Chapter 5.1, Problem 8E

Oven thermostats were tested by setting them to 350°F and measuring the actual temperature of the oven. In a sample of 67 thermostats, the average temperature was 348.2°F and the standard deviation was 5.1°F.

1. a. Find a 90% confidence interval for the mean oven temperature.
2. b. Find a 95% confidence interval for the mean oven temperature.
3. c. What is the confidence level of the interval (347.5, 348.9)?
4. d. How many thermostats must be sampled so that a 90% confidence interval specifies the mean to within ±0.8°F?
5. e. How many thermostats must be sampled so that a 95% confidence interval specifies the mean to within ±0.8°F?

To determine

Find the 90% confidence interval for the mean oven temperature.

Answer to Problem 8E

The 90% confidence interval for the mean oven temperature is $\underset{\_}{\left(347.175,349.225\right)}$.

Explanation of Solution

Given info:

$n=67$, $\overline{X}=348.2°F$, and $s=5.1°F$.

Calculation:

Confidence interval:

Software Procedure:

Step-by-step procedure to obtain the confidence interval using the MINITAB software:

• Choose Stat > Basic Statistics > 1-Sample Z.
• In Summarized data, enter the sample size as 67 and mean as 348.2.
• In Standard deviation, enter a value for s as 5.1.
• Check Options; enter Confidence level as 90%.
• Choose not equal in alternative.
• Click OK.

Output using the MINITAB software is given below:

From the output, the 90% confidence interval for the mean oven temperature is $\underset{\_}{\left(347.175,349.225\right)}$.

To determine

Find the 95% confidence interval for the mean oven temperature.

Answer to Problem 8E

The 95% confidence interval for the mean oven temperature is $\underset{\_}{\left(346.979,349.421\right)}$.

Explanation of Solution

Calculation:

Confidence interval:

Software Procedure:

Step-by-step procedure to obtain the confidence interval using the MINITAB software:

• Choose Stat > Basic Statistics > 1-Sample Z.
• In Summarized data, enter the sample size as 67 and mean as 348.2.
• In Standard deviation, enter a value for s as 5.1.
• Check Options; enter Confidence level as 90%.
• Choose not equal in alternative.
• Click OK.

Output using the MINITAB software is given below:

From the output, the 95% confidence interval for the mean oven temperature is $\underset{\_}{\left(346.979,349.421\right)}$.

To determine

Find the level of confidence for the given confidence interval.

Answer to Problem 8E

The level of confidence for the given confidence interval is 73.88%.

Explanation of Solution

Given info:

The confidence interval is $\left(347.5,348.9\right)$.

Calculation:

The value of $\frac{\sigma }{\sqrt{n}}$ is,

$\begin{array}{c}\frac{\sigma }{\sqrt{n}}=\frac{5.1}{\sqrt{67}}\\ =0.623\end{array}$

Confidence level:

Software Procedure:

Step-by-step procedure to obtain the confidence level using the MINITAB software:

• Choose Graph > Probability Distribution Plot choose View Probability> OK.
• From Distribution, choose ‘Normal’ distribution.
• Enter 348.2 under Mean and Enter 0.623 under Standard deviation.
• Click the Shaded Area tab.
• Choose X Value and Middle Tail for the region of the curve to shade.
• Enter the X value 1 as 347.5 and Enter the X value 2 as 348.9.
• Click OK.

Output using the MINITAB software is given below:

From the MINITAB output, the level of confidence for the given confidence interval is 0.7388 or 73.88%.

To determine

Find the number of thermostats must be sampled when the 90% confidence interval specifies the mean to within $\pm 0.8°F$.

Answer to Problem 8E

The number of thermostats must be sampled when the 90% confidence interval specifies the mean to within $\pm 0.8°F$ is 110.

Explanation of Solution

Calculation:

From Table A.2 of the Cumulative Normal distribution, the value of $z_{\frac{\alpha }{2}}$ with 90% level of confidence is 1.645.

The sample size is,

$\begin{array}{c}1.645\frac{5.1}{\sqrt{n}}=0.8\\ \frac{\text{8}\text{.3895}}{\sqrt{n}}=0.8\\ \sqrt{n}=\text{10}\text{.4869}\\ n=110\end{array}$

Thus, the number of thermostats must be sampled when the 90% confidence interval specifies the mean to within $\pm 0.8°F$ is 110.

To determine

Find the number of thermostats must be sampled when the 95% confidence interval specifies the mean to within $\pm 0.8°F$.

Answer to Problem 8E

The number of thermostats must be sampled when the 95% confidence interval specifies the mean to within $\pm 0.8°F$ is 157.

Explanation of Solution

Calculation:

From Table A.2 of the Cumulative Normal distribution, the value of $z_{\frac{\alpha }{2}}$ with 95% level of confidence is 1.96.

The sample size is,

$\begin{array}{c}1.96\frac{5.1}{\sqrt{n}}=0.8\\ \frac{\text{9}\text{.996}}{\sqrt{n}}=0.8\\ \sqrt{n}=\text{12}\text{.495}\\ n=157\end{array}$

Thus, the number of thermostats must be sampled when the 95% confidence interval specifies the mean to within $\pm 0.8°F$ is 157.