Question

This bonus problem relates to constructing a two-sided 95% confidence interval (C.I.) for the population mean, mu, and a two-sided 95% C.I. for the population variance, sigma^2, based on a random sample of size n from a normal population. You cannot change the sample size n here since the data collection process has completed. Please provide detailed justifications for your answer to each of the following two questions. (a) Here, assume the population variance is known, so that we can use the C.I. for mu that depends on the population standard deviation: Xbar plus/minus 1.96 times \sigma/\sqrt(n). Among all two-sided 95% C.I.'s for mu (that depends on the population standard deviation), why is the symmetric C.I. centering at the sample mean Xbar the shortest? (b) Among all two-sided 95% C.I.'s for sigma^2, which one is the shortest? Describe how you would find this shortest two-sided 95% C.I. for the population variance.

          This bonus problem relates to constructing a two-sided 95% confidence interval (C.I.) for the population mean, mu, and a two-sided 95% C.I. for the population variance, sigma^2, based on a random sample of size n from a normal population. You cannot change the sample size n here since the data collection process has completed.
Please provide detailed justifications for your answer to each of the following two questions.
(a) Here, assume the population variance is known, so that we can use the C.I. for mu that depends on the population standard deviation: Xbar plus/minus 1.96 times \sigma/\sqrt(n). Among all two-sided 95% C.I.'s for mu (that depends on the population standard deviation), why is the symmetric C.I. centering at the sample mean Xbar the shortest?
(b) Among all two-sided 95% C.I.'s for sigma^2, which one is the shortest? Describe how you would find this shortest two-sided 95% C.I. for the population variance.

This bonus problem relates to constructing a two-sided 95% confidence interval (C.I.) for the population mean, mu, and a two-sided 95% C.I. for the population variance, sigma^2, based on a random sample of size n from a normal population. You cannot change the sample size n here since the data collection process has completed.
Please provide detailed justifications for your answer to each of the following two questions.
(a) Here, assume the population variance is known, so that we can use the C.I. for mu that depends on the population standard deviation: Xbar plus/minus 1.96 times σ/√(()n). Among all two-sided 95% C.I.'s for mu (that depends on the population standard deviation), why is the symmetric C.I. centering at the sample mean Xbar the shortest?
(b) Among all two-sided 95% C.I.'s for sigma^2, which one is the shortest? Describe how you would find this shortest two-sided 95% C.I. for the population variance.

Added by Carlos W.

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