Jobs and productivity! How do retail stores rate? One way to...

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 1 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 " $q$ " 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.

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 1 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 " $q$ " 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.

Key Concepts

Sample Mean

The sample mean is the arithmetic average of the observed values in a dataset. It serves as an estimate of the population mean, and its accuracy increases with larger sample sizes. In the context of statistical inference, it is the central value from which confidence intervals and hypothesis tests are developed.

Population Mean

The population mean is the true average of the entire population from which the sample is drawn. Although it is usually unknown, statistical methods provide ways to estimate it using sample data, typically in the form of a confidence interval, which offers a range of plausible values for this parameter.

Standard Deviation and Known Population Standard Deviation

The standard deviation is a measure of the variability or spread in a set of data. When the population standard deviation is known, it simplifies the process of statistical inference as it permits the use of the Z-distribution for constructing confidence intervals. This assumption is critical in many textbook examples and industrial applications where historical data provide reliable estimates.

Confidence Interval

A confidence interval is a range of values, derived from the sample data, within which the true population parameter is expected to lie with a specified probability (confidence level). It provides an estimate of uncertainty around the sample statistic by accounting for variability in the data.

Confidence Level

The confidence level is the probability, expressed as a percentage, that the confidence interval constructed from the sample data contains the true population parameter. Common confidence levels include 80%, 90%, 95%, and 99%. A higher confidence level results in a wider interval, reflecting greater uncertainty while ensuring greater confidence that the interval captures the true parameter.

Margin of Error

The margin of error quantifies the maximum expected difference between the true population parameter and the sample estimate. It is determined by the product of the critical value from the appropriate distribution (such as the Z-distribution when the population standard deviation is known) and the standard error, which is the standard deviation divided by the square root of the sample size.

Interpretation of Confidence Intervals

Interpreting confidence intervals involves comparing observed values or additional data points against the interval to determine their plausibility relative to the population parameter. If a value falls outside the confidence interval, it can be considered significantly different from the mean under the specified level of confidence, which can guide decision-making in practical scenarios.

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