A sample of 40 CDs from a student's collection showed a mean length of 52.74 minutes with a standard deviation of 13.21 minutes. Construct a 95% confidence interval for the population standard deviation. (Round your answers to 4 decimal places.)
You might try a formula like this one:
s/(1 + 1.96/√2n) to s/(1 - 1.96/√2n)
Substitute and calculate. There may be other variations of similar formulas you can use as well.
To construct a 95% confidence interval for the population standard deviation, you can use the chi-square distribution.
Step 1: Identify the critical values.
Since the confidence level is 95%, we need to find the lower and upper critical values from the chi-square distribution. The degrees of freedom (df) is equal to the sample size minus 1, so df = 40 - 1 = 39. Look up the critical values for a chi-square distribution with 39 degrees of freedom, one-tailed, and a significance level of 0.025 (since 95% confidence interval corresponds to 100% - 95% = 5%, and dividing this by 2 gives us 0.025 for each tail).
The critical value for the lower tail is 22.1562, and the critical value for the upper tail is 60.078.
Step 2: Calculate the confidence interval.
The confidence interval for the population standard deviation is calculated using the formula:
CI = [√((n-1)s² / X²), √((n-1)s² / X²)]
where n is the sample size, s is the sample standard deviation, X is the chi-square critical value.
In this case, n = 40, s = 13.21, lower critical value X₁ = 22.1562, and upper critical value X₂ = 60.078.
Plugging the values into the formula, we get:
CI = [√((40-1) * 13.21² / 60.078), √((40-1) * 13.21² / 22.1562)]
Step 3: Calculate the confidence interval values.
Calculating the values using a calculator or software:
CI = [9.5071, 19.2250]
Therefore, the 95% confidence interval for the population standard deviation is [9.5071, 19.2250], rounded to 4 decimal places.