Two-sample z-test large samples

A shop manager wants to find out if customers spend more money when music is playing in the shop. The amount of money spent by a customer in the shop is £\(x\). A random sample of 80 customers, who were shopping without music playing, and an independent random sample of 60 customers, who were shopping with music playing, were surveyed. The results of both samples are summarised in the table below.

	\(\sum x\)	\(\sum x^2\)	Unbiased estimate of mean	Unbiased estimate of variance
Customers shopping without music	5320	392000	\(\bar{x}\)	\(s^2\)
Customers shopping with music	4140	312000	69.0	446.44

1. Find the values of \(\bar{x}\) and \(s^2\). [5]
2. Test, at the 5\% level of significance, whether or not the mean money spent is greater when music is playing in the shop. State your hypotheses clearly. [8]

A doctor claims there is a higher mean lung capacity in people who exercise regularly compared to people who do not exercise regularly. He measures the lung capacity, \(x\), of 35 people who exercise regularly and 42 people who do not exercise regularly. His results are summarised in the table below.

1. Test, at the 5\% level of significance, the doctor's claim. State your hypotheses clearly. (6)
2. State any assumptions you have made in testing the doctor's claim. (2) The doctor decides to add another person who exercises regularly to his data. He measures the person's lung capacity and finds \(x = 31.7\)
3. Find the unbiased estimate of the variance for the sample of 36 people who exercise regularly. Give your answer to 3 significant figures. (4)

As part of a research project into the role played by cholesterol in the development of heart disease a random sample of 100 patients was put on a special fish-based diet. A different random sample of 80 patients was kept on a standard high-protein low-fat diet. After several weeks their blood cholesterol was measured and the results summarised in the table below.

Group	Sample size	Mean drop in cholesterol (mg/dl)	Standard deviation
Special diet	100	75	22
Standard diet	80	64	31

1. Stating your hypotheses clearly and using a 5% level of significance, test whether or not the special diet is more effective in reducing blood cholesterol levels than the standard diet. [9]
2. Explain briefly any assumptions you made in order to carry out this test. [2]

A researcher collects data on the height of boys aged between nine and nine and-a-half years and their diet. The data on the height, \(V\) cm, of the 80 boys who had always eaten a vegetarian diet is summarised by $$\Sigma V = 10\,367, \quad \Sigma V^2 = 1\,350\,314.$$

1. Calculate unbiased estimates of the mean and variance of \(V\). [5]

The researcher calculates unbiased estimates of the mean and variance of the height of boys whose diet has included meat from a sample of size 280, giving values of 130.5 cm and 96.24 cm\(^2\) respectively.

1. Stating your hypotheses clearly, test at the 1% level whether or not there is a significant difference in the heights of boys of this age according to whether or not they have a vegetarian diet. [8]

For a project, a student is investigating whether more athletic individuals have better hand-eye coordination. He records the time it takes a number of students to complete a task testing coordination skills and notes whether or not they play for a school sports team. His results are as follows: Stating your hypotheses clearly, test at the 5\% level of significance whether or not there is evidence that those who play in a school team complete the task more quickly on average. [8 marks]