Jobs and productivity! How do banks rate? One way to answer...

Jobs and productivity! How do banks rate? One way to answer this question is to examine annual profits per employee. Forbes Top Companies, edited by J. T. Davis (John Wiley \& Sons), gave the following data about annual profits per employee (in units of 1 thousand dollars per employee) for representative companies in financial services. Companies such as Wells Fargo, First Bank System, and Key Banks were included. Assume $\sigma \approx 10.2$ thousand dollars. $\begin{array}{lllllllllll}42.9 & 43.8 & 48.2 & 60.6 & 54.9 & 55.1 & 52.9 & 54.9 & 42.5 & 33.0 & 33.6 \\ 36.9 & 27.0 & 47.1 & 33.8 & 28.1 & 28.5 & 29.1 & 36.5 & 36.1 & 26.9 & 27.8 \\ 28.8 & 29.3 & 31.5 & 31.7 & 31.1 & 38.0 & 32.0 & 31.7 & 32.9 & 23.1 & 54.9 \\ 43.8 & 36.9 & 31.9 & 25.5 & 23.2 & 29.8 & 22.3 & 26.5 & 26.7 & & \end{array}$ (a) Use a calculator or appropriate computer software to verify that, for the preceding data, $\bar{x} \approx 36.0$ (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. (c) Interpretation Let us say that you are the manager of a local bank with a large number of employees. Suppose the annual profits per employee are less than 30 thousand dollars per employee. Do you think this might be somewhat low compared with other successful financial institutions? Explain by referring to the confidence interval you computed in part (b). (d) Interpretation Suppose the annual profits are more than 40 thousand dollars per employee. As manager of the bank, 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 $90 \%$ confidence level.

Jobs and productivity! How do banks rate? One way to answer this question is to examine annual profits per employee. Forbes Top Companies, edited by J. T. Davis (John Wiley \& Sons), gave the following data about annual profits per employee (in units of 1 thousand dollars per employee) for representative companies in financial services. Companies such as Wells Fargo, First Bank System, and Key Banks were included. Assume $\sigma \approx 10.2$ thousand dollars. $\begin{array}{lllllllllll}42.9 & 43.8 & 48.2 & 60.6 & 54.9 & 55.1 & 52.9 & 54.9 & 42.5 & 33.0 & 33.6 \\ 36.9 & 27.0 & 47.1 & 33.8 & 28.1 & 28.5 & 29.1 & 36.5 & 36.1 & 26.9 & 27.8 \\ 28.8 & 29.3 & 31.5 & 31.7 & 31.1 & 38.0 & 32.0 & 31.7 & 32.9 & 23.1 & 54.9 \\ 43.8 & 36.9 & 31.9 & 25.5 & 23.2 & 29.8 & 22.3 & 26.5 & 26.7 & & \end{array}$
(a) Use a calculator or appropriate computer software to verify that, for the preceding data, $\bar{x} \approx 36.0$
(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.
(c) Interpretation Let us say that you are the manager of a local bank with a large number of employees. Suppose the annual profits per employee are less than 30 thousand dollars per employee. Do you think this might be somewhat low compared with other successful financial institutions? Explain by referring to the confidence interval you computed in part (b).
(d) Interpretation Suppose the annual profits are more than 40 thousand dollars per employee. As manager of the bank, 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 $90 \%$ confidence level.

Key Concepts

Sample Mean

The sample mean is a fundamental concept in statistics that represents the average of the data points in a sample. It is calculated by summing all the observations and dividing by the number of observations. This statistic serves as an unbiased point estimator of the population mean when the sample is representative, and it is often used as the basis for further statistical inference such as constructing confidence intervals.

Population Standard Deviation

The population standard deviation is a measure of variability that quantifies the average distance of the data points from the population mean. In many inferential statistical procedures, knowing the population standard deviation (or having a good estimate of it) allows for the use of the normal distribution to form confidence intervals or conduct hypothesis tests. It is especially useful when the underlying population distribution is assumed to be normal.

Standard Error

The standard error of the sample mean measures the dispersion of the sample mean estimates around the true population mean. It is calculated by dividing the population standard deviation by the square root of the sample size. This concept is critical because it reflects the reliability of the sample mean as an estimator of the population mean; a smaller standard error indicates a more precise estimate.

Confidence Interval

A confidence interval is a range of values, derived from the sample data, that is likely to contain the true population parameter with a specified level of confidence. The interval provides an estimated range of values which is believed, with a certain degree of certainty (such as 75% or 90%), to include the population mean. This is a key concept in inferential statistics as it quantifies the uncertainty associated with sampling.

Critical Value

The critical value is the multiplier used in the construction of a confidence interval, corresponding to the chosen confidence level. For normally distributed data (or with large samples), this value is derived from the standard normal (z) distribution. It sets the distance on either side of the sample mean such that the probability of the true mean being outside this range is the complement of the confidence level.

Margin of Error

The margin of error in a confidence interval quantifies the extent of uncertainty around the point estimate (the sample mean). It is computed by multiplying the critical value by the standard error. This value indicates how far the interval extends from the sample mean in both directions and is essential in understanding the precision of the estimate.

Interpretation of Confidence Intervals

Interpreting confidence intervals involves understanding that, with a specified level of confidence, the interval calculated from the sample data is expected to contain the true population parameter. This interpretation supports decision-making by providing a range within which the parameter likely lies, and it can help assess whether observed sample values (such as performance measures) are typical or anomalous relative to the estimated population parameter.

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