In a study published in the Journal of the American Medical Association, three groups of 18 - to 35 -year-old participants overate for an 8 -week period. The groups consumed different levels of protein in their diet. The low protein group's diet was \(5 \%\) protein, the normal protein group's diet was \(15 \%\) protein, and the high protein group's diet was \(25 \%\) protein. The study found that the low protein group gained considerably less weight than the normal protein group or the high protein group.

You are a scientist working at a health research firm. The firm wants you to replicate the experiment. You conduct a similar experiment over an 8-week period. The results of the experiment are shown below.

EXERCISES
In Exercises1-3, perform a two-sample t-test to determine whether the mean weight gains of the two indicated studies are different. Assume the populations are normally distributed and the population variances are equal. For each exercise, write your conclusions as a sentence. Use \(\alpha=0.05\).

1. Test the weight gains of the low protein group against those in the normal protein group.

2. Test the weight gains of the low protein group against those in the high protein group.

3. Test the weight gains of the normal protein group against those in the high protein group.

4. In which comparisons in Exercises 1- 3 did you find a difference in weight gains? Write a summary of your findings.

5. Construct a \(95 \%\) confidence interval for \(\mu_{1}-\mu_{2}\), where \(\mu_{1}\) is the mean weight gain in the normal protein group and \(\mu_{2}\) is the mean weight gain in the high protein group. Assume the populations are normally distributed and the population variances are equal.

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Weight gain (after 8 weeks) Low protein Normal protein High protein group group group x = 6.8 lb 2-13.5 lb x=14.2 lb S = 1.7 lb S = 2.5 lb $3 = 2.1 lb 11 "1 = 12 = n 16 n3 = 15