of years, which we can write in probability notation as:\n\n$$P(X > 1000) = 0.20$$","highlights":[{"text":"mean of 850 mm","location":"specification"},{"text":"in 20% of years, the rainfall is more than 1000 mm","location":"specification"}]},{"type":"text","order":1,"title":"Convert to a percentile","content":"To use the standard normal distribution, it is easiest to work with the cumulative probability (the area to the left). \n\nIf $20\\%$ of the distribution lies above $1000\\text{ mm}$, then the remaining $80\\%$ of the distribution lies below or equal to $1000\\text{ mm}$:\n\n$$P(X \\le 1000) = 1 - 0.20 = 0.80$$"},{"type":"text","order":2,"title":"Find the z-score","content":"Now we standardise the variable by transforming it to the standard normal distribution, $Z \\sim N(0, 1)$, where:\n\n$$Z = \\frac{X - \\mu}{\\sigma}$$\n\nUsing a calculator or standard normal tables, we find the $z$-score corresponding to the 80th percentile (where the area to the left is $0.80$):\n\n$$P(Z \\le z) = 0.80 \\implies z \\approx 0.84162$$","calculator":[{"gdc":{"expression":"invNorm(0.80, 0, 1)","instructions":"Go to Statistics -> Distributions -> Normal -> Inverse Normal. Set Area/Probability to 0.80, $\\mu = 0$, and $\\sigma = 1$."},"ti84":{"expression":"invNorm(0.80, 0, 1)","instructions":"Press [2nd] [VARS] (DISTR), select 3:invNorm(. Enter Area: 0.80, $\\mu$: 0, $\\sigma$: 1. Highlight Paste and press [ENTER]."},"desmos":{"expression":"inverseNormal(normaldist(0,1), 0.80)","instructions":"Type `inverseNormal(normaldist(0,1), 0.80)` into the input bar."},"description":"Find the z-score corresponding to a left-tail area of 0.80."}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$$\\htmlData{anchor=z-val}{\\color{@sky}{0.84162}} = \\frac{\\htmlData{anchor=x-val}{\\color{@amber}{1000}} - \\htmlData{anchor=mu-val}{\\color{@purple}{850}}}{\\htmlData{anchor=sigma-var}{\\color{@green}{\\sigma}}}$$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"z-val"},{"color":"amber","style":"text-color","anchor":"x-val"},{"color":"purple","style":"text-color","anchor":"mu-val"},{"color":"green","style":"text-color","anchor":"sigma-var"}],"connections":[{"to":"sigma-var","from":"z-val","color":"sky","label":"z-score","style":"dashed"}]},"order":3,"title":"Set up the equation","description":"Substitute the values of the $z$-score, $x = 1000$, and $\\mu = 850$ into the standardisation formula:"},{"type":"text","order":4,"title":"Solve for standard deviation","content":"Rearrange the equation to solve for the standard deviation, $\\sigma$:\n\n$$\\begin{aligned} 0.84162 &= \\frac{150}{\\sigma} \\\\ \\sigma &= \\frac{150}{0.84162} \\\\ \\sigma &\\approx 178.22 \\text{ mm} \\end{aligned}$$\n\nRounding to 3 significant figures, we get:\n\n$$\\sigma \\approx 178 \\text{ mm}$$"}]},{"id":"838e7fa7-1f7a-4e42-b729-39352ef32e07:gdc_numerical_solver","label":"Calculator Numerical Solver","steps":[{"type":"text","order":0,"title":"Define the variable and parameters","content":"Let $X$ represent the annual rainfall in mm.\n\nWe are given that $X$ follows a normal distribution:\n$$X \\sim N(\\mu, \\sigma^2)$$\n\nWhere:\n- The mean is $\\mu = 850$ mm\n- The standard deviation $\\sigma$ is unknown.\n\nWe are also told that in 20% of years, the rainfall is more than 1000 mm. This can be written as a probability:\n$$P(X > 1000) = 0.20$$","highlights":[{"text":"mean of 850 mm","location":"specification"},{"text":"in 20% of years, the rainfall is more than 1000 mm","location":"specification"}]},{"type":"text","order":1,"title":"Rewrite the probability equation","content":"To solve this using our calculator's numerical solver, we should express the equation in terms of the standard cumulative distribution function, which calculates the area to the left (less than a value).\n\nSince total probability is $1$:\n$$P(X \\le 1000) = 1 - P(X > 1000)$$\n$$P(X \\le 1000) = 1 - 0.20 = 0.80$$"},{"type":"text","order":2,"title":"Set up the calculator solver","content":"We can set up the numerical solver on a graphing calculator to find the unknown standard deviation $\\sigma$ (which we will enter as the variable $x$).\n\nThe probability equation we want to solve is:\n$$\\text{NormalCDF}(-\\infty, 1000, 850, x) = 0.80$$","calculator":[{"gdc":{"expression":"NormCD(-9e99, 1000, x, 850) = 0.80","instructions":"Go to the Equation/Solver menu, select 'Num Solve', and enter the equation: NormCD(-∞, 1000, x, 850) = 0.80 (Note: Casio format is NormCD(Lower, Upper, σ, μ)). Enter an initial estimate for x (e.g., 100) and press [SOLVE]."},"ti84":{"expression":"normalcdf(-1E99, 1000, 850, x) - 0.80 = 0","instructions":"Press [MATH] and select 'Solver...' or 'Numeric Solver...'. Enter the equation: E1: normalcdf(-1E99, 1000, 850, x) - 0.80 = 0. Set a reasonable guess for the standard deviation (e.g., x = 100) and press [ALPHA][SOLVE]."},"desmos":{"expression":"normaldist(850, s).cdf(1000) = 0.80","instructions":"Plot the cumulative normal distribution with mean 850 and standard deviation s as a slider: y = normaldist(850, s).cdf(1000). Set y = 0.80 and find the value of s where they intersect, or type the equation directly to solve."},"description":"Solve for the unknown standard deviation $x$ using a numerical solver or graph intersection."}]},{"type":"text","order":3,"title":"Solve and state the final answer","content":"Running the numerical solver on your graphing calculator yields the solution:\n$$\\sigma \\approx 178.225...$$\n\nRounding to 3 significant figures, we get:\n$$\\sigma = 178\\text{ mm}$$"}]}],"generatedAt":"2026-06-09T01:58:56.202Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"e7632857-5936-473b-b4fe-0d914aa99731","revisionId":"abb51267-6c5a-4bd2-99e5-7953700bb835","questionNumber":null,"questionType":"LA","specification":"A safety engineer tests the breaking strength of steel cables. The manufacturer claims the cables have a mean breaking strength of 80 kN. The engineer suspects the cables are weaker than claimed. Data from 200 cables are summarized below:\n\n$$n = 200 \\quad \\sum x = 15940 \\quad \\sum x^2 = 1271162$$","difficulty":3,"difficultyBand":"medium","tier":null,"totalMarks":8,"sourceTitle":null,"calculatorUse":null,"isCalculation":true,"parts":[{"id":"92de46cc-f257-4a9c-a14a-ec2efc3b3d7a","label":"a","content":"Find the mean and standard deviation of the sample.","markscheme":"- $\\bar{x} = \\frac{15940}{200} = 79.7$ B1\n- $\\sigma = \\sqrt{\\frac{1271162}{200} - (79.7)^2} = \\sqrt{3.72}$ M1\n- $\\sigma = 1.93$ (3sf) A1","marks":3,"order":0,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"Calculation of the sample mean.","code":"B"},{"id":"p2","marks":1,"description":"Correct use of variance formula $\\frac{\\sum x^2}{n} - \\bar{x}^2$.","code":"M"},{"id":"p3","marks":1,"description":"Final standard deviation to 3 significant figures.","code":"A"}],"finalAnswers":["79.7","1.93"],"generatedAt":"2026-05-06T14:05:35.758Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"population_sd","label":"Population Standard Deviation Formula","steps":[{"type":"text","order":0,"title":"Calculate the sample mean","content":"To find the sample mean, denoted by $\\bar{x}$, we use the formula:\n\n$$\\bar{x} = \\frac{\\sum x}{n}$$\n\nSubstituting our given values:\n\n$$\\bar{x} = \\frac{15940}{200} = 79.7$$\n\nThis earns the first mark **B1**."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$$\\sigma = \\sqrt{\\frac{\\htmlData{anchor=sum-x2}{\\color{@sky}{\\sum x^2}}}{\\htmlData{anchor=n-val}{\\color{@amber}{n}}} - (\\htmlData{anchor=mean-val}{\\color{@purple}{\\bar{x}}})^2}$$"},{"latex":"$$$\\sigma = \\sqrt{\\frac{\\htmlData{anchor=sum-x2-val}{\\color{@sky}{1271162}}}{\\htmlData{anchor=n-sub}{\\color{@amber}{200}}} - (\\htmlData{anchor=mean-sub}{\\color{@purple}{79.7}})^2}$$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"sum-x2"},{"color":"amber","style":"text-color","anchor":"n-val"},{"color":"purple","style":"text-color","anchor":"mean-val"}],"connections":[{"to":"sum-x2-val","from":"sum-x2","color":"sky","label":null,"style":"solid"},{"to":"n-sub","from":"n-val","color":"amber","label":null,"style":"solid"},{"to":"mean-sub","from":"mean-val","color":"purple","label":null,"style":"solid"}]},"order":1,"title":"Use standard deviation formula","description":"Now we calculate the sample standard deviation using the population standard deviation formula (biased estimator):\n\n$$\\sigma = \\sqrt{\\frac{\\sum x^2}{n} - \\bar{x}^2}$$\n\nSubstituting the values into this formula earns the method mark **M1**."},{"type":"text","order":2,"title":"Compute standard deviation","content":"Let's simplify the expression inside the square root step-by-step:\n\n$$\\frac{1271162}{200} = 6355.81$$\n\n$$(79.7)^2 = 6352.09$$\n\nSubtracting these values gives:\n\n$$\\sigma = \\sqrt{6355.81 - 6352.09} = \\sqrt{3.72}$$\n\n$$\\sigma \\approx 1.92873...$$\n\nRounding to 3 significant figures, we get:\n\n$$\\sigma = 1.93$$\n\nThis final accuracy earns you the mark **A1**."