Question

Joan is trying to estimate the cost of living in various cities. One measure is called the Cost of Living Index, which ranges from about 55 to 100 for the United States, with 100 corresponding to New York City. She collects a sample of 15 cities in South Carolina, and finds a sample mean Cost of Living Index of 91.41, and a sample standard deviation of 23. a. Find the critical t-score for a 90% confidence interval for the average Cost of Living Index in South Carolina. Round your answer to 2 decimal places. b. Find the margin of error for a 90% confidence interval for the average average Cost of Living Index in South Carolina. Round your answer to 2 decimal places. c. Fill in the blank: We can be % confident that the average Cost of Living Index for cities in South Carolina is between and .

Answer 1

a. The critical t-score for a 90% confidence interval is approximately 1.76. b. The margin of error for a 90% confidence interval is approximately 10.45. c. We can be 90% confident that the average Cost of Living Index for cities in South Carolina is between 80.96 and 101.86.

Step-by-step explanation:

a. To find the critical t-score for a 90% confidence interval, we need to determine the degrees of freedom (df). The df is equal to the sample size minus one, which in this case would be 15 - 1 = 14. Using a t-table or a t-distribution calculator, we can find the critical t-value for a 90% confidence level with 14 degrees of freedom. In this case, the critical t-score is approximately 1.7614.

b. The margin of error for a 90% confidence interval is calculated by multiplying the critical t-score by the standard error of the sample mean. The standard error of the sample mean is equal to the sample standard deviation divided by the square root of the sample size. In this case, the standard error would be 23 / sqrt(15) ≈ 5.9377. Therefore, the margin of error would be 1.7614 * 5.9377 ≈ 10.45.

c. To find the lower and upper bounds of the confidence interval, we subtract and add the margin of error to the sample mean. Therefore, the lower bound would be 91.41 - 10.45 ≈ 80.96, and the upper bound would be 91.41 + 10.45 ≈ 101.86. Therefore, we can be 90% confident that the average Cost of Living Index for cities in South Carolina is between 80.96 and 101.86.