- In a trial of \(\operatorname { diet } A\) a random sample of 80 participants were asked to record their weight loss, \(x \mathrm {~kg}\), after their first week of using the diet. The results are summarised by
$$\sum x = 361.6 \text { and } \sum x ^ { 2 } = 1753.95$$
- Find unbiased estimates for the mean and variance of weight lost after the first week of using diet \(A\).
The designers of diet \(A\) believe it can achieve a greater mean weight loss after the first week than a standard diet \(B\). A random sample of 60 people used diet \(B\). After the first week they had achieved a mean weight loss of 4.06 kg , with an unbiased estimate of variance of weight loss of \(2.50 \mathrm {~kg} ^ { 2 }\).
- Test, at the \(5 \%\) level of significance, whether or not the mean weight loss after the first week using \(\operatorname { diet } A\) is greater than that using diet \(B\). State your hypotheses clearly.
- Explain the significance of the central limit theorem to the test in part (b).
- State an assumption you have made in carrying out the test in part (b).