}]},{"id":"unbiased_sd","label":"Unbiased Sample Standard Deviation Formula","steps":[{"type":"text","order":0,"title":"Calculate the sample mean","content":"To find the mean breaking strength of the sample, we use the formula for the sample mean:\n\n$$\\bar{x} = \\frac{\\sum x}{n}$$\n\nGiven the values from the data:\n- $n = 200$\n- $\\sum x = 15940$\n\nSubstituting these values in, we get:\n\n$$\\bar{x} = \\frac{15940}{200} = 79.7\\text{ kN}$$\n\nThis earns the first mark **B1**."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$$s = \\sqrt{\\frac{\\sum x^2 - n\\bar{x}^2}{n - 1}}$$"},{"latex":"$$$s = \\sqrt{\\frac{\\htmlData{anchor=sumx2}{\\color{@sky}{1271162}} - \\htmlData{anchor=nval}{\\color{@amber}{200}}(\\htmlData{anchor=xbar}{\\color{@purple}{79.7}})^2}{\\htmlData{anchor=nsub}{\\color{@amber}{200}} - 1}}$$"},{"latex":"$$$s = \\sqrt{\\frac{1271162 - 1270418}{199}} = \\sqrt{\\frac{744}{199}} \\approx \\sqrt{3.7387}$$"},{"latex":"$$$s \\approx \\htmlData{anchor=final-val}{1.9336} \\approx 1.93\\text{ kN}$$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"sumx2"},{"color":"amber","style":"text-color","anchor":"nval"},{"color":"purple","style":"text-color","anchor":"xbar"}],"connections":[{"to":"final-val","from":"sumx2","color":"sky","label":"sum of x^2","style":"dashed"},{"to":"final-val","from":"xbar","color":"purple","label":"mean","style":"dashed"}]},"order":1,"title":"Calculate unbiased standard deviation","description":"Next, we calculate the unbiased sample standard deviation, $s$. We divide by $n - 1$ to correct for bias when estimating the population standard deviation from a sample."},{"type":"text","order":2,"title":"State the final results","content":"Thus, the calculated statistics for our sample of steel cables are:\n\n- **Sample mean**: $\\bar{x} = 79.7\\text{ kN}$\n- **Unbiased sample standard deviation**: $s \\approx 1.93\\text{ kN}$ (rounded to 3 significant figures)"}]}],"generatedAt":"2026-06-09T02:27:25.697Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]},{"id":"465afc4d-acc2-4c14-8deb-08a6c6b33aa2","label":"b","content":"Carry out a hypothesis test to test the engineer's suspicion at the 1% significance level. State your assumptions clearly.","markscheme":"- $H_0: \\mu = 80, H_1: \\mu < 80$ B1\n- $\\bar{X} \\sim N(80, \\frac{1.93^2}{200})$ M1\n- $P(\\bar{X} < 79.7) = 0.0139$ (using $\\sigma=1.93$) or $0.0139$ (using $\\sqrt{3.72}$) A1\n- $0.0139 > 0.01$, so do not reject $H_0$. M1\n- There is not enough evidence to support the claim that the cables are weaker than 80 kN. (Assumption: Sample is random and $n$ is large so CLT applies) A1","marks":5,"order":1,"markingRubric":{"totalMarks":5,"points":[{"id":"p1","marks":1,"description":"Correct hypotheses for a one-tailed test.","code":"B"},{"id":"p2","marks":1,"description":"Correct normal distribution model for the sample mean.","code":"M"},{"id":"p3","marks":1,"description":"Accurate calculation of probability/p-value.","code":"A"},{"id":"p4","marks":1,"description":"Comparison of p-value with 0.01.","code":"M"},{"id":"p5","marks":1,"description":"Correct conclusion in context with assumptions stated.","code":"A"}],"finalAnswers":["0.0139","Do not reject H0"],"generatedAt":"2026-05-06T14:05:35.758Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"pvalue","label":"p-value Approach","steps":[{"type":"text","order":0,"title":"State the hypotheses","content":"First, we define the parameter under investigation and set up our null and alternative hypotheses.\n\nLet the population mean breaking strength of the steel cables (in kN) be:\n$$\\mu$$\n\n* **Null Hypothesis:**\n $$H_0: \\mu = 80$$\n This represents the manufacturer's claim.\n\n* **Alternative Hypothesis:**\n $$H_1: \\mu < 80$$\n This represents the engineer's suspicion that the cables are weaker than claimed.\n\nThis is a one-tailed test because we are testing specifically for a reduction in breaking strength.","highlights":[{"text":"mean breaking strength of 80 kN","location":"specification"},{"text":"weaker than claimed","location":"specification"}]},{"type":"text","order":1,"title":"Calculate sample statistics","content":"We calculate the sample mean and the sample standard deviation from the given summary data.\n\n* **Sample Mean:**\n $$\\bar{x} = \\frac{\\sum x}{n} = \\frac{15940}{200} = 79.7\\text{ kN}$$\n\n* **Sample Standard Deviation:**\n $$\\sigma = \\sqrt{\\frac{\\sum x^2}{n} - \\bar{x}^2} = \\sqrt{\\frac{1271162}{200} - (79.7)^2} = \\sqrt{3.72} \\approx 1.93\\text{ kN}$$"},{"type":"text","order":2,"title":"State distribution and assumptions","content":"Since the sample size is large (with $$n = 200$$), we apply the **Central Limit Theorem (CLT)**. This allows us to assume that the sample mean distribution is approximately normal, even if the parent distribution of breaking strengths is not.\n\nUnder the null hypothesis, the distribution of the sample mean is:\n$$\\bar{X} \\sim N\\left(80, \\frac{1.93^2}{200}\\right)$$\n\n**Assumptions made:**\n1. The 200 cables represent a random sample.\n2. The sample size is large enough ($$n > 30$$) for the Central Limit Theorem to apply."},{"type":"text","order":3,"title":"Find the p-value","content":"Using the *p*-value approach, we find the probability of obtaining a sample mean as extreme as or more extreme than our observed sample mean under the null hypothesis:\n$$P(\\bar{X} \\le 79.7)$$\n\nUsing a normal distribution solver with:\n* Mean:\n $$\\mu = 80$$\n* Standard Error (SE):\n $$\\text{SE} = \\frac{1.93}{\\sqrt{200}} \\approx 0.1365$$\n\nWe get:\n$$p\\text{-value} \\approx 0.0139$$","calculator":[{"gdc":{"expression":"NormCD(-∞, 79.7, 1.93/√(200), 80)","instructions":"Go to the Probability/Statistics menu, select Distributions, then Normal Cumulative. Enter the lower bound as -∞, upper bound as 79.7, mean as 80, and standard deviation as 1.93/√(200)."},"ti84":{"expression":"normalcdf(-1E99, 79.7, 80, 1.93/√(200))","instructions":"Press [2nd] [VARS] (DISTR) and select 2:normalcdf(. Enter Lower: -1E99, Upper: 79.7, μ: 80, σ: 1.93/√(200). Paste and press [ENTER]."},"desmos":{"expression":"normaldist(80, 1.93/sqrt(200)).cdf(-\\infty, 79.7)","instructions":"Use the normal distribution function with mean 80 and standard deviation 1.93/sqrt(200), then find the cumulative probability up to 79.7."},"description":"Calculate the cumulative normal probability for the sample mean."}]},{"type":"text","order":4,"title":"Compare p-value and conclude","content":"Now we compare our calculated *p*-value to the significance level of 0.01:\n$$0.0139 > 0.01$$\n\nSince the *p*-value is greater than the 1% significance level, we do not reject the null hypothesis.\n\n**Conclusion in Context:**\nThere is insufficient evidence at the 1% significance level to support the engineer's suspicion. We conclude that there is not enough evidence to suggest that the mean breaking strength of the steel cables is less than 80 kN.","highlights":[{"text":"1% significance level","location":"specification"}]}]},{"id":"criticalvalue","label":"Critical Value Approach","steps":[{"type":"text","order":0,"title":"State the Hypotheses","content":"The first step is to set up our null hypothesis and alternative hypothesis.\n\nThe manufacturer claims the mean breaking strength is 80 kN, while the engineer suspects the cables are weaker (so the mean is less than 80 kN).\n\n$$\\begin{aligned} H_0: \\mu &= 80 \\\\ H_1: \\mu &< 80 \\end{aligned}$$\n\nThis is a **one-tailed test** at the 1% significance level.","highlights":[{"text":"mean breaking strength of 80 kN","location":"specification"},{"text":"weaker than claimed","location":"specification"}]},{"type":"text","order":1,"title":"Calculate Sample Statistics","content":"We find the sample mean and the sample standard deviation using the given summary statistics:\n\n* **Sample Mean:**\n $$\\bar{x} = \\frac{\\sum x}{n} = \\frac{15940}{200} = 79.7\\text{ kN}$$\n\n* **Sample Standard Deviation:**\n $$\\sigma = \\sqrt{\\frac{\\sum x^2}{n} - \\bar{x}^2} = \\sqrt{\\frac{1271162}{200} - (79.7)^2} = \\sqrt{3.72} \\approx 1.93\\text{ kN}$$"},{"type":"text","order":2,"title":"State Assumptions","content":"Before conducting the test, we must state our assumptions:\n\n1. **Random Sampling:** We assume the sample of 200 cables is a random representation of the population.\n2. **Central Limit Theorem (CLT):** Since the sample size is large ($$n = 200$$), we can assume by the Central Limit Theorem that the distribution of the sample mean is approximately normal:\n\n$$\\bar{X} \\sim N\\left(80, \\frac{1.93^2}{200}\\right)$$\n\nThis distribution has a standard error of:\n$$\\text{SE} = \\frac{1.93}{\\sqrt{200}} \\approx 0.1365$$"},{"type":"text","order":3,"title":"Find the Critical Z-Value","content":"Using the **Critical Value Approach**, we find the critical boundary for our test statistic.\n\nFor a **one-tailed test at the 1% significance level** (lower-tail), our critical value corresponds to the point where the area to the left under the standard normal curve is 0.01:\n\n$$P(Z < z_{\\text{crit}}) = 0.01 \\implies z_{\\text{crit}} \\approx -2.326$$","calculator":[{"gdc":{"expression":"invNorm(0.01, 0, 1)","instructions":"Go to Statistics -> Distributions -> Normal -> Inverse Normal. Set Area = 0.01, μ = 0, σ = 1."