5. A greengrocer is investigating the weights of two types of orange, type \(A\) and type \(B\). She believes that on average type \(A\) oranges weigh greater than 5 grams more than type \(B\) oranges. She collects a random sample of 40 type \(A\) oranges and 32 type \(B\) oranges and records the weight, \(x\) grams, of each orange.
The table shows a summary of her data.
| \(n\) | \(\bar { x }\) | \(\sum x ^ { 2 }\) |
| Type \(A\) oranges | 40 | 140.4 | 790258 |
| Type \(B\) oranges | 32 | 134.7 | 581430 |
- Calculate unbiased estimates for the variance of the weights of the population of type \(A\) oranges and the variance of the weights of the population of type \(B\) oranges.
- Test, at the \(5 \%\) level of significance, the greengrocer's belief. You should state the hypotheses and the critical value used for this test.
- Explain how you have used the fact that the sample sizes are large in your answer to part (b).