Suppose Twenty-Seven Communities Have an Average of Reported Cases of Larceny

Suppose twenty-seven communities have an average of reported cases of larceny per year. Assume that $\sigma$ is known to be 41.7 cases per year. Find an 80%, 90%, and 95% confidence interval for the population mean annual number of reported larceny cases in such communities. Compare the lengths of the confidence intervals. As the confidence levels increase, do the confidence intervals increase in length? $$\bar { x } = 133.5$$

A) The 80% confidence level has a confidence interval length of 6.1; the 90% confidence level has a confidence interval length of 5.1; and the 95% confidence level has a confidence interval length of 4.0. As the confidence level increases, the confidence interval length decreases.
B) The 80% confidence level has a confidence interval length of 106.8; the 90% confidence level has a confidence interval length of 137.2; and the 95% confidence level has a confidence interval length of 163.5. As the confidence level increases, the confidence interval length increases.
C) The 80% confidence level has a confidence interval length of 4.0; the 90% confidence level has a confidence interval length of 5.1; and the 95% confidence level has a confidence interval length of 6.1. As the confidence level increases, the confidence interval lengths increases.
D) The 80% confidence level has a confidence interval length of 163.5; the 90% confidence level has a confidence interval length of 137.2; and the 95% confidence level has a confidence interval length of 106.8. As the confidence level increases, the confidence interval lengths decreases.
E) The 80% confidence level has a confidence interval length of 20.5; the 90% confidence level has a confidence interval length of 26.4; and the 95% confidence level has a confidence interval length of 31.5. As the confidence level increases, the confidence interval lengths increases.