An engineering firm sets an aptitude test when applicants first apply for training. The times taken to complete the test are normally distributed with mean 40.5 minutes and standard deviation 7.5 minutes. Applicants who complete. the test in less than 30 minutes are immediately accepted for training. Those who take between 30 and 36 minutes are required to take a further test. All other applicants are rejected.

(a) For a randomly chosen applicant calculate the probability of (i) immediate acceptance for training, (ii) requirement to take a further test.

(b) Given that a randomly chosen applicant was not rejected after this first test, calculate, to three decimal places, the probability that the applicant was immediately accepted for training.

(c) On a certain occasion there were 100 applicants. Use a suitable distributional approximation to calculate the probability that more than 25 applicants were required to take a further test. (NEAB)