Question

construct 90, 95, and 99 percent confidence intervals for the population mean, and state the practical and probabilistic interpretations of each. Indicate which interpretation you think would be more appropriate to use. Explain why the three intervals that you construct are not of equal width. Indicate which of the three intervals you would prefer to use as an estimate of the population mean, and state the reason for your choice.

In a length of hospitalization study conducted by several cooperating hospitals, a random sample of 64 peptic ulcer patients was drawn from a list of all peptic ulcer patients ever admitted to the participating hospitals and the length of hospitalization per admission was determined for each. The mean length of hospitalization was found to be 8.25 days. The population standard deviation is known to be 3 days.

Answer 1

Given that mean = 8.25, population standard deviation = 3 and sample size n = 64

Calculation for 90% confidence interval.......{z score for 90% confidence interval is 1.64, using z distribution table}

{interval length is difference between lower and upper limits}

There is 0.90 probability that the population mean will lie between the calculated range

We can be 90% confident that the population mean will lie between the calculated range

Calculation for 95% confidence interval.......{z score for 95% confidence interval is 1.96, using z distribution table}

{interval length is difference between lower and upper limits}

There is 0.95 probability that the population mean will lie between the calculated range

We can be 95% confident that the population mean will lie between the calculated range

Calculation for 99% confidence interval.......{z score for 99% confidence interval is 2.58, using z distribution table}

{interval length is difference between lower and upper limits}

There is 0.99 probability that the population mean will lie between the calculated range

We can be 99% confident that the population mean will lie between the calculated range

I think practical interpretation is more appropriate to use as compared to probabilistic interpretation.

Three intervals are not of equal width because the margin of error is different for each confidence level due to the change in the critical value. I will prefer 95% confidence interval because it is neither narrow like 90% confidence level nor wide like 99% confidence interval.