Question

Using Correct Distribution. In Exercises 5–8, assume that we want to construct a confidence interval. Do one of the following, as appropriate: (a) Find the critical value$t_{\frac{\alpha }{2}}$, (b) find the critical value $z_{\frac{\alpha }{2}}$, or (c) state that neither the normal distribution nor the t distribution applies.

Miami Heat Salaries Confidence level is 95%, $\sigma$is not known, and the normal quantile plot of the 17 salaries (thousands of dollars) of Miami Heat basketball players is as shown.

Short answer

Neither the normal distribution nor the t distribution is applicable.

Step-by-step solution

Given information

A sample of salaries is of Miami Heat Basketball players is considered with a sample size of 17. A normal quantile plot is constructed for the values.

Appropriate method

The following points are used to decide the appropriate method for constructing the confidence interval estimate of the population mean:

• Ifis unknown and the sample is either from a normally distributed population or the sample size is greater than 30, use the t-distribution$\left(t_{\frac{\alpha }{2}}\right)$

• Ifis known and the sample is either from a normally distributed population or the sample size is greater than 30, use the standard normal distribution$\left(z_{\frac{\alpha }{2}}\right)$.

• If the sample is not from a normally distributed population, do not use any of the above two distributions. Rather, use the bootstrap sampling method.

Interpretation of the normal quantile plot

It can be observed from the given graph that the points are not close to the straight line. This implies that the sample data are not from a normally distributed population. The distribution is skewed.

Deciding the correct distribution

Since the sample is not from a normally distributed population,neither the normal distribution nor the t distribution can be applied.

Thus, option (c) is correct.