Use the given confidence level and sample data to find a confidence interval for the population standard sigma. Assume that a simple random sample has been selected from a population that has a normal distribution. salaries of college graduates who took a geology course in college 98% confidence; n= 91, x= $70,200, s=$19,912

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To find a confidence interval for the population standard deviation (sigma), we can use the chi-square distribution. Here's how you can calculate it:

Step 1: Identify the confidence level and the provided sample data.
- Confidence level: 98% (which corresponds to an alpha level of 0.02)
- Sample size: n = 91
- Sample mean: x = $70,200
- Sample standard deviation: s = $19,912

Step 2: Determine the degrees of freedom.
The degrees of freedom (df) for estimating the population standard deviation can be calculated using the formula df = n - 1. In this case, df = 91 - 1 = 90.

Step 3: Find the critical values for the chi-square distribution.
Since we have a two-tailed test and a confidence level of 98%, we need to find the critical values that leave a 1% (0.5% on each tail) probability in the tails. Using a chi-square table or calculator, the critical values for a 1% probability in each tail and 90 degrees of freedom are approximately 54.572 and 127.596, respectively.

Step 4: Calculate the confidence interval.
The confidence interval formula for the population standard deviation (sigma) is given by:
Lower confidence limit (LCL) = sqrt((n - 1) * s^2 / X^2)
Upper confidence limit (UCL) = sqrt((n - 1) * s^2 / X^2)

Applying the formula:
LCL = sqrt((90 * 19912^2) / 127.596) ≈ $16,277.33
UCL = sqrt((90 * 19912^2) / 54.572) ≈ $26,999.11

Therefore, the 98% confidence interval for the population standard deviation (sigma) is approximately $16,277.33 to $26,999.11.