Question

A city wants to estimate the standard deviation of the time it takes a bus to travel between two stops in a city. To develop an estimate for the standard deviation, the city has collected a random sample of the times required for 11 trips. The sample standard deviation is 4.6 minutes. Based on these data, what is the 95% confidence interval estimate for the true population standard deviation.

Answer 1

The 95% confidence interval estimate for the true population standard deviation is (1.634, 7.879) minutes.

Step-by-step explanation:

To estimate the population standard deviation, we can use the sample standard deviation as our point estimate. In this case, the sample standard deviation is 4.6 minutes. To find the 95% confidence interval estimate for the true population standard deviation, we need to compute the lower and upper bounds of the interval.

Using the chi-square distribution, we can find the critical values for the lower and upper bounds. For a 95% confidence level, with a sample size of 11 (n-1 = 10 degrees of freedom), the critical values are 3.247 (lower bound) and 20.483 (upper bound).

Therefore, the 95% confidence interval estimate for the true population standard deviation is (sqrt((n-1)*s²)/upper bound, sqrt((n-1)*s²)/lower bound) = (sqrt(10*4.6²)/20.483, sqrt(10*4.6²)/3.247) = (1.634, 7.879) minutes.

Answer 2

The 95% confidence interval estimate for the true population standard deviation is [5.06, 8.81] minutes.

Explanation:

To estimate the 95% confidence interval for the true population standard deviation, we can use the following formula:

Confidence interval = [√(n-1)s2/X2α/2, √(n-1)s2/X21-α/2]

where n is the sample size, s is the sample standard deviation, X^2α/2 is the chi-square critical value with n-1 degrees of freedom at the lower tail, and X^21-α/2 is the chi-square critical value with n-1 degrees of freedom at the upper tail.

Substituting the given values, we get:

Confidence interval = [√(11-1)(4.6)^2/19.68, √(11-1)(4.6)^2/3.82]

Confidence interval = [5.06, 8.81]

Therefore, the 95% confidence interval estimate for the true population standard deviation is [5.06, 8.81] minutes.