To find the 90% confidence interval for the population mean based on the given sample, we will follow these steps:
- Calculate the sample mean (
). - Calculate the sample standard deviation (
). - Determine the critical value (
) for a 90% confidence level. - Calculate the margin of error.
- Construct the confidence interval.
Step 1: Calculate the sample mean ( )
The sample data is:
First, calculate the sum of the sample values:
Then, the sample mean (
Step 2: Calculate the sample standard deviation ( )
We will first calculate the sum of squared deviations from the mean:
Each term
- For 77:
- For 44.4:
- For 33.6:
- For 63.1:
- For 68.1:
- For 70.6:
- For 52:
- For 43.5:
- For 26.4:
- For 72.6:
- For 30.9:
- For 29.4:
Now summing these squared deviations:
Now, calculate the sample variance (
Then take the square root to find the sample standard deviation (
Step 3: Determine the critical value ( )
For a 90% confidence level with
Step 4: Calculate the margin of error (ME)
The margin of error is calculated as:
Step 5: Construct the confidence interval
Finally, the confidence interval is given by:
Calculating the endpoints:
Thus, the 90% confidence interval is:
Final result with two decimal places as requested: