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Jobs and productivity! How do banks rate? One way to answer this question is to examine annual profits per employee. The following is data about annual profits per employee (in units of 1 thousand dollars per employee) for representative companies in financial services. Assume $\sigma \approx 10.7$ thousand dollars.
\begin{tabular}{lllllllllll}
27.8 & 29.9 & 46.7 & 25.7 & 29.2 & 53.0 & 36.9 & 50.9 & 42.5 & 33.0 & 33.6 \\
36.9 & 27.0 & 47.1 & 33.8 & 28.1 & 28.5 & 29.1 & 36.5 & 36.1 & 26.9 & 27.8 \\
28.8 & 29.3 & 31.5 & 31.7 & 31.1 & 38.0 & 32.0 & 31.7 & 32.9 & 23.1 & 54.9 \\
43.8 & 36.9 & 31.9 & 25.5 & 23.2 & 29.8 & 22.3 & 26.5 & 26.7 & &
\end{tabular}
(a) Use a calculator or appropriate computer software to find $\bar{x}$ for the preceding data. (Round your answer to two decimal places.) 33.30 thousand dollars
(b) Let us say that the preceding data are representative of the entire sector of (successful) financial services corporations. Find a $75 \%$ confidence interval for $\mu$, the average annual profit per employee for all successful banks. (Round your answers to two decimal places.)
lower limit
32
thousand dollars
upper limit
35
thousand dollars
Answer
The 75% confidence interval for the average annual profit per employee for all successful banks is approximately \(\boxed{[31.40, 35.20]}\) thousand dollars.
Step 1 ：We are given that the sample mean (x̄) is 33.30 thousand dollars and the standard deviation (σ) is 10.7 thousand dollars. The sample size (n) can be calculated by counting the number of data points.
Step 2 ：The formula for a confidence interval is \(x̄ ± Z * (σ/√n)\), where Z is the Z-score corresponding to the desired confidence level. For a 75% confidence interval, the Z-score is approximately 1.15.
Step 3 ：Calculate the sample size by counting the number of data points. The sample size (n) is 42.
Step 4 ：Plug the values into the formula to find the lower and upper limits of the confidence interval. The lower limit is approximately 31.40 thousand dollars and the upper limit is approximately 35.20 thousand dollars.
Step 5 ：The 75% confidence interval for the average annual profit per employee for all successful banks is approximately \(\boxed{[31.40, 35.20]}\) thousand dollars.
