Assume that we want to construct a confidence interval. Do one of the following, as appropriate: (a) find the critical value $t_{\alpha / 2}$, (b) find the critical value $z_{\alpha / 2}$, or (c) state that neither the normal distribution nor the $t$ distribution applies. The confidence level is $95 \%, \sigma$ is not known, and the histogram of 55 player salaries (in thousands of dollars) of football players on a team is as shown. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. $t_{\alpha / 2}=$ (Round to two decimal places as needed.) B. $z_{\alpha / 2}=$ (Round to two decimal places as needed.)
Solution
Step 1 ：Since the population standard deviation \(\sigma\) is not known and the sample size is less than 30, we should use the t-distribution to construct the confidence interval. Therefore, we need to find the critical value \(t_{\alpha / 2}\) for a 95% confidence level.
Step 2 ：The degrees of freedom for a t-distribution is given by the sample size minus 1. In this case, the degrees of freedom is \(55 - 1 = 54\).
Step 3 ：The value of \(\alpha\) is \(1 - 0.95 = 0.05\). Since we are constructing a two-tailed test, we need to divide \(\alpha\) by 2, which gives us \(0.025\).
Step 4 ：Using these values, we can find the critical value \(t_{\alpha / 2}\).
Step 5 ：The critical value \(t_{\alpha / 2}\) for a 95% confidence level with 54 degrees of freedom is approximately \(2.00\).
Step 6 ：Final Answer: \(\boxed{2.00}\)
