Question

Officer Salaries. Refer to Problem $5.$

a. Use the answer you obtained in Problem $5\left(b\right)$and Definition $3.11$on page $140$ to find the mean of the variable $\hat{x}$Interpret your answer.

b. Can you obtain the mean of the variable ix without doing the calculation in part (a)? Explain your answer.

Short answer

Part (a) The mean of the variable $\left(\overline{x}\right)$is $18\$1000$s

Part (b) Yes.

Step-by-step solution

Part (a) Step 1: Given information

From the population of six officers, $15$ size $4$ samples are possible. The first column in the following table lists them.

Part (a) Step 2: Concept

Formula used:${\mu }_{\overline{x}}=\frac{\sum \overline{x}}{N}$

Part (a) Step 3: Calculation

Obtain the mean of the variable $\left(\overline{x}\right)$

$$\begin{gathered} {\mu }_{\overline{x}}=\frac{\sum \overline{x}}{N} \\ =\frac{\left(\begin{array}{l}14+15+16+16+17+18+17+\\ 18+19+20+18+19+20+21+22\end{array}\right)}{15} \\ =\frac{270}{15} \\ =18 \end{gathered}$$

Thus, for sample size $4$, the mean $\left({\mu }_{\overline{x}}\right)$ of all potential sample mean wages is $18$ $\$1000s$.

Part (b) Step 1: Explanation

Yes, the mean of the variable $\left(\overline{x}\right)$ may be calculated because the mean for sampling sample means is the same as the population mean $\left(\mu \right)$ As a result, the population mean is $\$18.0$ thousand dollars.