Estimating the Mean With Unknown Variance

Java Applet Simulation of the mean estimation experiment

The Distribution of the Standard Score

Let us return to the problem of estimating the unknown mean of the normal distribution when the standard deviation is also unknown. In Section 3, we modified the procedure for the case when the standard deviation is known by using confidence bounds of the form

where the quantile z is appropriate for the type of interval and the confidence level. This was a purely ad-hoc procedure, and we saw that these confidence bounds did not work well for small samples.

If you go back and look at the derivation in Section 2, you will see that what we really need to know is the distribution of the standard score when the sample standard deviation is used:

1. Show that

2. Show that Z has the standard normal distribution.

3. Show that V has the chi-square distribution with n - 1 degrees of freedom.

4. Show that Z and V are independent. (Hint: the sample mean and sample standard deviation of a sample from the normal distribution are independent.)

5. Conclude from Exercises 1-4 that T has the student t distribution with n - 1 degrees of freedom.

Confidence Bounds

For a number p in (0, 1), we will denote the p'th quantile of the t distribution with n degrees of freedom by t_n,p. Thus, by definition, if random variable T has the t distribution with n degrees of freedom then

P(T < t_n,p) = p.

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

We can now easily derive the confidence bounds

6. Show that

7. Show that the expression in Exercise 6 can be equivalently written as

From Exercise 7, it follows that

is a 1 - a confidence interval for the distribution mean. Note that the length of this confidence interval is random, because it depends on the sample variance. This is in contrast with the case in which the variance of the underlying distribution is known, where the length of the confidence interval is fixed.

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

9. Use a derivation similar to Exercises 6 and 7 to show that a 1 - a confidence upper bound for the distribution mean is

The Simulation

In the simulation of the mean estimation experiment, you can choose from a list box whether to use population standard deviation s or the sample standard deviation S in the construction of the confidence interval. You can choose from another list box whether to use quantiles from the normal distribution or from the t distribution . In either case, the density of the chosen distribution is shown in the middle graph. The quantiles are recorded and the interval defined by the quantiles is shown as a blue bar in the middle graph. When you run the simulation, the value of the appropriate standard score is recorded in the third table and plotted as a red line on the horizontal axis. The event that this line falls in the critical interval is equivalent to the event that the confidence interval successfully captured the mean (and thus the success indicator variable I takes the value 1).

10. In the mean estimation experiment select Use S and Use t quantiles. Select the normal distribution with mean 0 and standard deviation 2, and select two-sided intervals. For each of the following sample sizes and confidence levels, run the experiment 1000 times with an update frequency of 10. Note the size and location of the confidence intervals an how well the proportion of successful intervals approximates the theoretical confidence level.

1. n = 5, 80%.
2. n = 5, 90%.
3. n = 10, 90%.
4. n = 30, 90%.

11. In the mean estimation experiment select use S and use z. Select the normal distribution with mean 0 and standard deviation 2, and select two-sided intervals. For each of the following sample sizes and confidence levels, run the experiment 1000 times with an update frequency of 10. Note the size and location of the confidence intervals an how well the proportion of successful intervals approximates the theoretical confidence level.

1. n = 5, 80%.
2. n = 5, 90%.
3. n = 10, 90%.
4. n = 30, 90%.

In Exercise 10, you are using the correct procedure and thus you should have noticed good agreement between the proportion of successful intervals and the theoretical confidence level in all cases. In Exercise 11, you are using our incorrect, ad hoc procedure. When the sample size is small, you should have noticed that the proportion of successful intervals was consistently smaller than the theoretical confidence level.

It's easy to understand the observed behavior mathematically. The t distribution has larger variance than the standard normal distribution. Thus the t quantiles for a given confidence level are larger in absolute value than the z quantiles for that confidence level and hence the interval constructed using the t quantiles is larger than the interval constructed using the z quantiles. On the other hand, the t distribution converges to the standard normal distribution as n increases and thus the difference is slight when the sample size is large.

Non Normal Distributions

When the distribution from which we are sampling is not normal, the procedure of this section is still used to obtain approximate confidence bounds. The procedure works well as long as the sample size is large and the distribution is not too far from normal.

12. In the mean estimation experiment, select Use S and Use t. Select the gamma distribution with shape parameter 1 and scale parameter 1. Select two-sided intervals and confidence level 0.90. For each of the following sample sizes, run the experiment 1000 times with an update frequency of 10. Note how well the proportion of successful intervals approximates the theoretical confidence level.

1. n = 5.
2. n = 10.
3. n = 30.

13. In the mean estimation experiment, select Use S and Use t. Select the gamma distribution with shape parameter 5 and scale parameter 1. Select two-sided intervals and confidence level 0.90. For each of the following sample sizes, run the experiment 1000 times with an update frequency of 10. Note how well the proportion of successful intervals approximates the theoretical confidence level.

1. n = 5.
2. n = 10.
3. n = 30.

14. In the mean estimation experiment, select Use S and Use t. Select the Poisson distribution with mean 1. Select two-sided intervals and confidence level 0.90. For each of the following sample sizes, run the experiment 1000 times with an update frequency of 10. Note how well the proportion of successful intervals approximates the theoretical confidence level.

1. n = 5.
2. n = 10.
3. n = 30.

Interval Estimation