Concept explainers
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.
- a. Find a 90% confidence interval for the
mean oven temperature. - b. Find a 95% confidence interval for the mean oven temperature.
- c. What is the confidence level of the interval (347.5, 348.9)?
- d. How many thermostats must be sampled so that a 90% confidence interval specifies the mean to within ±0.8°F?
- e. How many thermostats must be sampled so that a 95% confidence interval specifies the mean to within ±0.8°F?
a.
Find the 90% confidence interval for the mean oven temperature.
Answer to Problem 8E
The 90% confidence interval for the mean oven temperature is
Explanation of Solution
Given info:
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
b.
Find the 95% confidence interval for the mean oven temperature.
Answer to Problem 8E
The 95% confidence interval for the mean oven temperature is
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
c.
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
Calculation:
The value of
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%.
d.
Find the number of thermostats must be sampled when the 90% confidence interval specifies the mean to within
Answer to Problem 8E
The number of thermostats must be sampled when the 90% confidence interval specifies the mean to within
Explanation of Solution
Calculation:
From Table A.2 of the Cumulative Normal distribution, the value of
The sample size is,
Thus, the number of thermostats must be sampled when the 90% confidence interval specifies the mean to within
e.
Find the number of thermostats must be sampled when the 95% confidence interval specifies the mean to within
Answer to Problem 8E
The number of thermostats must be sampled when the 95% confidence interval specifies the mean to within
Explanation of Solution
Calculation:
From Table A.2 of the Cumulative Normal distribution, the value of
The sample size is,
Thus, the number of thermostats must be sampled when the 95% confidence interval specifies the mean to within
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