},"ti84":{"expression":"invNorm(0.01, 0, 1)","instructions":"Press [2nd] [VARS] (DISTR), select 3:invNorm(. Enter Area: 0.01, μ: 0, σ: 1, and Paste."},"desmos":{"expression":"inverseNormal(normaldist(0,1), 0.01)","instructions":"Type `inverseNormal(normaldist(0,1), 0.01)` in an input cell."},"description":"Find the critical value $z_{\\text{crit}}$ for a lower-tail area of $0.01$."}]},{"type":"text","order":4,"title":"Calculate the Test Statistic","content":"Now we calculate the *z*-score (test statistic) for our observed sample mean:\n\n$$z = \\frac{\\bar{x} - \\mu_0}{\\frac{\\sigma}{\\sqrt{n}}} = \\frac{79.7 - 80}{\\frac{1.93}{\\sqrt{200}}} \\approx \\frac{-0.3}{0.1365} \\approx -2.198$$"},{"type":"text","order":5,"title":"Compare and Conclude","content":"We compare our calculated test statistic to the critical value:\n\n* **Calculated *z*-score:**\n $$-2.198$$\n* **Critical *z*-value:**\n $$-2.326$$\n\nSince $$-2.198 > -2.326$$, our test statistic **does not fall in the critical region** (it is not further into the left tail than the critical value).\n\nTherefore, we **do not reject the null hypothesis**.\n\n**Conclusion in Context:**\nThere is insufficient evidence at the 1% significance level to support the engineer's suspicion that the cables are weaker than claimed."}]}],"generatedAt":"2026-06-09T02:31:44.690Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"92de46cc-f257-4a9c-a14a-ec2efc3b3d7a:population_sd","label":"(a) Population Standard Deviation Formula","steps":[{"type":"text","order":0,"title":"Calculate the sample mean","content":"To find the sample mean, denoted by $\\bar{x}$, we use the formula:\n\n$$\\bar{x} = \\frac{\\sum x}{n}$$\n\nSubstituting our given values:\n\n$$\\bar{x} = \\frac{15940}{200} = 79.7$$\n\nThis earns the first mark **B1**."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$$\\sigma = \\sqrt{\\frac{\\htmlData{anchor=sum-x2}{\\color{@sky}{\\sum x^2}}}{\\htmlData{anchor=n-val}{\\color{@amber}{n}}} - (\\htmlData{anchor=mean-val}{\\color{@purple}{\\bar{x}}})^2}$$"},{"latex":"$$$\\sigma = \\sqrt{\\frac{\\htmlData{anchor=sum-x2-val}{\\color{@sky}{1271162}}}{\\htmlData{anchor=n-sub}{\\color{@amber}{200}}} - (\\htmlData{anchor=mean-sub}{\\color{@purple}{79.7}})^2}$$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"sum-x2"},{"color":"amber","style":"text-color","anchor":"n-val"},{"color":"purple","style":"text-color","anchor":"mean-val"}],"connections":[{"to":"sum-x2-val","from":"sum-x2","color":"sky","label":null,"style":"solid"},{"to":"n-sub","from":"n-val","color":"amber","label":null,"style":"solid"},{"to":"mean-sub","from":"mean-val","color":"purple","label":null,"style":"solid"}]},"order":1,"title":"Use standard deviation formula","description":"Now we calculate the sample standard deviation using the population standard deviation formula (biased estimator):\n\n$$\\sigma = \\sqrt{\\frac{\\sum x^2}{n} - \\bar{x}^2}$$\n\nSubstituting the values into this formula earns the method mark **M1**."},{"type":"text","order":2,"title":"Compute standard deviation","content":"Let's simplify the expression inside the square root step-by-step:\n\n$$\\frac{1271162}{200} = 6355.81$$\n\n$$(79.7)^2 = 6352.09$$\n\nSubtracting these values gives:\n\n$$\\sigma = \\sqrt{6355.81 - 6352.09} = \\sqrt{3.72}$$\n\n$$\\sigma \\approx 1.92873...$$\n\nRounding to 3 significant figures, we get:\n\n$$\\sigma = 1.93$$\n\nThis final accuracy earns you the mark **A1**."}]},{"id":"92de46cc-f257-4a9c-a14a-ec2efc3b3d7a:unbiased_sd","label":"(a) Unbiased Sample Standard Deviation Formula","steps":[{"type":"text","order":0,"title":"Calculate the sample mean","content":"To find the mean breaking strength of the sample, we use the formula for the sample mean:\n\n$$\\bar{x} = \\frac{\\sum x}{n}$$\n\nGiven the values from the data:\n- $n = 200$\n- $\\sum x = 15940$\n\nSubstituting these values in, we get:\n\n$$\\bar{x} = \\frac{15940}{200} = 79.7\\text{ kN}$$\n\nThis earns the first mark **B1**."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$$s = \\sqrt{\\frac{\\sum x^2 - n\\bar{x}^2}{n - 1}}$$"},{"latex":"$$$s = \\sqrt{\\frac{\\htmlData{anchor=sumx2}{\\color{@sky}{1271162}} - \\htmlData{anchor=nval}{\\color{@amber}{200}}(\\htmlData{anchor=xbar}{\\color{@purple}{79.7}})^2}{\\htmlData{anchor=nsub}{\\color{@amber}{200}} - 1}}$$"},{"latex":"$$$s = \\sqrt{\\frac{1271162 - 1270418}{199}} = \\sqrt{\\frac{744}{199}} \\approx \\sqrt{3.7387}$$"},{"latex":"$$$s \\approx \\htmlData{anchor=final-val}{1.9336} \\approx 1.93\\text{ kN}$$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"sumx2"},{"color":"amber","style":"text-color","anchor":"nval"},{"color":"purple","style":"text-color","anchor":"xbar"}],"connections":[{"to":"final-val","from":"sumx2","color":"sky","label":"sum of x^2","style":"dashed"},{"to":"final-val","from":"xbar","color":"purple","label":"mean","style":"dashed"}]},"order":1,"title":"Calculate unbiased standard deviation","description":"Next, we calculate the unbiased sample standard deviation, $s$. We divide by $n - 1$ to correct for bias when estimating the population standard deviation from a sample."},{"type":"text","order":2,"title":"State the final results","content":"Thus, the calculated statistics for our sample of steel cables are:\n\n- **Sample mean**: $\\bar{x} = 79.7\\text{ kN}$\n- **Unbiased sample standard deviation**: $s \\approx 1.93\\text{ kN}$ (rounded to 3 significant figures)"}]},{"id":"465afc4d-acc2-4c14-8deb-08a6c6b33aa2:pvalue","label":"(b) p-value Approach","steps":[{"type":"text","order":0,"title":"State the hypotheses","content":"First, we define the parameter under investigation and set up our null and alternative hypotheses.\n\nLet the population mean breaking strength of the steel cables (in kN) be:\n$$\\mu$$\n\n* **Null Hypothesis:**\n $$H_0: \\mu = 80$$\n This represents the manufacturer's claim.\n\n* **Alternative Hypothesis:**\n $$H_1: \\mu < 80$$\n This represents the engineer's suspicion that the cables are weaker than claimed.\n\nThis is a one-tailed test because we are testing specifically for a reduction in breaking strength.","highlights":[{"text":"mean breaking strength of 80 kN","location":"specification"},{"text":"weaker than claimed","location":"specification"}]},{"type":"text","order":1,"title":"Calculate sample statistics","content":"We calculate the sample mean and the sample standard deviation from the given summary data.\n\n* **Sample Mean:**\n $$\\bar{x} = \\frac{\\sum x}{n} = \\frac{15940}{200} = 79.7\\text{ kN}$$\n\n* **Sample Standard Deviation:**\n $$\\sigma = \\sqrt{\\frac{\\sum x^2}{n} - \\bar{x}^2} = \\sqrt{\\frac{1271162}{200} - (79.7)^2} = \\sqrt{3.72} \\approx 1.93\\text{ kN}$$"},{"type":"text","order":2,"title":"State distribution and assumptions","content":"Since the sample size is large (with $$n = 200$$), we apply the **Central Limit Theorem (CLT)**. This allows us to assume that the sample mean distribution is approximately normal, even if the parent distribution of breaking strengths is not.\n\nUnder the null hypothesis, the distribution of the sample mean is:\n$$\\bar{X} \\sim N\\left(80, \\frac{1.93^2}{200}\\right)$$\n\n**Assumptions made:**\n1. The 200 cables represent a random sample.\n2. The sample size is large enough ($$n > 30$$) for the Central Limit Theorem to apply."},{"type":"text","order":3,"title":"Find the p-value","content":"Using the *p*-value approach, we find the probability of obtaining a sample mean as extreme as or more extreme than our observed sample mean under the null hypothesis:\n$$P(\\bar{X} \\le 79.7)$$\n\nUsing a normal distribution solver with:\n* Mean:\n $$\\mu = 80$$\n* Standard Error (SE):\n $$\\text{SE} = \\frac{1.93}{\\sqrt{200}} \\approx 0.1365$$\n\nWe get:\n$$p\\text{-value} \\approx 0.0139$$","calculator":[{"gdc":{"expression":"NormCD(-∞, 79.7, 1.93/√(200), 80)","instructions":"Go to the Probability/Statistics menu, select Distributions, then Normal Cumulative. Enter the lower bound as -∞, upper bound as 79.7, mean as 80, and standard deviation as 1.93/√(200)."},"ti84":{"expression":"normalcdf(-1E99, 79.7, 80, 1.93/√(200))","instructions":"Press [2nd] [VARS] (DISTR) and select 2:normalcdf(. Enter Lower: -1E99, Upper: 79.7, μ: 80, σ: 1.93/√(200). Paste and press [ENTER]."},"desmos":{"expression":"normaldist(80, 1.93/sqrt(200)).cdf(-\\infty, 79.7)","instructions":"Use the normal distribution function with mean 80 and standard deviation 1.93/sqrt(200), then find the cumulative probability up to 79.7."},"description":"Calculate the cumulative normal probability for the sample mean."}]},{"type":"text","order":4,"title":"Compare p-value and conclude","content":"Now we compare our calculated *p*-value to the significance level of 0.01:\n$$0.0139 > 0.01$$\n\nSince the *p*-value is greater than the 1% significance level, we do not reject the null hypothesis.