Question

For each of the following sets of numbers, calculate a 95% confidence interval for the mean (σ not known); before going through the steps in calculating the confidence interval, the sample mean ( X ¯ ) and standard deviation (s) must first be calculated.

a) 12, 11, 16, 9, 14, 10

b) 3.50, 2.25, 3.30, 4.75, 2.60, 4.00, 3.80

c) 24, 31, 28, 26, 19, 33, 22, 17, 25

d) 7, 4, 13, 8, 6, 10, 9, 5, 17, 10, 7, 12

Answer 1

To create a 95% confidence interval for the mean with an unknown population standard deviation, calculate the sample mean and standard deviation, identify the t-score for the desired confidence level and degrees of freedom, and apply these to find the confidence interval by adding/subtracting the margin of error from the sample mean.

Step-by-step explanation:

To calculate a 95% confidence interval for the mean when the population standard deviation (σ) is not known, we use the t-distribution.

The sample mean (μ), sample standard deviation (s), and the size of the sample (n) are required for this calculation.

The steps involved include finding the sample mean and standard deviation, determining the degrees of freedom (df = n - 1), and then using a t-table or calculator to find the t-score that corresponds to the 95% confidence level.

Once we have the t-score, we calculate the margin of error (EBM) by multiplying the t-score with (s / √n).

Finally, the confidence interval is found by adding and subtracting the margin of error from the sample mean.

For example, if we have a sample with the following values (a): 12, 11, 16, 9, 14, 10, we calculate the mean and standard deviation of this sample, find the appropriate t-score for df = 5 at 95% confidence, and then calculate the margin of error to construct the confidence interval.