Answer :
To find the 99.9% confidence interval for the population mean [tex]\(\mu\)[/tex] using the given sample data, we can follow a series of steps:
1. Calculate the Sample Mean:
- The sample mean is the average of all the sample values. For this sample, the mean is approximately [tex]\(67.68\)[/tex].
2. Determine the Sample Standard Deviation:
- The sample standard deviation measures the spread of the sample data around the mean. For this sample, it is approximately [tex]\(15.93\)[/tex].
3. Find the Sample Size:
- The sample size [tex]\(n\)[/tex] is the total number of data points, which is 63 in this case.
4. Locate the Critical z-value:
- For a 99.9% confidence level and a two-tailed test, we use the z-value that corresponds to the middle 99.9% of the standard normal distribution. The critical z-value is approximately [tex]\(3.291\)[/tex].
5. Calculate the Margin of Error:
- The margin of error (ME) is calculated using the formula:
[tex]\[
\text{ME} = z \times \left(\frac{\text{sample standard deviation}}{\sqrt{n}}\right)
\][/tex]
- For this problem, the margin of error is approximately [tex]\(6.60\)[/tex].
6. Determine the Confidence Interval:
- Finally, we calculate the confidence interval using the sample mean and the margin of error:
[tex]\[
\text{Lower Limit} = \text{sample mean} - \text{margin of error} \approx 61.07
\][/tex]
[tex]\[
\text{Upper Limit} = \text{sample mean} + \text{margin of error} \approx 74.28
\][/tex]
Putting all this together, the 99.9% confidence interval for the population mean [tex]\(\mu\)[/tex] is approximately:
[tex]\[ 61.07 < \mu < 74.28 \][/tex]
This interval suggests that we are 99.9% confident that the true population mean lies between 61.07 and 74.28.
1. Calculate the Sample Mean:
- The sample mean is the average of all the sample values. For this sample, the mean is approximately [tex]\(67.68\)[/tex].
2. Determine the Sample Standard Deviation:
- The sample standard deviation measures the spread of the sample data around the mean. For this sample, it is approximately [tex]\(15.93\)[/tex].
3. Find the Sample Size:
- The sample size [tex]\(n\)[/tex] is the total number of data points, which is 63 in this case.
4. Locate the Critical z-value:
- For a 99.9% confidence level and a two-tailed test, we use the z-value that corresponds to the middle 99.9% of the standard normal distribution. The critical z-value is approximately [tex]\(3.291\)[/tex].
5. Calculate the Margin of Error:
- The margin of error (ME) is calculated using the formula:
[tex]\[
\text{ME} = z \times \left(\frac{\text{sample standard deviation}}{\sqrt{n}}\right)
\][/tex]
- For this problem, the margin of error is approximately [tex]\(6.60\)[/tex].
6. Determine the Confidence Interval:
- Finally, we calculate the confidence interval using the sample mean and the margin of error:
[tex]\[
\text{Lower Limit} = \text{sample mean} - \text{margin of error} \approx 61.07
\][/tex]
[tex]\[
\text{Upper Limit} = \text{sample mean} + \text{margin of error} \approx 74.28
\][/tex]
Putting all this together, the 99.9% confidence interval for the population mean [tex]\(\mu\)[/tex] is approximately:
[tex]\[ 61.07 < \mu < 74.28 \][/tex]
This interval suggests that we are 99.9% confident that the true population mean lies between 61.07 and 74.28.
