8.1 Testing the Difference Between Means (Independent Samples, 1 and 2 Known)

• Key Concepts:– Dependent vs Independent Samples– Sampling Distribution of the Difference of the

Sample Means– Two-Sample Hypothesis Testing

8.1 Testing the Difference Between Means (Independent Samples, 1 and 2 Known)

• Our work to this point has been limited to only one population. In this section, we will be studying samples from two populations.– Before we do anything, we need to analyze the nature of the

samples.

• Scenario #1A company claims its new fuel additive will improve the fuel economy of any vehicle. A consumer group plans to test the company’s claim. What would be the best way for the consumer group to conduct their study?

8.1 Testing the Difference Between Means (Independent Samples, 1 and 2 Known)

• Scenario #2The National Association of Colleges and Employers (NACE) conducts surveys of salary offers to new college graduates and publishes the results in Salary Survey. The starting salaries of 35 randomly selected liberal arts graduates and 32 criminal justice graduates are shown below (data are in thousands of dollars). Is there enough evidence to conclude that the mean staring salaries of liberal arts graduates and criminal justice graduates differ? (Weiss, 7th ed. #10.11 p. 490)

8.1 Testing the Difference Between Mean (Independent Samples, 1 and 2 Known )

Liberal Arts

32.0 28.4 33.2 30.8 27.1

26.0 27.1 27.1 30.0 31.8

31.3 27.4 31.3 32.0 31.1

29.5 28.3 30.4 28.5

30.9 30.0 29.5 30.6

31.5 28.2 26.7 34.2

31.0 29.2 30.7 31.3

31.4 31.9 32.6 30.2

Criminal Justice

28.1 26.2 28.9 26.2

28.3 29.5 30.5 25.5

26.7 29.8 28.4 27.1

28.8 28.4 27.9 29.8

28.1 26.8 24.6 28.4

29.1 26.7 28.5 28.7

29.3 28.3 27.4 24.4

28.5 25.0 27.9 27.4

8.1 Testing the Difference Between Means (Independent Samples, 1 and 2 Known)

• Definitions:– Two samples are independent if the sample selected from one

population is not related to the sample selected from the second population. Two samples are dependent if each member of one sample corresponds to a member of the other sample. Dependent samples are also called paired samples or matched samples.

• It is important to make this distinction because there are hypothesis tests designed specifically for each type of sampling.

8.1 Testing the Difference Between Means (Independent Samples, 1 and 2 Known)

• Sampling Distribution of the Difference of the Sample Means– If our variable of interest is normally distributed on each

population (or n 30 for each sample) and both 1 and 2 are known, then:

1 2 1 2

1 2 1 2

1 2

2 22 2 1 2

1 2

1 2 is normal (or approximately normal)

X X X X

X X X X n n

X X

8.1 Testing the Difference Between Means (Independent Samples, 1 and 2 Known)

• When we conduct a two-sample hypothesis test, we work under the assumption that there is no difference between the parameters of the two populations. Ho will take one of three forms:

Ho: µ1 = µ2

Ho: µ1 ≤ µ2

Ho: µ1 ≥ µ2

• Our test statistic will be the standardized version of the difference of the sample means.– Guidelines for the Two-Sample z-Test for the Difference Between Means

are given on page 421.

8.1 Testing the Difference Between Means (Independent Samples, 1 and 2 Known)

• We’re all set!– #18 p. 425 (Repair Costs: Washing Machines)– #24 p. 426 (Spending More Time Sleeping?)