Q 6RP. - Elementary Statistics

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.

Step-by-Step Solution

Answer

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

Part (b) Yes.

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.