\n\n**Conclusion in Context:**\nThere is insufficient evidence at the 1% significance level to support the engineer's suspicion. We conclude that there is not enough evidence to suggest that the mean breaking strength of the steel cables is less than 80 kN.","highlights":[{"text":"1% significance level","location":"specification"}]}]},{"id":"465afc4d-acc2-4c14-8deb-08a6c6b33aa2:criticalvalue","label":"(b) Critical Value Approach","steps":[{"type":"text","order":0,"title":"State the Hypotheses","content":"The first step is to set up our null hypothesis and alternative hypothesis.\n\nThe manufacturer claims the mean breaking strength is 80 kN, while the engineer suspects the cables are weaker (so the mean is less than 80 kN).\n\n$$\\begin{aligned} H_0: \\mu &= 80 \\\\ H_1: \\mu &< 80 \\end{aligned}$$\n\nThis is a **one-tailed test** at the 1% significance level.","highlights":[{"text":"mean breaking strength of 80 kN","location":"specification"},{"text":"weaker than claimed","location":"specification"}]},{"type":"text","order":1,"title":"Calculate Sample Statistics","content":"We find the sample mean and the sample standard deviation using the given summary statistics:\n\n* **Sample Mean:**\n $$\\bar{x} = \\frac{\\sum x}{n} = \\frac{15940}{200} = 79.7\\text{ kN}$$\n\n* **Sample Standard Deviation:**\n $$\\sigma = \\sqrt{\\frac{\\sum x^2}{n} - \\bar{x}^2} = \\sqrt{\\frac{1271162}{200} - (79.7)^2} = \\sqrt{3.72} \\approx 1.93\\text{ kN}$$"},{"type":"text","order":2,"title":"State Assumptions","content":"Before conducting the test, we must state our assumptions:\n\n1. **Random Sampling:** We assume the sample of 200 cables is a random representation of the population.\n2. **Central Limit Theorem (CLT):** Since the sample size is large ($$n = 200$$), we can assume by the Central Limit Theorem that the distribution of the sample mean is approximately normal:\n\n$$\\bar{X} \\sim N\\left(80, \\frac{1.93^2}{200}\\right)$$\n\nThis distribution has a standard error of:\n$$\\text{SE} = \\frac{1.93}{\\sqrt{200}} \\approx 0.1365$$"},{"type":"text","order":3,"title":"Find the Critical Z-Value","content":"Using the **Critical Value Approach**, we find the critical boundary for our test statistic.\n\nFor a **one-tailed test at the 1% significance level** (lower-tail), our critical value corresponds to the point where the area to the left under the standard normal curve is 0.01:\n\n$$P(Z < z_{\\text{crit}}) = 0.01 \\implies z_{\\text{crit}} \\approx -2.326$$","calculator":[{"gdc":{"expression":"invNorm(0.01, 0, 1)","instructions":"Go to Statistics -> Distributions -> Normal -> Inverse Normal. Set Area = 0.01, μ = 0, σ = 1."},"ti84":{"expression":"invNorm(0.01, 0, 1)","instructions":"Press [2nd] [VARS] (DISTR), select 3:invNorm(. Enter Area: 0.01, μ: 0, σ: 1, and Paste."},"desmos":{"expression":"inverseNormal(normaldist(0,1), 0.01)","instructions":"Type `inverseNormal(normaldist(0,1), 0.01)` in an input cell."},"description":"Find the critical value $z_{\\text{crit}}$ for a lower-tail area of $0.01$."}]},{"type":"text","order":4,"title":"Calculate the Test Statistic","content":"Now we calculate the *z*-score (test statistic) for our observed sample mean:\n\n$$z = \\frac{\\bar{x} - \\mu_0}{\\frac{\\sigma}{\\sqrt{n}}} = \\frac{79.7 - 80}{\\frac{1.93}{\\sqrt{200}}} \\approx \\frac{-0.3}{0.1365} \\approx -2.198$$"},{"type":"text","order":5,"title":"Compare and Conclude","content":"We compare our calculated test statistic to the critical value:\n\n* **Calculated *z*-score:**\n $$-2.198$$\n* **Critical *z*-value:**\n $$-2.326$$\n\nSince $$-2.198 > -2.326$$, our test statistic **does not fall in the critical region** (it is not further into the left tail than the critical value).\n\nTherefore, we **do not reject the null hypothesis**.\n\n**Conclusion in Context:**\nThere is insufficient evidence at the 1% significance level to support the engineer's suspicion that the cables are weaker than claimed."}]}],"generatedAt":"2026-06-09T02:31:59.690Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":null,"walkthroughVideo":null}],"subjectCode":"a_level_maths__aqa","summaries":[{"id":"98a4ecf3-1b45-4da6-98f8-de02feb29a22","difficultyBand":"easy","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"c3756de7-18bb-4d1e-ac9e-59e394daaaa3","difficultyBand":"easy","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"f0aed593-f5f7-4705-9aef-04a022654dfc","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"02674204-2cdc-4c40-8f13-bc9d3881945f","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"05a57420-67e4-4f76-a8bb-08933d1bf60e","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"14cf52b2-80d9-4d81-b1c4-0225bf209f4f","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"200eebc2-2782-43e8-b030-ffd0bcfb91c5","difficultyBand":"easy","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"25bb853c-1f47-4470-9b92-caea608d36e3","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"2edff2e2-4ade-4c75-aeb5-36628213da93","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"41f09e76-d2c5-4be5-b804-4665cee1d010","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"4b6e6880-1866-46c1-85f5-5386a61ecfab","difficultyBand":"easy","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"4d52f7d9-f06f-4aca-8669-bdcbb234e16e","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"5c4e601b-be09-46a1-94ba-87d09203da07","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"5d32b5dc-d2ce-4901-ab65-b8f12f82ea87","difficultyBand":"hard","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"6406eca4-5dc1-49fd-9f4f-d4bd0b6008f5","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"6da601c4-d779-4087-9a6b-00d062ce1117","difficultyBand":"medium","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"743750a5-bc94-4ec3-9903-a972017a817a","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"82a3b5d2-d686-4ffb-95cc-f2da0e6c1d4d","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"87feee78-6c15-4553-b383-f1188bae3c09","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"885e3cc0-89fe-4f1f-bb32-7dbac855de91","difficultyBand":"easy","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"98dd04b1-c937-4fb4-8d13-d4ffa64fdccb","difficultyBand":"easy","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"9917378a-197b-4cb9-8a67-be7f308dd525","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"9be06b31-7c63-4901-85e8-b460d672d42e","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"f154074d-527f-443a-b133-b1e5e94f4525","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"fed5653b-ef1d-476b-b5e1-206f0decf5a1","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"85fb6113-6973-4e63-9720-b5cc0e9444d3","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"d1473c00-c866-4ae0-a241-f416cdea76fb","difficultyBand":"easy","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"035e1680-ddb3-46bd-816a-d519c1a55a14","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"09425f5e-08cc-40e5-a561-3e2bb7eb0603","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"0f4b8f1d-95c1-48dc-ac73-6c1cf162995e","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"1150d53b-b489-40c3-953b-37c5d0b41132","difficultyBand":"easy","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"19d62578-7a81-4a6f-809b-99cfbf8a3ea8","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"1e8db0e0-15b3-44d9-96eb-d0d7beb961c6","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"29bb5001-1a57-437d-b156-393f5bbec7dd","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"2ca25c5c-f3ac-4ed9-ad41-4280ef08b7df","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"47956445-935c-4526-9b2d-3429a78c7785","difficultyBand":"hard","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"7c4e6c6e-52b7-4d6d-a13b-1054bcdca824","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"7ccb239f-3032-4487-977b-935f2e3fb11d","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"7d92aa3b-f8f7-4684-9a41-a1e6f245892a","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"896a22ca-d1d5-4cec-86e4-a363f8100636","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"8a5e0e97-6b75-4590-a6c0-b5e9578b6c9b","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"8af8a385-9151-49bb-9996-f90ae4cdb287","difficultyBand":"easy","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"9184a2a3-c7af-4600-8fb1-fd84d34f77cb","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"99a39844-8347-4cb5-ab66-b12124949859","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"a58f8ad8-58bb-407f-be5b-6077621b5479","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"c090314a-cca7-448b-bbbf-6932237f0c02","difficultyBand":"easy","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"c9bc38b6-74af-4cc5-a637-d3c05add2d3b","difficultyBand":"easy","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"cf18f8cb-42af-4c28-bb2e-60c9807fe625","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"d3944ef9-4980-4247-a820-c41c7f91d762","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"f77833b0-8eb5-4e87-8539-8df69acfedcb","difficultyBand":"medium","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"7a0b7927-1163-4ab2-bc39-d0751ed7f28e","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"d2691fdf-804b-4945-b788-07b00586bf5d","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"ddb7e0c7-f902-4249-8ff1