Question

Use the given degree of confidence and sample data to find a confidence interval for the population standard deviation σ. Assume that the population has a normal distribution. Round the confidence interval limits to the same number of decimal places as the sample standard deviation.

The mean replacement time for a random sample of 20 washing machines is 9.4 years and the standard deviation is 2.5 years. Construct a 99% confidence interval for the standard deviation, σ, of the replacement times of all washing machines of this type.

A. 1.8 yr < σ < 3.9 yr
B. 1.8 yr < σ < 4.2 yr
C. 1.8 yr < σ < 5.3 yr
D. 1.7 yr < σ < 4.8 yr

Answer 1

To construct a 99% confidence interval for the standard deviation, σ, of the replacement times of all washing machines of this type, we calculate the lower and upper bounds using the Chi-Square distribution. The 99% confidence interval is option C (1.8 yr < σ < 4.2 yr).

Step-by-step explanation:

To construct a confidence interval for the population standard deviation, we need to use the Chi-Square distribution. Given the sample mean replacement time of 9.4 years and the sample standard deviation of 2.5 years, we can calculate the lower and upper bounds of the confidence interval using the Chi-Square distribution.

The degrees of freedom for this problem is equal to the sample size minus 1, which is 20 - 1 = 19. For a 99% confidence interval, we need to find the critical values from the Chi-Square distribution table or use a Chi-Square calculator. The lower bound is found using the chi-square distribution with 19 degrees of freedom, and a cumulative probability of 0.005 (1% divided by 2), which equals 8.907. The upper bound is found using the chi-square distribution with 19 degrees of freedom, and a cumulative probability of 0.995 (1% divided by 2), which equals 35.172. Hence, the 99% confidence interval for the population standard deviation is (1.8 yr < σ < 4.2 yr), rounding to one decimal place.