Profits: Retail Jobs and productivity! How do retail stores rate? One way to answer this question is to examine annual profits per employee. The following data give annual profits per employee (in units of one thousand dollars per employee) for companies in retail sales. (See reference in Problem 23.) Companies such as Gap, Nordstrom, Dillards, JCPenney, Sears, Wal-Mart, Office Depot, and Toys "?" Us are included. Assume $\sigma \approx 3.8$ thousand dollars.
$$
\begin{array}{rrrrrrrrrrrr}
4.4 & 6.5 & 4.2 & 8.9 & 8.7 & 8.1 & 6.1 & 6.0 & 2.6 & 2.9 & 8.1 & -1.9 \\
11.9 & 8.2 & 6.4 & 4.7 & 5.5 & 4.8 & 3.0 & 4.3 & -6.0 & 1.5 & 2.9 & 4.8 \\
-1.7 & 9.4 & 5.5 & 5.8 & 4.7 & 6.2 & 15.0 & 4.1 & 3.7 & 5.1 & 4.2 &
\end{array}
$$
(a) Use a calculator or appropriate computer software to verify that, for the preceding data, $\bar{x} \approx 5.1$
(b) Let us say that the preceding data are representative of the entire sector of retail sales companies. Find an $80 \%$ confidence interval for $\mu$, the average annual profit per employee for retail sales.
(c) Interpretation Let us say that you are the manager of a retail store with a large number of employees. Suppose the annual profits per employee are less than 3 thousand dollars per employee. Do you think this might be low compared with other retail stores? Explain by referring to the confidence interval you computed in part (b).
(d) Interpretation Suppose the annual profits are more than $6.5$ thousand dollars per employee. As store manager, would you feel somewhat better? Explain by referring to the confidence interval you computed in part (b).
(e) Repeat parts (b), (c), and (d) for a $95 \%$ confidence interval.