5. A student believes that there is a difference in the mean lengths of English and French films. He goes to the university video library and randomly selects a sample of 120 English films and a sample of 70 French films. He notes the length, \(x\) minutes, of each of the films in his samples. His data are summarised in the table below.
| \(\Sigma x\) | \(\Sigma x ^ { 2 }\) | \(s ^ { 2 }\) | \(n\) |
| English films | 10650 | 956909 | 98.5 | 120 |
| French films | 6510 | 615849 | 151 | 70 |
- Verify that the unbiased estimate of the variance, \(s ^ { 2 }\), of the lengths of English films is 98.5 minutes \({ } ^ { 2 }\)
- Stating your hypotheses clearly, test, at the 1\% level of significance, whether or not the mean lengths of English and French films are different.
- Explain the significance of the Central Limit Theorem to the test in part (b).
- The university video library contained 724 English films and 473 French films. Explain how the student could have taken a stratified sample of 190 of these films.