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(a) The UK national average height for men in 2021 is known to be 1.78 cm. A sample of 15 men in 2022 has a mean height of 180 cm with a standard deviation of 1.5 cm. Is this evidence that men are getting taller? Show your method and explain your answer.

(b) The average score in an economics module in school A is compared to the average score in school B. The mean, standard deviation and sample size for each school are reported below:
School A School B
Mean 68 63
Standard deviation 9 11
Sample 100 80

i. Calculate the 95% confidence interval for the mean score in School A. Comment on what this tells you about the differences between the two schools.
ii. Use a significance level of 5% to test the null hypothesis that there is no difference in the means between the two schools.

(c) From a sample of 300 Finance graduates, 67% are found to be employed six months after graduation. For a sample of 120 Biology graduates 42% are found to be employed six months after graduation. Use this information to estimate the 99% confidence interval for the true difference between the two employment rates.

(d) A new gym has been built on campus in order to improve the time students spend doing physical exercise. The time spent in physical exercise, measured in minutes per week, by a sample of 10 students before and after the new gym was built are as follows:

Before 50 100 70 80 90 30 40 60 90 80
After 60 100 90 70 60 70 80 90 100 100

Conduct a statistical test to determine whether there is a significant improvement in physical exercise using a 5% significance level. Clearly formulate the null and alternative hypothesis in this case. What do you conclude?

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