Tests based on independent samples. (Population variances are unknown and equal)

(Population variances are unknown and equal)

Many times it may not be possible to take large samples from populations to make inferences about the difference between two population means. This section discusses how to test a hypothesis about the difference between two population means when samples are small , and independent. Our main assumption in this case is that the two populations from which the two samples are drawn are (approximately) normally distributed. If this assumption is true, and we know the population variances, we can still use the normal distribution to make inferences about when samples are small and independent. However, we usually do not know the population variances and . In such cases, we replace the normal distribution by the Student’s t distribution to make inferences about for small and independent samples. In this section we will make one more assumption that the variances of the two populations are equal. When the variances of the two populations are equal, we can use for both and . Since is unknown, we replace it by its point estimator , which is called pooled sample variance.

Now assume that we have independent random samples of size and

observations from normally distributed populations with means and and a common variance. The sample variances and are used to compute a pooled variance estimator

The value of the test statistic for is computed as

and the following tests have a significance level

1. To test either null hypothesis

or

against the alternative

the decision rule is

Reject if

2. To test either null hypothesis

or

against the alternative

the decision rule is

Reject if

3. To test the null hypothesis

against the two sided alternative

the decision rule is

Reject if or

Here, is the number for which

where the random variable follows a Student’s t distribution with degrees of freedom.

Example:

A sample of 12 cans of Brand A diet soda gave a mean number of calories of 22 per can with a standard deviation of 2 calories. Another sample of 15 cans of Brand B diet soda gave the mean number of calories of 24 per can with a standard deviation of 3 calories. At the 1% significance level, are the mean number of calories per can different for these two brands of diet soda?

Assume that the calories per can of diet soda are normally distributed for each of the two brands and that the variances for the two populations are equal.

Solution:

Let and be the mean number of calories per can for diet soda of Brand A and Brand B, respectively, and let and be the means of respective samples. From the given information,

; ; ;

; ;

The significance level is .

We are to test for the difference in the mean number of calories per can for two brands. The null and alternative hypotheses are

(the mean number of calories are not different)

(the mean number of calories are different)

The decision rule is

Reject if or

and .

The pooled estimate is

The test statistic is then computed as

Because the value of test statistic for falls in the nonrejection region (Fig.1.10), we fail to reject the null hypothesis. Consequently we conclude that there is no difference between the mean number of calories per can for the two brands of diet soda. The difference in and observed for two samples may have occurred due to sampling error only.

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