A weight loss program claims that its clients have lost, on average, more than 40 pounds over the course of a 4 -month period. A random sample of 30 participants in the weight loss program shows an average loss of 44 pounds in a 4 -month period, with a standard deviation of 11 pounds.

a. For a significance level of \(\alpha=0.01\), test whether you would support the program's claims.

b. What sample size is needed in order to find a mean difference of at least 3 pounds over a 4-month period?

c. For a sample of size \(n=30\), determine the minimum mean difference in weight loss that can be detected provided that such a difference exists.