The mean room rate for two adults for a random sample of 26 three-star hotels in Cincinnati has a sample standard deviation of \(\$ 31\). Assume the population is normally distributed.

(a) Construct a 99\% confidence interval for the population variance.

(b) Construct a \(99 \%\) confidence interval for the population standard deviation.

(c) A travel analyst claims that the standard deviation of the mean room rate for two adults at three-star hotels in Cincinnati is at most \(\$ 30\). At \(\alpha=0.01\), can you reject the travel analyst's claim? Interpret the decision in the context of the original claim.