1.8.2. Tests based on independent samples

(Known variance or large sample size)

Let us consider the case where we have independent random samples from two normally distributed populations. The first population has mean and variance and we obtain a random sample of size . The second population has mean and variance and we obtain a random sample of size .

We know that if the sample means are denoted and , then the random variable

has a standard normal distribution. If the population variances are known, tests for the difference between the population means can be based on this result. 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

Remark: If the sample sizes are large ( ) then a good approximation at significance level can be made if the population variances and are replaced by the sample variances and .

In addition the central limit theorem leads to good approximations even if the populations are not normally distributed.

Example:

According to the Bureau of Labor Statistics, last year university instructors earned an average $440 per month and college instructors earned an average of $420 per month. Assume that these mean earnings have been calculated for samples of 400 and 600 instructors taken from the two populations, respectively. Further assume that the standard deviations of monthly earnings of the two populations are $50 and $63, respectively. Test at 1% significance level if the mean monthly earnings of the two groups of the instructors are different.

Solution:

From the information given above,

; ; ;

; ; ;

where the subscript x refers to university instructors and y-to college instructors. Let

= mean monthly earnings of all university instructors

= mean monthly earnings of all college instructors.

We are to test if the two population means are different. The null and alternative hypotheses are

(the monthly earnings are not different)

(the monthly earnings are different).

The decision rule is

Reject if or

First of all we find the value of . Since , the value of is (approximately) 2.58 and .

The value of the test statistic is computed as follows:

.

and the value of test statistic falls in the rejection region, we reject the null hypothesis . Therefore, we conclude that the mean monthly earnings of the two groups of instructors are different.

Note that we can not say for sure that two means are different. All we can say is that the evidence from the two samples is very strong that the corresponding population means are different.

Exercises

1. The following information is obtained from two independent samples selected from two populations

Test at the 1% significance level if the two population means are the same against the alternative that they are different.

2. Daily wage is $13.62 for transportation workers and $11.61 for factory workers. Assume that these two estimates are based on random samples of 1000 and 1200 workers taken, respectively, from the two populations. Also assume that the standard deviations of the two populations are $1.85 and $1.40, respectively.

a) Test at the 5% significance level if the mean daily wage of transportation workers and factory workers are the same against the alternative that it is higher for transportation workers.

b) What will your decision be in part a) if the probability of making a Type I error is zero. Explain.

3. A consulting firm was asked by a large insurance company to investigate if business majors were better salespersons. A sample of 40 salespersons with a business degree showed that they sold an average of 10 insurance policies per week with a standard deviation of 1.80. Another sample of 45 salespersons with a degree other than business showed that they sold an average of 8.5 insurance policies per week with a standard deviation of 1.35. Using the 1% significance level, can you conclude that person with a business degree are better salespersons than those who have a degree in another area?

4. The management at the bank A claims that the mean waiting time for all customers at its branches is less than that at the bank B, which is main competitor. They took a sample of 200 customers from the bank A and found that they waited an average of 4.60 minutes with a standard deviation of 1.2 minutes before being served. Another sample of 300 customers taken from the bank B showed that these customers waited an average of 4.85 minutes with a standard deviation of 1.5 minutes before being served.

a) Test at the 2.5% significance level if the claim of the management of the bank A is true.

b) Calculate the p-value. Based on this p-value, would you reject the null hypothesis if What if

5. A production line is designed on the assumption that the difference in mean assembly times for two operations is 5 minutes. Independent tests for the two assembly operations show the following results:

Operation A Operation B

minutes minutes

minutes minutes

For , test the hypothesis that the difference between the mean assembly times is 5 minutes.

6. An investigation was carried out to determine if women employees are as well paid as their male counterparts. Random samples of 75 males and 64 females are selected. Their mean salaries were 45 530 and 44 620, standard deviations were 780 and 750, correspondingly. If you were to test the null hypothesis that the mean salaries are equal against the two sided alternative, what would be the conclusion of your test with ?

7. For a random sample of 125 state companies, the mean number of job changes was 1.91 and the standard deviation was 1.32. For a random sample of 86 private companies, the mean number of job changes was 0.21 and the standard deviation was 0.53. Test the null hypothesis that the population means are equal against the alternative that the mean number of job changes is higher in state companies than for private companies.

Answers

1. ;reject ;2.a) ; reject ;b)do not reject ; 3. ; reject ; 4. a) ; reject ;b) p-value=0.0197;

do not reject at ; reject at ; 5. ; reject ; 6. ; reject ; 7. ; reject at any level.