2. (a) Dietitians are interested in the amount of nutrition people get from their food. A dietitian carried out a pilot study which measured the calcium intake in a group of n = 15 women aged between 25 and 50. The recommended intake of Calcium is 1000mg per day. In the study the dietician found the sample mean was 648mg and sample standard deviation was 476mg. On the basis of the information here, carry out a t-test at the significance level a = 0.05 to assess whether or not on average women are getting the recommended amount of calcium in their diet. (b) Later it was discovered that the dietitian had presented only a subset of the results of the study. In fact the study looked at 5 nutrients with recommended intakes as follows. Variable Calcium Recommended Intake 1000 mg Iron 15mg Protein 60g Vitamin A 800 μg Vitamin C 75 mg For the full study, the sample means and sample standard deviations were: mean s.d. Calcium Iron Protein A C 70 499 75 476 5 27 340 65 649 12 For the purpose of this question, the sample covariance matrix S has not been provided. However, the following intermediate results may be useful, -0.00257 0.5255 0.1077 -0.0049 0.0193 [-0.0001] 0.6666 0.0214 -0.0001 0.0003 where is the sample mean vector and 40 is the vector of recommended intakes. Your task is to carry out Hotelling's T2 test on the full data at a significance level a = 0.05. In particular, show that T2 = 53.6. State carefully the assumptions behind the test, and the hypothesis being tested. If the null hypothesis rejected, construct simultaneous confidence intervals to explain the reasons for rejection. In particular, show that the simultaneous confidence interval for calcium takes the form (56, 1242), where the values have been rounded to the nearest integer. (c) Compare the conclusions from part (a) to the conclusions from part (b), Discuss which analysis is is more appropriate and why.