6. 8. Confidence interval for the difference between the population means: unknown population variances that are assumed to be equal

Let us consider the confidence interval estimation procedure for the difference between the means of the populations when the population have normal distributions with equal variances, i.e., . We will again be assuming that independent random samples are selected from the populations. In this case the sampling distribution of is normal regardless of the sample sizes involved. The mean of the sampling distribution is .

Because of the equal variances , we can write

If the variance is known, then confidence interval population means can be found easily. However, if is unknown, the two samples variances,

and , can be combined to compute the following estimate of :

The process of combining the results of the two independent samples to provide an estimate of the population variance is referred to as polling, and is referred to as polled estimator of .

Definition:

Suppose that two samples of sizes and are selected from normally distributed population with means and , and a common, but unknown variance . If sample means are and , sample variances are and , then confidence interval for is given by

were S is given by

and is the number for which

.

The random variable t follows to the Student’s t distribution with degrees of freedom.

Example:

Independent random samples of checking account balances for customers at two branches of National Bank show the following results:

Bank branches	Number of checking accounts	Sample mean balance	Sample standard deviation
Bank B	12 10

Find a 90 % confidence interval estimate for the difference between the mean checking account balances of the two branches.

Solution:

confidence interval is

.

Thus, the interval estimation becomes

At a 90 % level of confidence the interval estimate for the difference in mean account balances of two branches of Bank is to .

The fact that the interval includes a negative range of values indicates that the actual difference in the two means may be negative.

Thus could be actually be larger than .

Exercises

1. The following information was obtained from two independent samples selected from two normally distributed populations with unknown but equal variances.

; ;

; ;

Construct a 99 % confidence interval for .

2. The National Bank is interested in estimating the difference between the mean credit card balances of its two branches. Independent random samples of credit card customers provide the following results:

Branch A Branch B

Construct an interval estimate of the difference between mean balances. Use a confidence coefficient of 0.99.

3. Production quantities of two workers are shown below. Each data value indicates the amount of items produced during a randomly selected 1 day period

Worker 1: 20; 18; 21; 22; 20

Worker 2: 22; 18; 20; 23; 24

Develop a 90 % confidence interval estimate for the difference between the mean production rates of the two workers.

4. The mean salary of male professors at the universities was 54 340 tg and that of female professors was 48 080 tg. For convenience, assume that these two means are based on random samples of 28 male and 26 female professors. Assume that the standard deviations of the two samples are

3 100 tg and 2 800 tg, respectively.

Construct a 90 % confidence interval for the difference between the two population means.

5. The following summary statistics are recorded for independent random samples from two populations:

Sample 1 Sample 2

Stating any assumptions that you need, determine a 98 % confidence interval for .

6. A company is interested in buying one of two different kinds of machines. Company tested the two machines for production purposes. The first machine was run for 8 hours and produced an average 123 items per hour with a standard deviation of 9 items. The second machine was run for 10 hours and produced an average of 114 items per hour with a standard deviation of 6 items. Assume that the production per hour for each machines approximately normally distributed. Also assume that the standard deviation of the hourly production of the two populations is equal.

Then find a 95 % confidence interval for the difference between the two population means.

Answers

1. (2.21; 9.29); 2. ($37.57; $212.43); 3. (Worker 2- Worker 1): 0.87; units to 3.27 units); 4. (4909.87; to 7610.13); 5. (-2.84; 25.84); 6. (1.50 to 16.50 items).