1.8.3. Tests based on independent samples

(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.

Exercises

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

; ; ;

; ;

a) Test at 1% significance level if the two population means are different.

b) Test at 5% significance level if is different than .

2. The following summary statistics are recorded for independent random samples from two normally distributed populations with equal variances

Sample 1 Sample 2

Test the null hypothesis against the alternative that with .

3. Salary surveys of marketing and management majors show the following starting annual salary data

Marketing majors management majors

Consider the test of the hypothesis that the mean annual salaries are the same for both majors. For can you conclude that a difference exists in the mean annual salary for the two majors?

4. A professor took two samples, one of 21 males and another of 15 females from university students who were enrolled in business statistics at the same university. He found that the mean score of male students in a mid-term examination in statistics was 75.3 with a standard deviation of 6.4, and the mean score of female students was 78.3 with a standard deviation of 7.3. Assume that the scores of all male and all female students are normally distributed with equal but unknown standard deviations.

Test at the 2.5 significance level if the mean score in business statistics for all male and female students are the same against the alternative that male students have lower score than that for all female students.

5. The management of a supermarket wanted to investigate if the male customers spend less money on average, than the female customers. A sample of 16 male customers who shopped at this supermarket showed that they spent an average of $55 with a standard deviation of $12.50. Another sample of 22 female customers who shopped at the supermarket showed that they spent an average of $63 with a standard deviation of $14.5. Assume that the amounts of money spent at this supermarket by all male and female customers are normally distributed with equal but unknown population variance. Test at the 5% significance level if the mean amount spent by all male and female customers are the same against the alternative that male customers at this supermarket spend less than that of female customers.

6. A bank has two branches. The quality department wanted to check if the customers are equally satisfied with the service provided at these two branches. Randomly selected customers asked to measure the satisfaction of services (on scale of 1 to 11, 1 being the lowest and 11 being the highest).

A random sample of six customers from the branch A produced following data:

9.50; 8.60; 8.59; 6.50; 4.79; 4.29

An independent random sample of six customers selected from the branch B produced following data:

10.21; 9.66; 7.67; 5.12; 4.88; 3.12

Stating any assumptions you need to make, test against two sided alternative the null hypothesis that the two populations mean satisfaction index for all customers for the two branches are the same.

7. Given that , , , and , ,

. Test against with .

8. A researcher wants to test the mean GPA (grade point averages) of all male and all female university students. She took a random sample of 28 male students and 24 female students. She found that GPA’s of the two groups to be 2.62 and 2.74, respectively, with the corresponding standard deviations equal to 0.43 and 0.38. Test at the 5% significance level if the mean GPA’s of the two populations are equal against two sided alternative.

Assume that the GPA’s of all male and female students are normally distributed with equal but unknown standard deviations.

Answers

1. a) ; reject ; b) ; reject ;

2. ; is not rejected; 3. ; accept ;

4. ; accept ; 5. ; reject ; 6. We assume that the values are normally distributed with equal variance; ; Fail to reject at 20% significance level; 7. ; reject ;

8. ; accept .