Estimating the Variance

Java Applet Simulation of the mean estimation experiment

Confidence Bounds

We will now consider the problem of constructing confidence bounds for the variance of a normal distribution. It is natural to start with a special multiple of the sample variance

because we have shown that this statistic has the chi-square distribution with n - 1 degrees of freedom.

If p is a number in (0, 1), we will denote the p'th quantile of the chi-squared distribution with n degrees of freedom by

For selected values of n and p, these quantiles are given in the table of the chi-squared distribution.

1. Show that

2. Show that the expression in Exercise 1 can be equivalently written as

It follows from Exercise 2 that

is a 1 - a confidence interval for the distribution variance. Note that, in general, this confidence interval is not symmetric about the sample variance. This contrasts with confidence intervals for the distribution mean, which are always symmetric about the sample mean.

3. Show that a 1 - a confidence interval for the distribution standard deviation can be obtained by taking square roots of the confidence bounds above.

4. Use a derivation similar to Exercises 1 and 2 to show that a 1 - a confidence lower bound for the distribution variance is

5. Show that a 1 - a confidence lower bound for the distributin standard deviation can be obtained by taking the square root of the confidence bound in Exercise 4.

6. Use a derivation similar to Exercises 1 and 2 to show that a 1 - a confidence upper bound for the distribution variance is

7. Show that a 1 - a confidence upper bound for the distribution standard deviation can be obtained by taking the square root of the confidence bound in Exercise 6.

8. In the interval estimate experiment, select the normal distribution with mean 0 and standard deviation 5. Run the experiment 100 times, updating after every run. For each run, compute the 0.95 confidence interval for the variance. Over the 100 runs, compute the relative frequency of successes and compare with the confidence level.

Non Normal Distributions

Even when the underlying distribution is not normal, the procedure of this section is still used to obtain approximate confidence bounds for the variance. If the distribution is not too far from normal, the procedure is sufficiently robust and usually works well.

9. In the interval estimate experiment, select the Gamma distribution with shape parameter 1 and scale parameter 1. Run the experiment 100 times, updating after every run. For each run, compute the 0.95 confidence interval for the variance. Over the 100 runs, compute the relative frequency of successes and compare with the confidence level.

10. In the interval estimate experiment, select the Gamma distribution with shape parameter 5 and scale parameter 1. Run the experiment 100 times, updating after every run. For each run, compute the 0.90 confidence interval for the variance. Over the 100 runs, compute the relative frequency of successes and compare with the confidence level.

11. In the interval estimate experiment, select the Poisson distribution with mean 1. Run the experiment 100 times, updating after every run. For each run, compute the 0.90 confidence interval for the variance. Over the 100 runs, compute the relative frequency of successes and compare with the confidence level.

Interval Estimation