Chapter 8.1, Problem 6E

(a)

To determine

To calculate:The shape, center and the spread of the sampling distribution of $\overline{x}$ .

Answer to Problem 6E

The shape of sampling distribution of the sample mean $\overline{x}$ is approximately normally distributed with mean $\mu$ and standard deviation 0.057.

Explanation of Solution

Given information:

Number of trucks tested for NOX emission test is n = 50

Standard deviation $\sigma$ = 0.04

Formula Used:

  $\text{Population}\text{standard}\text{deviation}=\frac{\text{Standard}\text{deviation}}{\text{Square}\text{root}\text{of}\text{sample}\text{size}}$

Given, number of trucks tested for NOX emission test is n = 50

Standard deviation $\sigma$ = 0.04

For the shape of the sampling distribution, use the Central Limit Theorem (CLT).

The shape of sampling distribution of the sample mean $\overline{x}$ is approximately normally distributed.

For the center, from given information of the sampling distribution of the sample mean is $\mu$ .

And for the spread, use the formula.

  $\begin{array}{c}\text{Population}\text{standard}\text{deviation}=\frac{\text{Standard}\text{deviation}}{\text{Square}\text{root}\text{of}\text{sample}\text{size}}\\ {\sigma }_{\overline{x}}=\frac{\sigma }{\sqrt{n}}\\ =\frac{0.4}{\sqrt{50}}\\ \approx 0.057\end{array}$

Hence, the shape is approximately normally distributed with center $\mu$ and spread is 0.057.

(b)

To determine

To draw:The sampling distribution of the mean and mark three standard deviation value on each side of the mean.

Explanation of Solution

Given information:

Number of trucks tested for NOX emission test is n = 50

Standard deviation $\sigma$ = 0.04

Given, number of trucks tested for NOX emission test is n = 50

Standard deviation $\sigma$ = 0.04

  

Below figure shows the sampling distribution of $\overline{x}$ with mean $\mu$ as center and three standard deviation values on each sides.

  

(c)

To determine

Tocalculate: The distance m of the mean of the sampling distribution.

Answer to Problem 6E

Distance m of the mean of the sampling distribution is 0.114.

Explanation of Solution

Given information:

Number of trucks tested for NOX emission test is n = 50

Standard deviation $\sigma$ = 0.04 Population standard deviation ${\sigma }_x=0.057$

Given, number of trucks tested for NOX emission test is n = 50

Standard deviation $\sigma$ = 0.04 Population standard deviation ${\sigma }_x=0.057$

By 68-95-00.7% rule, about 95% of all values of $\overline{x}$ lie within a distance m of the mean i.e. twice the population standard deviations of the mean.

  $\begin{array}{c}m=2\sigma \\ =2\times 0.057\\ =0.114\end{array}$

  

Hence, distance m of the mean of the sampling distribution is 0.114.

(d)

To determine

The percent captured $\mu$ by the sample intervals.

Answer to Problem 6E

Sample intervals capture 95% of $\mu$ .

Explanation of Solution

Given information:

Number of trucks tested for NOX emission test is n = 50

Standard deviation $\sigma$ = 0.04

From 6c, 95% of all values of $\overline{x}$ lie within a distance m of the mean of the sampling distribution.

Hence 95% of all possible sample capture $\mu$ .