Final answer:
The 90% confidence interval for the mean temperature is (80.07, 85.59) degrees Fahrenheit, and the 95% confidence interval is (79.36, 86.3) degrees Fahrenheit. The 95% interval is wider because higher confidence levels require larger margins of error.
Step-by-step explanation:
To construct confidence intervals for the population mean when the population standard deviation is known, we use the z-distribution. The formula for the confidence interval is given by the mean ± (z*population standard deviation/√n), where n is the sample size, and z is the z-score corresponding to the desired confidence level.
For a sample size of 65, with a sample mean temperature of 82.83°F and a population standard deviation of 13.96°F, the confidence intervals are calculated as follows:
- Find the z-score for each confidence level. For a 90% confidence level, the z-score is approximately 1.645. For a 95% level, it's about 1.96.
- Calculate the margin of error (EBM) using the formula EBM = z*(population standard deviation/√n).
- Add and subtract this margin of error from the sample mean to find the confidence intervals.
The 90% confidence interval is calculated as 82.83 ± (1.645*13.96/√65) which gives (80.07, 85.59), rounded to two decimal places. The 95% confidence interval is 82.83 ± (1.96*13.96/√65) which gives (79.36, 86.3), rounded to two decimal places.
The 95% confidence interval is wider than the 90% interval because it requires a larger z-score to ensure that we capture the mean with more confidence.