-6e618897006d","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"00886745-74b2-4ad3-a037-2126ffb9bf5f","difficultyBand":"medium","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"02f688ab-2fd6-4bb5-844e-9b35e2483359","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"12bcd9af-a36c-439a-a55f-bd829d4f7655","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"30c9bcbf-112a-41cd-9d5d-5e29f60989a0","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"42483a63-52b2-4450-8b72-7a119c30f512","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"55219d83-4aa1-4aa4-a59c-259583094efe","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"60de1b51-b614-4db3-a137-2629ed3db8d4","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"716aa561-b2c1-4cca-b202-fa54ed4dcbe1","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"76907e7f-e407-486d-9358-b227c51f8cd2","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"77dd596c-a583-4140-a86b-df32fb21df03","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"8fe79d9e-8fc8-4a96-8f74-9cfc230758a4","difficultyBand":"easy","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"b102b064-4cee-44a2-a200-fa5e63cfa132","difficultyBand":"medium","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"b640c066-49d9-4812-b788-2259ab54bc23","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"b7c0bc63-edd0-4320-b416-a2b2c3f98c3e","difficultyBand":"easy","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"c38d64e5-443c-4967-91b2-6203ba0fdbb9","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"d26a8b1f-153d-4d10-bf28-ad8593b4701d","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"e4fc7119-8b36-4819-8ae5-f2fac44c1d58","difficultyBand":"easy","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"e7632857-5936-473b-b4fe-0d914aa99731","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"ec5bb75a-5fa7-4c35-bdb0-f6a1aeb7eaec","difficultyBand":"easy","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"ece4cf01-279f-45d5-adb6-ca7a9acdfc3b","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"fbc59c51-58dc-4c04-a6cf-a72a8c95372d","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"fc991a56-9980-4385-80ee-98ef05eba51c","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"fe427f82-49da-48b1-b34d-c455721cd209","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"d5a759f2-2b88-457d-aae3-201b49d485d7","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"e98fe578-69f4-4204-80eb-53368018d6d1","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"665f474b-de8c-4741-bd5e-7d9bc2f34314","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"74576567-53f0-4e83-ae89-43b3201d17c7","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"7926a85f-aa62-4c8e-b954-fe1694a01bfa","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"9e144df1-0625-46e9-b3a2-25c78f422a93","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"9ea2e058-ff1b-40bf-a888-dec2fa4b354c","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"da1a7fd1-39c2-40df-8d21-1f3e5daf72ea","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"ad630ec3-4d98-4ef6-b12b-b1740826d4e2","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"443a4ce4-da2d-45a2-9807-c58d3f7879f6","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"169cdf99-9c88-4cfe-b494-b6e784afa669","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"e2419178-e6f5-44ff-98f9-7bfe1c0196ba","difficultyBand":"medium","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"e6383e78-6a1e-4574-9848-9bfbe841a15c","difficultyBand":"medium","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"8d895476-ea18-4692-9904-61f2f309c8c1","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"001df448-ed6d-409d-ab38-7c46fa5ccd19","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"00d44451-371d-4da3-9636-97d6e54cc369","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"029c92f5-b6db-4a5e-a962-2d74ff6aaf3f","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"0314e997-98fa-402c-b7c3-f9e650b646af","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"03166f25-bc4b-413c-9068-0f48775b76bb","difficultyBand":"medium","calculatorUse":null,"totalMarks":12,"questionType":"LA","isCalculation":true},{"id":"042cd133-39a5-4a42-982b-6d848c59b345","difficultyBand":"medium","calculatorUse":null,"totalMarks":12,"questionType":"LA","isCalculation":true},{"id":"049b9ffd-56cd-429f-8dff-2cb4b9acf51e","difficultyBand":"medium","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"050ad8ee-f36b-4469-a03d-732f3fa7a1f2","difficultyBand":"medium","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"06715a47-a93f-46e3-b243-098c347f023b","difficultyBand":"hard","calculatorUse":null,"totalMarks":15,"questionType":"LA","isCalculation":true},{"id":"069e8d38-45fc-4109-ada0-608f32d2d022","difficultyBand":"medium","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"06c51ecb-e1c0-4db8-a246-407642f53d66","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"07441850-6db1-4db3-8d10-82c7682bc42a","difficultyBand":"medium","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"08323025-74e6-444d-8ed0-888b0f5e33e0","difficultyBand":"hard","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":false},{"id":"0d439058-f1a6-4e01-a4d2-60a8d52965d5","difficultyBand":"medium","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"0d827c47-efa6-4ab8-aacc-82cdc121e017","difficultyBand":"medium","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"0fb5b0a5-7520-4b43-8755-3e33815eb3d9","difficultyBand":"medium","calculatorUse":null,"totalMarks":14,"questionType":"LA","isCalculation":true},{"id":"10052755-9f87-40f2-9572-df1ce94c354c","difficultyBand":"medium","calculatorUse":null,"totalMarks":11,"questionType":"LA","isCalculation":true},{"id":"132a6f67-d93b-4552-bf95-d9617e550b44","difficultyBand":"hard","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"13606d3a-46dc-4e02-afd0-40adf3ece4ef","difficultyBand":"medium","calculatorUse":null,"totalMarks":11,"questionType":"LA","isCalculation":true},{"id":"18d6b372-1a91-45d4-aa80-774b57593c66","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"19f93b26-acaf-453a-aa9d-e51eaa72fae8","difficultyBand":"medium","calculatorUse":null,"totalMarks":11,"questionType":"LA","isCalculation":true},{"id":"1a3d981f-79d3-49c7-8f17-9f7f77a4a1b8","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"1c41c9b7-0b64-4e85-8b6a-d757f545f5b2","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"1cf1d723-f3ed-4824-94ce-1a88eec06ce5","difficultyBand":"medium","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"1d4eb2eb-4963-4c91-9b4f-d92f0f3d5362","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":false},{"id":"1dbae8db-c5b8-4a2b-ba32-05bdec60b738","difficultyBand":"easy","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"1f19a9c4-4198-40e1-bd7a-fe36e5cf157f","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"1f7ac050-2952-4e9f-8084-bacf2137e8f2","difficultyBand":"hard","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"1f8468cc-c2b0-45c6-9f87-c4c01612ed16","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"20172fa2-4965-4b2c-8f43-50869da68d37","difficultyBand":"medium","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"2070b61f-5171-4ed8-aa89-06ebcea38a92","difficultyBand":"medium","calculatorUse":null,"totalMarks":14,"questionType":"LA","isCalculation":true},{"id":"23f61c1a-400a-4ccd-9d52-7c6e6e626fb9","difficultyBand":"medium","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"247acd3e-a117-4984-92ba-2e8e40fdc304","difficultyBand":"medium","calculatorUse":null,"totalMarks":14,"questionType":"LA","isCalculation":true},{"id":"24f9b598-50d7-4d2f-b798-a148f27ee17f","difficultyBand":"medium","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"25483535-358c-4b5b-a43c-16614792a046","difficultyBand":"hard","calculatorUse":null,"totalMarks":13,"questionType":"LA","isCalculation":true},{"id":"273de15e-0c40-4b52-902e-33899729c334","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":false},{"id":"29b42647-f8ff-4fc1-b8af-878deda15ec3","difficultyBand":"medium","calculatorUse":null,"totalMarks":11,"questionType":"LA","isCalculation":true},{"id":"2b4ec197-79ef-46cc-81fa-93a60fa78644","difficultyBand":"medium","calculatorUse":null,"totalMarks":11,"questionType":"LA","isCalculation":true},{"id":"2c0ea56a-620a-4d07-86db-786ade080fbd","difficultyBand":"medium","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"2d1670c3-bcd0-4f95-b9bb-67983eb0d9b7","difficultyBand":"medium","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"2d1d378a-8cfc-4b49-a436-fcbd0f7564c2","difficultyBand":"medium","calculatorUse":null,"totalMarks":12,"questionType":"LA","isCalculation":true},{"id":"2d511bc8-cd55-4366-bd87-af1d100bbbc4","difficultyBand":"medium","calculatorUse":null,"totalMarks":11,"questionType":"LA","isCalculation":true},{"id":"2d5d161b-95dd-4123-bca5-1a1f9657ea50","difficultyBand":"hard","calculatorUse":null,"totalMarks":12,"questionType":"LA","isCalculation":true},{"id":"2e9c3a5b-7573-45fb-9d74-433fe655d3ce","difficultyBand":"hard","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