In a large city, 45% of residents use public transport for their daily commute. A random sample of 40 residents is taken. Let the random variable XXX represent the number of residents in the sample who use public transport.
Find (i) P(X=22)P(X = 22)P(X=22) (ii) P(X<15)P(X < 15)P(X<15) (iii) the smallest value of kkk such that P(X≤k)>0.6P(X \le k) > 0.6P(X≤k)>0.6
A new larger random sample of 500 residents is taken.
(i) Using a suitable normal approximation, find the probability that at least 215 of these 500 residents use public transport. (ii) Justify why a normal approximation is appropriate in part (b)(i).
