the following data values are a simple random sample form a population that is normally distributed, with ²= 25.0: 47,33,42,34, and 41. Construct and interpret the 95% and 99% confidence intervals for the population mean.

Formula:

CI95 = mean + or - 1.96(sd divided by √n)
...where + or - 1.96 represents the 95% confidence interval using a z-table, sd = standard deviation, √ = square root, and n = sample size.

For 99%, substitute + or - 1.96 in the above formula to reflect the 99% confidence interval.

You already have standard deviation. Find the mean of the data listed. Substitute values into the formula and go from there.

I hope this will help get you started.

To construct confidence intervals for the population mean, you can use the formula:

Confidence Interval = Sample Mean ± (Critical Value * Standard Error)

Where:
- Sample Mean is the average of the data values
- Critical Value depends on the desired confidence level and is obtained from the standard normal distribution table (Z-table)
- Standard Error is calculated as the standard deviation of the sample divided by the square root of the sample size

Let's calculate the confidence intervals step by step:

1. Sample Mean:
Add up all the data values and divide them by the sample size (5), which gives us:
(47 + 33 + 42 + 34 + 41) / 5 = 197 / 5 = 39.4

2. Standard Error:
Since the population standard deviation (sigma, σ) is given as 25.0, we can calculate the standard error by dividing σ by the square root of the sample size:
Standard Error = 25 / √5 ≈ 11.1803

3. Critical Values:
For a 95% confidence level, we need to find the critical value associated with the two-tailed test. The critical value for a 95% confidence level is approximately 1.96.
For a 99% confidence level, the critical value is approximately 2.58.

4. Confidence Interval for the 95% Confidence Level:
Using the formula:
Confidence Interval = Sample Mean ± (Critical Value * Standard Error)
Confidence Interval = 39.4 ± (1.96 * 11.1803)
Confidence Interval ≈ 39.4 ± 21.8847
Confidence Interval ≈ (17.5153, 61.2847)

Interpretation: We are 95% confident that the true population mean falls within the range of 17.5153 to 61.2847.

5. Confidence Interval for the 99% Confidence Level:
Using the formula:
Confidence Interval = Sample Mean ± (Critical Value * Standard Error)
Confidence Interval = 39.4 ± (2.58 * 11.1803)
Confidence Interval ≈ 39.4 ± 28.8203
Confidence Interval ≈ (10.5797, 68.2203)

Interpretation: We are 99% confident that the true population mean falls within the range of 10.5797 to 68.2203.

Note: The wider the confidence interval, the higher the confidence level.