State circumstances under which it would be necessary to calculate a pooled estimate of variance when carrying out a two-sample hypothesis test.
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 \(\mu _ { 1 } - \mu _ { 2 }\).