Answer :
Sure! To find the 99.5% confidence interval for the population mean [tex]\(\mu\)[/tex] based on the given sample, follow these steps:
1. Calculate the Sample Mean ([tex]\(\bar{x}\)[/tex]):
- Add up all the sample data values provided.
- Divide the sum by the total number of data points.
2. Find the Sample Standard Deviation (s):
- Subtract the sample mean from each data point and square the result.
- Sum up all these squared differences.
- Divide this sum by one less than the number of data points (this gives us the sample variance).
- Take the square root of the sample variance to find the standard deviation.
3. Determine the Standard Error (SE):
- Divide the sample standard deviation by the square root of the number of data points. This gives the standard error of the mean.
4. Find the Critical Value:
- Determine the critical value for a 99.5% confidence interval using the t-distribution because the sample size is relatively small and we do not know the population standard deviation.
- The critical value is found using the t-distribution table, with [tex]\( n-1 \)[/tex] degrees of freedom (where [tex]\( n \)[/tex] is the number of data points).
5. Calculate the Margin of Error (ME):
- Multiply the standard error by the critical value. This gives the margin of error.
6. Determine the Confidence Interval:
- Subtract the margin of error from the sample mean to get the lower bound of the confidence interval.
- Add the margin of error to the sample mean to get the upper bound of the confidence interval.
7. Result:
- Based on the calculations, the 99.5% confidence interval for the population mean [tex]\(\mu\)[/tex] is approximately [tex]\( 73.53 < \mu < 85.74 \)[/tex].
This interval tells us that we are 99.5% confident that the true population mean falls within this range.
1. Calculate the Sample Mean ([tex]\(\bar{x}\)[/tex]):
- Add up all the sample data values provided.
- Divide the sum by the total number of data points.
2. Find the Sample Standard Deviation (s):
- Subtract the sample mean from each data point and square the result.
- Sum up all these squared differences.
- Divide this sum by one less than the number of data points (this gives us the sample variance).
- Take the square root of the sample variance to find the standard deviation.
3. Determine the Standard Error (SE):
- Divide the sample standard deviation by the square root of the number of data points. This gives the standard error of the mean.
4. Find the Critical Value:
- Determine the critical value for a 99.5% confidence interval using the t-distribution because the sample size is relatively small and we do not know the population standard deviation.
- The critical value is found using the t-distribution table, with [tex]\( n-1 \)[/tex] degrees of freedom (where [tex]\( n \)[/tex] is the number of data points).
5. Calculate the Margin of Error (ME):
- Multiply the standard error by the critical value. This gives the margin of error.
6. Determine the Confidence Interval:
- Subtract the margin of error from the sample mean to get the lower bound of the confidence interval.
- Add the margin of error to the sample mean to get the upper bound of the confidence interval.
7. Result:
- Based on the calculations, the 99.5% confidence interval for the population mean [tex]\(\mu\)[/tex] is approximately [tex]\( 73.53 < \mu < 85.74 \)[/tex].
This interval tells us that we are 99.5% confident that the true population mean falls within this range.