"30a502f3-8098-4547-b07f-8ca3bcb20c44","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":false},{"id":"311729eb-3de8-4eb0-86e0-420290cd918a","difficultyBand":"hard","calculatorUse":null,"totalMarks":12,"questionType":"LA","isCalculation":true},{"id":"3166c633-f89c-47a3-8069-2a34f76cfe6f","difficultyBand":"medium","calculatorUse":null,"totalMarks":11,"questionType":"LA","isCalculation":true},{"id":"319bb490-f33c-4ed9-9aba-c92d31a20c1d","difficultyBand":"medium","calculatorUse":null,"totalMarks":14,"questionType":"LA","isCalculation":true},{"id":"320e8cab-d83f-49c7-8483-d45fd9e1b593","difficultyBand":"medium","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"32cb0260-7c47-410d-a4ed-fc00a1af8e6c","difficultyBand":"medium","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"33a27906-8888-47b3-a701-e7fd3a279526","difficultyBand":"hard","calculatorUse":null,"totalMarks":11,"questionType":"LA","isCalculation":true},{"id":"35314f92-7206-450a-ab57-fdf5f5c4c5aa","difficultyBand":"medium","calculatorUse":null,"totalMarks":12,"questionType":"LA","isCalculation":true},{"id":"35ad8051-83a9-4b3b-a425-62370e18bf8c","difficultyBand":"hard","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"36a53472-8076-45cd-8ebd-f6d77e010495","difficultyBand":"hard","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"38403e30-9db1-4322-8eb9-c5264976f2bc","difficultyBand":"hard","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"3a300b59-370f-4a9a-9970-b9750af7129e","difficultyBand":"medium","calculatorUse":null,"totalMarks":11,"questionType":"LA","isCalculation":true},{"id":"3a6761da-2396-4aa0-8eda-cf3d9590ccdf","difficultyBand":"hard","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"3aa74ecd-c78a-438c-ad6f-bc8af4591ab2","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"3b81255f-af68-4867-8a97-3bd9fdb722c9","difficultyBand":"medium","calculatorUse":null,"totalMarks":12,"questionType":"LA","isCalculation":true},{"id":"3be7b1eb-f2de-436f-8915-6510d66ab26b","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"3ce5bd26-4cef-40e7-84f3-f74000b55cb0","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":false},{"id":"3eba53a7-549b-49bc-a646-9b4d9b9f0d9d","difficultyBand":"medium","calculatorUse":null,"totalMarks":12,"questionType":"LA","isCalculation":true},{"id":"4040a14e-3b32-4be6-bff0-57f3a68c20ad","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"4132381c-03b6-4a87-911c-39e2a2260152","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"41437ad6-6566-4491-8da8-59dd07a0a921","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"429a3aa2-7ee1-4f25-8295-fe5b62bf5f7c","difficultyBand":"hard","calculatorUse":null,"totalMarks":12,"questionType":"LA","isCalculation":true},{"id":"4366e85a-8f94-4ec3-9c38-e04fddad2577","difficultyBand":"medium","calculatorUse":null,"totalMarks":11,"questionType":"LA","isCalculation":true},{"id":"43fb91da-5715-431e-beb8-d2541146362c","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"44317895-74c8-4ab9-9bac-aae3bbfb1d6d","difficultyBand":"medium","calculatorUse":null,"totalMarks":13,"questionType":"LA","isCalculation":true},{"id":"4607d608-bda3-4a6a-b190-65c706647c7d","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"47a38e33-829f-4df8-b876-fe65266a0ce2","difficultyBand":"hard","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"4909143d-6bc4-4fdd-83cf-5685e43a4ea9","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"49167f95-52dd-40d9-b4ec-7e335f1e6ab1","difficultyBand":"hard","calculatorUse":null,"totalMarks":13,"questionType":"LA","isCalculation":true},{"id":"4ad4bdae-b53f-4f99-a4f8-93a38cd1ca15","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"4b553e0d-8cbc-49b8-b075-0935a160a117","difficultyBand":"hard","calculatorUse":null,"totalMarks":11,"questionType":"LA","isCalculation":true},{"id":"4ceea461-3f70-43af-a3da-8ff13dbc1dbc","difficultyBand":"hard","calculatorUse":null,"totalMarks":13,"questionType":"LA","isCalculation":true},{"id":"4eb8098d-e3f7-41e1-912d-9d09c8830f0c","difficultyBand":"medium","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":false},{"id":"4ebec8ad-3ef0-4fcb-933c-c204d7f7db2a","difficultyBand":"hard","calculatorUse":null,"totalMarks":13,"questionType":"LA","isCalculation":true},{"id":"513fc607-f2dd-423d-b91d-3aa6cad2648e","difficultyBand":"hard","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"52556f94-e838-4aab-8d2d-f6a6912144e9","difficultyBand":"hard","calculatorUse":null,"totalMarks":13,"questionType":"LA","isCalculation":true},{"id":"52aa9523-71ae-4e4a-82b9-8cec53e1667f","difficultyBand":"hard","calculatorUse":null,"totalMarks":12,"questionType":"LA","isCalculation":true},{"id":"532453f9-b9a2-4916-8f2b-1b52e1c3b051","difficultyBand":"hard","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"54180f67-f93e-4a57-ba13-45c9babcbb99","difficultyBand":"easy","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"54940e81-cec1-40b2-9041-dc6a9b16dbb7","difficultyBand":"medium","calculatorUse":null,"totalMarks":12,"questionType":"LA","isCalculation":true},{"id":"54e00aed-c174-47e4-9538-cd2261a81df9","difficultyBand":"hard","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":false},{"id":"5737b7a2-d326-43e2-ae1c-0bb7719c7072","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":false},{"id":"58418fd9-3dce-4f2f-b99b-a50bf4a71084","difficultyBand":"medium","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"59311d1c-ca2c-455c-b9b2-c750bf1be981","difficultyBand":"medium","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"599c5870-3162-4cbb-9d6e-128c98656c79","difficultyBand":"medium","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"5afce8bb-195a-43f1-985d-8b0c53850f7a","difficultyBand":"easy","calculatorUse":null,"totalMarks":1,"questionType":"MC","isCalculation":false},{"id":"5b13a951-8527-4d59-845a-67a9548c41f0","difficultyBand":"medium","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"5c3257c1-4f2b-4d2c-ad46-79421649b2c8","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"5c64d1a2-316e-4489-b2dd-50a82e885b06","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":false},{"id":"5c8545be-7612-4900-a690-1872a3bcaada","difficultyBand":"hard","calculatorUse":null,"totalMarks":18,"questionType":"LA","isCalculation":true},{"id":"5deb2d60-46af-46a6-8530-fb7345f19c92","difficultyBand":"medium","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"5e0e6d1f-ddb4-4a6f-804f-009adecedd03","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"5e10b3c3-4a66-48ea-a04d-fb84c5d5a7a6","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"5e1278ee-e1ec-413c-aca4-dad9f77f5ca3","difficultyBand":"medium","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"5e550d84-3f6f-4760-93b4-105be79d5e76","difficultyBand":"medium","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"5ec6f776-4fb4-4aa1-9a59-c4a4bb486a5e","difficultyBand":"hard","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"5ede7072-018c-4d2e-9348-d694060c9730","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"5efc7e52-2829-4a35-a3bd-e3c6425d2aa8","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"5f219421-ce71-4f2d-a4bc-f3f0e0b00960","difficultyBand":"medium","calculatorUse":null,"totalMarks":11,"questionType":"LA","isCalculation":true},{"id":"5f5fd762-2c01-4677-80dd-259fef8c59c4","difficultyBand":"hard","calculatorUse":null,"totalMarks":12,"questionType":"LA","isCalculation":true},{"id":"5f7716e6-4731-4a63-b7f1-89083da72e0f","difficultyBand":"hard","calculatorUse":null,"totalMarks":15,"questionType":"LA","isCalculation":true},{"id":"60c62b14-68e5-4790-913f-184148c81039","difficultyBand":"medium","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"6235e3fb-f0d0-4b25-8e03-23c5b80f971e","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":false},{"id":"63ba0ea3-ef0e-40f6-b13c-dd0e37218822","difficultyBand":"hard","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"687a7af7-dfe7-4f40-ae63-710adbab7882","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"69c8acd1-53c0-4a45-ae97-c4903312d813","difficultyBand":"medium","calculatorUse":null,"totalMarks":11,"questionType":"LA","isCalculation":true},{"id":"6a467884-cc91-45b2-a8f1-3d28faa0523c","difficultyBand":"medium","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"6aff8c09-6349-4f6b-acfb-1a7eb20c105f","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":false},{"id":"6f91fef7-89de-4199-90ad-fdfd928b9804","difficultyBand":"hard","calculatorUse":null,"totalMarks":14,"questionType":"LA","isCalculation":true},{"id":"70c3a231-96c5-408e-a68f-33c659330ba7","difficultyBand":"medium","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"716c4950-c5a6-44a1-b20d-4993d5be135c","difficultyBand":"medium","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"71dd5319-1cc1-4ba4-9b15-ec8af6d3e678","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"71fd8cb8-251d-4749-91c9-89a78fdd2ee8","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"73176b0a-e696-4ba2-842c-d48ada433773","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"732806f8-5b02-450b-a9b9-12c40c05459d","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