A sample of size 100 is taken resulting in a standard deviation of 3. Construct a 95% confidence interval for the population variance.

To construct a confidence interval for the population variance, we can use the chi-square distribution.

The formula for a confidence interval for the population variance is:
[ (n-1) * s^2 / chi-square upper, (n-1) * s^2 / chi-square lower ]

Where:
- n is the sample size
- s is the sample standard deviation
- chi-square upper and chi-square lower are the critical values of the chi-square distribution with (n-1) degrees of freedom, corresponding to the desired confidence level.

Since you have a sample size of 100 and a sample standard deviation of 3, we can find the critical values for a 95% confidence level using the chi-square distribution table or a statistical software.

To find the critical values, you would need to find the chi-square value that has 95% of the area under the curve to the left and the remaining 5% of the area to the right. Since the chi-square distribution is skewed, we typically use a two-tailed test by dividing the significance level (0.05) by 2, giving us 0.025 in each tail.

For a sample size of 100 minus 1 (99) degrees of freedom, the chi-square critical values are approximately 73.361 and 128.422.

Finally, we can substitute the values into the formula:
[ (100-1) * 3^2 / 128.422 , (100-1) * 3^2 / 73.361 ]
This gives us the 95% confidence interval for the population variance.

Note: The lower and upper bounds of the confidence interval will always be positive because variance cannot be negative.