Chapter 5, Problem 84E

To determine

To find: The probability that sample mean $\overline{x}$ is greater than 85.

Answer to Problem 84E

Solution: The probability is 0.00017.

Explanation of Solution

Calculation: The sampling distribution of sample mean $\overline{x}$ is distributed normally with mean ${\mu }_{\overline{x}}=\mu$ and standard deviation ${\sigma }_{\overline{x}}=\sigma /\sqrt{n}$, where a random sample is drawn from a population with $\mu$ and $\sigma$, and the sample size is relatively large.

The value of the mean and the standard deviation of the sampling distribution is calculated as:

$\begin{array}{c}{\mu }_{\overline{x}}=\mu \\ =80.63\end{array}$

And value of ${\sigma }_{\overline{x}}$ for the sampling distribution of $\overline{x}$ can be calculated as:

$\begin{array}{c}{\sigma }_{\overline{x}}=\frac{\sigma }{\sqrt{n}}\\ =\frac{27.32}{\sqrt{500}}\\ =1.2218\end{array}$

The sampling distribution of $\overline{x}$ is approximately normal with mean ${\mu }_{\overline{x}}=80.63$ and standard deviation ${\sigma }_{\overline{x}}=1.2218$.

The probability that $\overline{x}$ is greater than 85 is the area to the right of the point 85 under the normal curve. To find the area under the normal curve, first, convert the value of sample mean $\overline{x}=85$ to a Z-score, and then obtain the area by subtracting the area left to the Z-score from 1. The Z-score for the value 85 is calculated as:

$\begin{array}{c}Z=\frac{\overline{x}-{\mu }_{\overline{x}}}{{\sigma }_{\overline{x}}}\\ =\frac{85-80.63}{1.2218}\\ =3.58\end{array}$

The area left to the particular Z-score can be obtained using Excel. The following procedure is carried out in Excel:

Step 1: Open the Excel spreadsheet.

Step 2: Type $‘=\text{NORM}\text{.DIST}\left(\right)\text{’}$ in one of the cells and specify $x=3.58$, $\text{Mean = 0}$, $\text{standard deviation = 1}$, and cumulative as TRUE. The screenshot is shown below:

The area under the standard normal curve to the left of $Z=3.58$ is obtained as 0.99938 from Excel. Therefore, the required probability is calculated as:

$1-0.99983=0.00017$

Hence, the probability that $\overline{x}$ is greater than 85 is 0.00017.