"77a90cfa-d243-4906-ace9-9cc10cb4ac0c","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"789559e8-f99d-4cc3-aa12-bdf43fc71647","difficultyBand":"medium","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"7a4ad297-1972-4c4b-8ab5-f985e35749bb","difficultyBand":"medium","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"7a8e8c5c-66e9-4ac9-a261-f724f5dbe49e","difficultyBand":"hard","calculatorUse":null,"totalMarks":12,"questionType":"LA","isCalculation":true},{"id":"7a9febd5-b1f5-4d7c-bb26-d4a026b7b218","difficultyBand":"medium","calculatorUse":null,"totalMarks":11,"questionType":"LA","isCalculation":true},{"id":"7ae8b506-b713-433c-a423-8d476cccdbe6","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"7b4b5870-36a0-427d-818b-114af0c90c40","difficultyBand":"medium","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"7d1b93f9-2ca4-4309-86cc-6ce2e04dd3d5","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"7fcb9ad7-a074-420b-add6-087fb88d2924","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"8092d85f-04e7-45f6-9f7e-ac2f6c0656ac","difficultyBand":"medium","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"8363b9e7-f313-4820-8163-78e90f452d80","difficultyBand":"medium","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"83aed595-1481-449c-a899-fe9c672e8027","difficultyBand":"hard","calculatorUse":null,"totalMarks":11,"questionType":"LA","isCalculation":true},{"id":"85bc9c99-4615-48df-94de-f0da31a9f863","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":false},{"id":"88dbf433-b218-4b12-ad0e-45370b3a9c7a","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"8a6dc187-e6fd-4247-85c9-7c19b2b7e352","difficultyBand":"hard","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"8bacb129-a230-4618-b6cb-7b4fab5000c9","difficultyBand":"medium","calculatorUse":null,"totalMarks":12,"questionType":"LA","isCalculation":true},{"id":"8c0689fa-0639-42aa-85c1-b33a4e47929a","difficultyBand":"medium","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"8d1332ba-002b-49dd-9037-70f1c04f81e1","difficultyBand":"hard","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"8d1f7b0e-c50a-4ef2-8b1c-9e73b6587f86","difficultyBand":"hard","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"8e197a93-a2f0-4bc4-a357-5bd45d1ba5a1","difficultyBand":"medium","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"8e7f9011-7d54-40ba-b5d7-16757cab7f28","difficultyBand":"medium","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"92acf677-a1bc-486f-a518-421bcafa564d","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"944a4f38-4696-477b-a7ca-796598045629","difficultyBand":"medium","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"952eae34-17a5-4b5a-b034-cd8869cb1978","difficultyBand":"hard","calculatorUse":null,"totalMarks":11,"questionType":"LA","isCalculation":true},{"id":"96a491fa-2746-4971-9a19-4488010c77c2","difficultyBand":"hard","calculatorUse":null,"totalMarks":13,"questionType":"LA","isCalculation":true},{"id":"97a164e0-6e5b-4bcb-82a4-788aec7d19c1","difficultyBand":"medium","calculatorUse":null,"totalMarks":12,"questionType":"LA","isCalculation":true},{"id":"982820e0-ee09-4a08-a0ce-eb0a4a64f3bc","difficultyBand":"medium","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"9833e40f-984c-4e02-b55f-6e35e7949462","difficultyBand":"hard","calculatorUse":null,"totalMarks":12,"questionType":"LA","isCalculation":true},{"id":"98f480a7-cc8e-4241-964d-bea0dadbb737","difficultyBand":"medium","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"98fe3718-7636-43b9-9f18-b598d56b6646","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"992faf6e-21d0-4a74-86b7-0f90ed9a98c9","difficultyBand":"medium","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"9a0e5710-46e9-4a4b-9a58-90a4434fb31a","difficultyBand":"hard","calculatorUse":null,"totalMarks":11,"questionType":"LA","isCalculation":true},{"id":"9a699f34-d9be-4852-9e24-f88d64841219","difficultyBand":"medium","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"9b02f3b1-faaa-414a-a0c4-109fd2582728","difficultyBand":"medium","calculatorUse":null,"totalMarks":13,"questionType":"LA","isCalculation":true},{"id":"9c5dfbfd-7b56-47f6-b5b9-04dd4c398a83","difficultyBand":"medium","calculatorUse":null,"totalMarks":12,"questionType":"LA","isCalculation":true},{"id":"9c8798b1-7599-44e6-bf57-d015819f2cce","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":false},{"id":"9e68f472-edd7-4e41-b170-96c320cc0ef4","difficultyBand":"medium","calculatorUse":null,"totalMarks":11,"questionType":"LA","isCalculation":true},{"id":"9e90f689-b6d9-4bd7-a615-a7b1f6d1031d","difficultyBand":"hard","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"a07ed724-4cdb-4ed5-84da-3b9d249928cc","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"a1b2cc73-baea-42af-b56a-4b3cbe16ca10","difficultyBand":"hard","calculatorUse":null,"totalMarks":11,"questionType":"LA","isCalculation":true},{"id":"a1c228ac-fb71-4cc4-b198-a351fcf4dea2","difficultyBand":"hard","calculatorUse":null,"totalMarks":12,"questionType":"LA","isCalculation":true},{"id":"a1e4157b-b454-49cc-b844-74c06bebd93d","difficultyBand":"medium","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"a2531632-2ada-4305-a648-077f45278c67","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"a54cfc0a-222a-45c7-9121-9bf7539346c9","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":false},{"id":"a7108931-fbea-4056-be65-db0499e2a516","difficultyBand":"medium","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"a7a39a3b-684b-4126-b099-6a27c46ba72e","difficultyBand":"medium","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"a7a579df-cae6-442f-baff-94031e005940","difficultyBand":"medium","calculatorUse":null,"totalMarks":13,"questionType":"LA","isCalculation":true},{"id":"a7c5203b-1037-4685-8775-3044aa855718","difficultyBand":"hard","calculatorUse":null,"totalMarks":11,"questionType":"LA","isCalculation":true},{"id":"a85bf278-0859-4fb4-9da5-12b4dcc69aa9","difficultyBand":"medium","calculatorUse":null,"totalMarks":12,"questionType":"LA","isCalculation":true},{"id":"a86bceed-7d02-4c2f-a99c-56f96b946e09","difficultyBand":"medium","calculatorUse":null,"totalMarks":12,"questionType":"LA","isCalculation":true},{"id":"a98bf99a-68ff-4921-b92a-ba00b3d10436","difficultyBand":"hard","calculatorUse":null,"totalMarks":12,"questionType":"LA","isCalculation":true},{"id":"a9df5bd8-4000-4b29-8f6f-9b7ba98f53f2","difficultyBand":"medium","calculatorUse":null,"totalMarks":12,"questionType":"LA","isCalculation":true},{"id":"a9f8e0f5-712f-48bd-899b-035d9798e862","difficultyBand":"easy","calculatorUse":null,"totalMarks":1,"questionType":"MC","isCalculation":true},{"id":"ac861318-d1ee-4ecf-b75e-16cd986f9cbc","difficultyBand":"medium","calculatorUse":null,"totalMarks":11,"questionType":"LA","isCalculation":true},{"id":"aff272e8-e324-4cf0-8c03-7f22b3de7efd","difficultyBand":"medium","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"b0895922-6547-48f2-971b-ae27a3cbd7d6","difficultyBand":"medium","calculatorUse":null,"totalMarks":13,"questionType":"LA","isCalculation":true},{"id":"b1568ac6-5f42-4f44-981c-77f28e5e7b03","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"b2d07747-b85e-4198-8f23-4a584482f2ec","difficultyBand":"hard","calculatorUse":null,"totalMarks":11,"questionType":"LA","isCalculation":true},{"id":"b32e43bc-ae99-49df-869e-940a412f170f","difficultyBand":"medium","calculatorUse":null,"totalMarks":12,"questionType":"LA","isCalculation":true},{"id":"b33ee934-1c09-4c51-88f9-422722b0d9a2","difficultyBand":"medium","calculatorUse":null,"totalMarks":12,"questionType":"LA","isCalculation":true},{"id":"b43057ce-22e0-43c2-83fa-f1873a08e005","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"b4ae0dea-1997-4e62-b07e-6a1fd0d39e31","difficultyBand":"hard","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"b5455858-666f-42c8-9c95-399a9776d1b1","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"b65dda4d-bdfb-474b-8ec9-91f10f30c464","difficultyBand":"hard","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"b6a90258-2133-4bcc-9df4-76f083774e0d","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"b6e8037e-10cf-4336-a39e-fa265f110136","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"b8586a24-dd1b-4174-8625-514f70090144","difficultyBand":"hard","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"beac452b-d6dc-4243-a581-d5dc63e2419c","difficultyBand":"medium","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"bebdef69-8e4c-45aa-84ba-db96b3929505","difficultyBand":"medium","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"c00a6011-c765-4a54-bfaa-af9036016b19","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"c0d6e50e-32de-430a-ba6c-9099b6de8f99","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":false},{"id":"c176367c-3fb1-4987-a7ce-dc69e7b995bc","difficultyBand":"medium","calculatorUse":null,"totalMarks":11,"questionType":"LA","isCalculation":true},{"id":"c384d198-48ef-4da9-8096-e34de312cc3f","difficultyBand":"medium","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"c4db02cf-85db-4270-8161-6a23b6b42dfc