on":"gpt-5.6-terra","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":{"percent":32}},{"id":"597901b7-e35d-4fde-913a-4818719b4400","revisionId":"a4f75249-803d-43c5-a473-bac524aa606e","questionNumber":"4","questionType":"LA","specification":"","difficulty":1,"difficultyBand":"medium","totalMarks":5,"sourceTitle":"Question Set 7: Hypothesis Testing","calculatorUse":null,"parts":[{"id":"61ad2093-f4e5-43b6-8851-ff8a37cb9fb2","label":"0","content":"The chance of a webpage visitor clicking on a link is 0.15. Angela believes that increasing the font size for the link will increase the proportion of visitors that click on the link.\n\nAfter increasing the font size a sample of 90 visitors is taken and 20 clicked on the link.\n\nStating your hypotheses clearly test, at the 5% level of significance, whether or not there is evidence to support Angela's belief.","markscheme":"$$H_0 : p = 0.15$\n$H_1 : p > 0.15$ B1\n\n$X \\sim B(90, 0.15)$\n$P(X \\geq 20) = 1 - P(X \\leq 19)$\n$= 1 - 0.957$\n$= 0.043$ M1A1\n\n$0.043 < 0.05$ M1\n\nThere is evidence to support Angela's belief. A1","marks":5,"order":0,"markingRubric":{"totalMarks":5,"points":[{"id":"p1","marks":1,"description":"States hypotheses $H_0: p = 0.15$ and $H_1: p > 0.15$.","code":"B"},{"id":"p2","marks":1,"description":"Identifies distribution $B(90, 0.15)$.","code":"M"},{"id":"p3","marks":1,"description":"Calculates $P(X \\geq 20) \\approx 0.043$.","code":"A"},{"id":"p4","marks":1,"description":"Compares p-value to 0.05.","code":"M"},{"id":"p5","marks":1,"description":"Concludes there is evidence to support the belief.","code":"A"}],"finalAnswers":[],"generatedAt":"2026-05-12T02:14:06.553Z","modelVersion":"gemini-3-flash-preview"},"explanation":{"methods":[{"id":"one_tailed_binomial_test","label":"One-tailed binomial test","steps":[{"type":"text","order":0,"title":"State the hypotheses","content":"Let $p$ be the true proportion of visitors who click the link after its font size has increased.\n\n$$H_0:p=0.15\\qquad H_1:p>0.15$$\n\nAngela predicts an **increase**, so the alternative hypothesis uses $>$. Stating both hypotheses correctly earns **B1**."},{"type":"hyper_latex","block":{"nodes":[{"latex":"X\\sim B(90,0.15)"},{"latex":"P(X\\geq20)=1-P(X\\leq19)"},{"latex":"P(X\\leq19)=0.957"},{"latex":"P(X\\geq20)=1-0.957"},{"latex":"P(X\\geq20)=0.043"}]},"order":1,"title":"Find the p-value","description":"Assume the null hypothesis is true. As 20 clicks were observed and the alternative is an increase, find the upper-tail probability. Using the binomial model and calculating this upper-tail probability earns M1A1."},{"type":"text","order":2,"title":"Compare with 5%","content":"Compare the p-value with the significance level:\n\n$$0.043<0.05$$\n\nThe p-value is below $5\\%$, so reject $H_0$. This correct comparison earns **M1**."},{"type":"text","order":3,"title":"Give the conclusion","content":"There is evidence at the $5\\%$ significance level to support Angela's belief that increasing the font size increases the proportion of visitors who click the link. This contextual conclusion earns **A1**."}]}],"generatedAt":"2026-07-21T03:59:21.564Z","modelVersion":"gpt-5.6-terra","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":{"percent":52}},{"id":"7f8e5bb9-86fc-452a-b884-4080e1e618c7","revisionId":"8dab1085-dfbd-4083-9fb5-9e5ec5bbe049","questionNumber":"5","questionType":"LA","specification":"Billy plays darts. He assumes that each dart throw is independent and he knows from past experience that the probability of hitting the bullseye is 0.2","difficulty":1,"difficultyBand":"medium","totalMarks":6,"sourceTitle":"Question Set 7: Hypothesis Testing","calculatorUse":null,"parts":[{"id":"ffe25b01-9838-4b21-bf04-347c5a30741a","label":"a","content":"Billy throws 9 darts. Calculate the probability that at least 3 of the darts will hit the bullseye.","markscheme":"$$X \\sim B(9, 0.2)$\n$P(X \\geq 3) = 1 - P(X \\leq 2)$\n$= 1 - 0.7382$\n$=

Hypothesis Testing

Question 4

Hypothesis Testing Questions

Hypothesis Testing

Question 4

Hypothesis Testing Questions