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

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 .

Дата добавления: 2015-08-05