You are working as an assistant to the dean of institutional research at your university. She wants to survey members of the alumni association who obtained their baccalaureate degrees 5 years ago to learn what their starting salaries were in their first full-time job after receiving their degrees. A sample of 100 alumni is to be randomly selected from the list of 2,500 graduates in that class. If her goal is to construct a 95% confidence interval estimate for the population mean starting salary, why is it unlikely that you will be able to use Equations (8.1) on page 255 for this purpose? Explain.
What equations?
Equations (8.1) on page 255 refer to the formula used to calculate the confidence interval for the population mean when the population standard deviation is known. However, in this case, it is unlikely that we will know the population standard deviation of the starting salaries.
To use Equation (8.1), we need to have the population standard deviation. Typically, the population standard deviation is not known in practice, especially when dealing with specific subsets of a population, such as alumni who obtained their degrees 5 years ago. In this case, we would need to estimate the population standard deviation from the sample data.
Since the population standard deviation is unknown, we need to use a different formula known as the t-distribution formula to construct the confidence interval. The t-distribution takes into account the uncertainty caused by the estimation of the population standard deviation.
Therefore, in order to construct a 95% confidence interval estimate for the population mean starting salary for the alumni, it is unlikely that we can use Equations (8.1) on page 255. Instead, we should use the appropriate formulas and procedures based on the t-distribution, which takes into account the uncertainty of the sample standard deviation.