Pooled variance estimation

4 A company has two different machines, \(X\) and \(Y\), each of which fills empty cups with coffee. The manager is investigating the volumes of coffee, \(x\) and \(y\), measured in appropriate units, in the cups filled by machines \(X\) and \(Y\) respectively. She chooses a random sample of 50 cups filled by machine \(X\) and a random sample of 40 cups filled by machine \(Y\). The volumes are summarised as follows. $$\sum x = 15.2 \quad \sum x ^ { 2 } = 5.1 \quad \sum y = 13.4 \quad \sum y ^ { 2 } = 4.8$$ The manager claims that there is no difference between the mean volume of coffee in cups filled by machine \(X\) and the mean volume of coffee in cups filled by machine \(Y\). Test the manager's claim at the \(10 \%\) significance level.

2 The children at two large schools, \(P\) and \(Q\), are all given the same puzzle to solve. A random sample of size 10 is taken from the children at school \(P\). Their individual times to complete the puzzle give a sample mean of 9.12 minutes and an unbiased variance estimate of 2.16 minutes \({ } ^ { 2 }\). A random sample of size 12 is taken from the children at school \(Q\). Their individual times, \(x\) minutes, to complete the puzzle are summarised by $$\sum x = 99.6 \quad \sum ( x - \bar { x } ) ^ { 2 } = 21.5$$ where \(\bar { x }\) is the sample mean. Times to complete the puzzle are assumed to be normally distributed with the same population variance. Test at the \(5 \%\) significance level whether the population mean time taken to complete the puzzle by children at school \(P\) is greater than the population mean time taken to complete the puzzle by children at school \(Q\).

8

1. State circumstances under which it would be necessary to calculate a pooled estimate of variance when carrying out a two-sample hypothesis test.
2. An investigation into whether passive smoking affects lung capacity considered a random sample of 20 children whose parents did not smoke and a random sample of 22 children whose parents did smoke. None of the children themselves smoked. The lung capacity, in litres, of each child was measured and the results are summarised as follows. For the children whose parents did not smoke: \(n _ { 1 } = 20 , \Sigma x _ { 1 } = 42.4\) and \(\Sigma x _ { 1 } ^ { 2 } = 90.43\).
   For the children whose parents did smoke: \(\quad n _ { 2 } = 22 , \Sigma x _ { 2 } = 42.5\) and \(\Sigma x _ { 2 } ^ { 2 } = 82.93\).
   The means of the two populations are denoted by \(\mu _ { 1 }\) and \(\mu _ { 2 }\) respectively.
   (a) State conditions for which a \(t\)-test would be appropriate for testing whether \(\mu _ { 1 }\) exceeds \(\mu _ { 2 }\).
   (b) Assuming the conditions are valid, carry out the test at the \(1 \%\) significance level and comment on the result.
   (c) Calculate a 99\% confidence interval for \(\mu _ { 1 } - \mu _ { 2 }\).

6 The independent random variables \(X\) and \(Y\) have distributions with the same variance \(\sigma ^ { 2 }\). Random samples of 5 observations of \(X\) and \(n\) observations of \(Y\) are made and the results are summarised by $$\Sigma x = 5.5 , \quad \Sigma x ^ { 2 } = 15.05 , \quad \Sigma y = 8.0 , \quad \Sigma y ^ { 2 } = 36.4$$ Given that the pooled estimate of \(\sigma ^ { 2 }\) is 3 , find the value of \(n\).

3. Star Farm produces duck eggs. Xander takes a random sample of 20 duck eggs from Star Farm and their widths, \(x \mathrm {~cm}\), are recorded. Xander's results are summarised as follows. $$\sum x = 92.0 \quad \sum x ^ { 2 } = 433.4974$$

1. Calculate unbiased estimates of the mean and the variance of the width of duck eggs produced by Star Farm. Yinka takes an independent random sample of 30 duck eggs from Star Farm and their widths, \(y \mathrm {~cm}\), are recorded. Yinka's results are summarised as follows. $$\sum y = 142.5 \quad \sum y ^ { 2 } = 689.5078$$
2. Treating the combined sample of 50 duck eggs as a single sample, estimate the standard error of the mean.
   (5) Research shows that the population of duck egg widths is normally distributed with standard deviation 0.71 cm . The farmer claims that the mean width of duck eggs produced by Star Farm is greater than 4.5 cm .
3. Using your combined mean, test, at the \(5 \%\) level of significance, the farmer's claim. State your hypotheses clearly.