","difficultyBand":"hard","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"c5d20f99-b62a-4455-a1a4-4ddff33b5e43","difficultyBand":"medium","calculatorUse":null,"totalMarks":11,"questionType":"LA","isCalculation":true},{"id":"c8f586f0-0555-4662-85f6-133d6fa07da6","difficultyBand":"medium","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"c91bb431-0596-41ef-9418-1fc555115018","difficultyBand":"medium","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"ca98322f-1bfc-420a-800c-d189f7dbe24c","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"cad8f818-ecb4-44b1-9ec3-f85c045fbec6","difficultyBand":"medium","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"cb85e3e6-3f73-4630-8c52-b79b86f8e668","difficultyBand":"medium","calculatorUse":null,"totalMarks":11,"questionType":"LA","isCalculation":true},{"id":"cbe82aee-369e-492a-9bad-5597b6446e4a","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"d13c1e2e-1e55-4cf6-a067-e7bfca55d5fa","difficultyBand":"medium","calculatorUse":null,"totalMarks":11,"questionType":"LA","isCalculation":true},{"id":"d4305308-ddaa-47fb-870b-307d006fa4bb","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"d5eb0490-f379-4473-b83a-2f2e46c3b085","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":false},{"id":"d64e41c1-ba80-468e-a81b-e5191f5c72f9","difficultyBand":"medium","calculatorUse":null,"totalMarks":11,"questionType":"LA","isCalculation":true},{"id":"d6bc146f-ca4a-4cf1-b90a-4f94f8c773b7","difficultyBand":"medium","calculatorUse":null,"totalMarks":12,"questionType":"LA","isCalculation":true},{"id":"d76825fe-bd21-4352-bd9e-1e317cc5a2aa","difficultyBand":"medium","calculatorUse":null,"totalMarks":11,"questionType":"LA","isCalculation":true},{"id":"da9a97b4-e0ae-4057-bb69-98d6d781b0a8","difficultyBand":"hard","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"dabfc759-1867-419a-a1b1-d7bb90016e38","difficultyBand":"medium","calculatorUse":null,"totalMarks":14,"questionType":"LA","isCalculation":true},{"id":"dc5d3065-7fe2-434f-8b3c-008e4a88c722","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":false},{"id":"dc7b3958-fda0-4482-8830-b186dc4db5f2","difficultyBand":"medium","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"dd96201e-bfff-4e85-b972-864cd0ecf42c","difficultyBand":"hard","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"ddf667f9-a6fb-4268-8511-2f1aa291181c","difficultyBand":"medium","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"de935550-6bd8-4ff4-a75b-2c0408fc75c9","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"deb39fc0-9975-486c-b608-8a5a787a7624","difficultyBand":"hard","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"df43786f-f7e9-4d1d-b3df-15b5746950d7","difficultyBand":"hard","calculatorUse":null,"totalMarks":15,"questionType":"LA","isCalculation":true},{"id":"e0780569-734c-420a-8c5b-2e71155a3b30","difficultyBand":"hard","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"e0fa8f96-c952-4044-a979-a7a7fa3ead0f","difficultyBand":"medium","calculatorUse":null,"totalMarks":12,"questionType":"LA","isCalculation":true},{"id":"e14b7610-4cab-4fc7-a2d8-ca10209aa310","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"e24fc455-a876-4358-9c6d-94d9f1146988","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"e3219802-5c7a-4613-9c83-60857117fe4e","difficultyBand":"hard","calculatorUse":null,"totalMarks":12,"questionType":"LA","isCalculation":true},{"id":"e3ffec2f-c59e-4f1d-af44-49eb0bbf81fb","difficultyBand":"medium","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"e4c723e5-6089-4663-863f-5144b1fc1ef0","difficultyBand":"hard","calculatorUse":null,"totalMarks":15,"questionType":"LA","isCalculation":true},{"id":"e5f9ce76-c7de-4137-b159-cb2443fa3c5c","difficultyBand":"medium","calculatorUse":null,"totalMarks":12,"questionType":"LA","isCalculation":true},{"id":"e6e57b35-73bf-44d7-aca5-2cf986ea1f1a","difficultyBand":"medium","calculatorUse":null,"totalMarks":11,"questionType":"LA","isCalculation":true},{"id":"e774c766-b2a3-41a7-9f6f-309bccf9d8c1","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"e7cc8d15-976e-42bb-b613-3dbc2f1a2bec","difficultyBand":"hard","calculatorUse":null,"totalMarks":12,"questionType":"LA","isCalculation":true},{"id":"ea63485e-14a2-4955-9e83-2a97a78d62fb","difficultyBand":"medium","calculatorUse":null,"totalMarks":11,"questionType":"LA","isCalculation":true},{"id":"eafed14d-a267-4264-a6fe-e4f5391d935c","difficultyBand":"medium","calculatorUse":null,"totalMarks":12,"questionType":"LA","isCalculation":true},{"id":"eb26faeb-da45-4440-aace-336222dd4cbd","difficultyBand":"medium","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"ec300f8b-362a-436a-9e16-9234e09d7256","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"ec68bfb6-bce5-4312-b111-5b6460e9778f","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"ec73d848-d615-4aff-a148-8a8830b86567","difficultyBand":"hard","calculatorUse":null,"totalMarks":12,"questionType":"LA","isCalculation":true},{"id":"ecb08b49-5aa5-450a-b46d-90a2be961f46","difficultyBand":"medium","calculatorUse":null,"totalMarks":13,"questionType":"LA","isCalculation":true},{"id":"edadd429-ff3c-438b-97bc-98ba46828958","difficultyBand":"medium","calculatorUse":null,"totalMarks":13,"questionType":"LA","isCalculation":true},{"id":"edf3d81b-fab2-4714-8c64-5c39f334cdad","difficultyBand":"medium","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"eeb2fffd-ab92-454c-960e-f378454c6c0d","difficultyBand":"hard","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"f009adbf-4855-4dff-9f08-b78a985261f8","difficultyBand":"medium","calculatorUse":null,"totalMarks":11,"questionType":"LA","isCalculation":true},{"id":"f0468348-6d5f-482a-98b7-21834ac3969c","difficultyBand":"medium","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":false},{"id":"f22d55f0-3df4-421d-ab6b-b7ad3b0293d1","difficultyBand":"medium","calculatorUse":null,"totalMarks":12,"questionType":"LA","isCalculation":true},{"id":"f23fcef0-c91d-44bb-a3f0-3f0d4d910d02","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":false},{"id":"f2cdf91d-a455-43af-8aaf-cdab977b6aeb","difficultyBand":"hard","calculatorUse":null,"totalMarks":15,"questionType":"LA","isCalculation":true},{"id":"f2d53d3f-fba9-4fe0-acba-955813d23104","difficultyBand":"medium","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"f31f42a3-f15f-48b8-ace9-b5a10d5c3636","difficultyBand":"medium","calculatorUse":null,"totalMarks":13,"questionType":"LA","isCalculation":false},{"id":"f3232bce-11c7-4407-9c50-b56bb546b0e8","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"f345c255-9e15-4994-b998-1c9e60b36786","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"f3af69c5-3cd5-4f9d-b214-35217d008d0a","difficultyBand":"medium","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"f3c72e5d-77a3-430f-9e20-2c1e14b283f6","difficultyBand":"medium","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"f406e257-c95f-4517-b0b6-0eda9ffaa16c","difficultyBand":"medium","calculatorUse":null,"totalMarks":11,"questionType":"LA","isCalculation":true},{"id":"f492e371-ced2-4b1f-82cc-020a4c9f142b","difficultyBand":"medium","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"f4eb808b-7c7f-4141-b6f9-8701e52a3c6e","difficultyBand":"hard","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"f4fbf5bf-5a71-4294-9c33-6ad3599b23d2","difficultyBand":"medium","calculatorUse":null,"totalMarks":14,"questionType":"LA","isCalculation":true},{"id":"f86ad542-a49a-4c62-b2ce-cf8526fad554","difficultyBand":"easy","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"f951522f-7538-4169-b448-cc647820ef15","difficultyBand":"medium","calculatorUse":null,"totalMarks":11,"questionType":"LA","isCalculation":true},{"id":"fafb5a33-a9ca-426b-9607-1bc34edf7ad5","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"fbc69200-2d1c-4360-8230-6143e15a4543","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"fc8e7416-85c9-4f08-a4f4-e233bb72b8ba","difficultyBand":"medium","calculatorUse":null,"totalMarks":11,"questionType":"LA","isCalculation":true},{"id":"fcc969e8-a27e-4e4c-92fc-c4de2284e52b","difficultyBand":"medium","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"fd4fbefe-bf76-4d6e-adba-f7e03e21d042","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"fdcedaa9-075f-4535-99a1-25d9200c01f7","difficultyBand":"medium","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"fdf54474-1201-4ec3-9b1d-40f33df73db5","difficultyBand":"medium","calculatorUse":null,"totalMarks":13,"questionType":"LA","isCalculation":true},{"id":"fe384b2c-3cd9-441a-ae65-20a998d1d126","difficultyBand":"hard","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true}],"title":"2.4.2 The Normal distribution (A-level only)","topicId":"f9661294-95d4-42ef-9ff4-0bf0c7ac3086","topicSidebar":"$L3c","topicSlug":"the-normal-distribution","topicTitle":"2.4.2 The Normal distribution (A-level only)"}]}],"$L3d","$L3e"]}]

2.4.2 The Normal distribution (A-level only)

Question 47

2.4.2 The Normal distribution (A-level only) Questions

2.4.2 The Normal distribution (A-level only)

Question 47

2.4.2 The Normal distribution (A-level only) Questions