A machine fills 1000 m` tetra packs of milk. To model the net volume of a given tetra pack, a random variable Y G(, ), with unknown parameters and , is used.

1. Let

998, 1001, 998, 997, 996, 1001, 1002, 997, 999, 1000, 997, 998, 1006, 1000, 995

be observed data for the net volume (in ml) of 15 tetra packs of milk. The sample standard deviation is given by s = 2.78. Construct a 90% confidence interval for . Show all your work for full credit.

2.From time to time, a random sample of tetra packs of milk is selected, the variance of the net volumes of these tetra packs is calculated and a (two-sided) 90% confidence interval for the population variance ^2 is constructed. If at least one of the two bounds of that confidence interval lies outside of the "acceptable range" [2.5, 8.5], then the machine should be adjusted.

Suppose we have a random sample of size n = 31, with a sample variance 4.7. Based on this sample, should the machine be adjusted? Justify your answer for full credit.

3.Let Y represent the net volume of a future randomly selected tetra pack of milk from this machine. Use the observed data from (a) to construct a 90% prediction interval for Y .