IMPORTANT: SHOW FULL STEPS OF CALCULATION

(a) Student A is asked to solve 18 crossword puzzles. The time it takes for A to solve each puzzle follows a normal distribution with a mean of 16 minutes and a standard deviation of 3.5 minutes. At the same time, Student B is asked to solve 20 different crossword puzzles of a similar standard. The time it takes for B to solve each puzzle follows a normal distribution with a mean of 15.5 minutes and a standard deviation of 3.8.

Is this evidence that there is a difference between the mean time it takes A and B to solve a crossword puzzle of this standard?

i) Choose an appropriate type of statistical test and explain the reason.

ii) Clearly state the null and alternative hypotheses for the test.

iii) Conduct the hypothesis test and write conclusion.

(b)

i) Among 43 people in executive-level positions, 10 of them are women. If the proportion of female executives (denoted by p) is equal to 0.35, calculate the probability of estimating p to0.03.

(ii) If p = 0.35, find the number of executive-level people that need to be recorded in order to estimate p0.01 with probability 0.98.