1. Chapter 10 Hypothesis Testing On The Comparisons of Two Population Means

2. Chapter Outline
Inferences About the Difference Between Two Population Means: s 1 and s 2 Known

3. Interval Estimation of m 1 – m 2
Inferences About the Difference Between Two Population Means: s 1 and s 2 Known
Interval Estimation of m 1 – m 2
Hypothesis Tests About m 1 – m 2

4. Estimating the Difference Between Two Population Means
So far, we have studied the estimation of ONE population mean based on the distribution of a sample drawn from the population. Similarly, when we want to estimate the difference between two population means, we can pick a sample from each population and analyze the sampling distribution of the difference between the two sample means.
More specifically, to estimate 1 - 2, we will select a simple random sample of size n1 from population 1 and a simple random sample of size n2 from population 2. The point estimator of the difference between the means of the populations 1 and 2 is simply

5. Sampling Distribution of
Expected Value
Standard Deviation (Standard Error)
where: 1 = standard deviation of population 1
2 = standard deviation of population 2
n1 = sample size from population 1
n2 = sample size from population 2

6. Interval Estimation of 1 - 2: s 1 and s 2 Known
Interval Estimate
where:
1 -  is the confidence coefficient

7. Interval Estimation of 1 - 2: s 1 and s 2 Known
Example: Textbook Spending
To compare the amount of money students spent on textbooks in the Fall semester between two universities, independent random samples were chosen and revealed the following sample information:
University A University B
Sample Size
Sample Mean $ $250
We assume that the population standard deviations of students’ spending on textbooks are known, i.e. s A=$20 and s B=$23 .

8. Interval Estimation of 1 - 2: s 1 and s 2 Known
Example: Textbook Spending
Let us develop a 95% confidence interval estimate of the difference in the average spending on textbooks between the two universities.
Point estimate of 1 - 2 = =
= $10
where:
1 = average spending on textbooks of students
in University A
2 = average spending on textbooks of students
in University B

9. Interval Estimation of 1 - 2: s 1 and s 2 Known
Example: Textbook Spending
or $0.97 to $19.03
We are 95% confident that the difference in
the average spending on textbooks by students
between the two universities is from $0.97 to $19.03.

10. Hypothesis Tests About 1 - 2: s 1 and s 2 Known
Hypotheses
Left-tailed
Right-tailed
Two-tailed
Test Statistic

11. Hypothesis Tests About 1 - 2: s 1 and s 2 Known
Example: Textbook Spending
Can we conclude, using  = .05, that, on the average, students at University A spent the same amount on textbooks as the students at University B?

12. Hypothesis Tests About 1 - 2: s 1 and s 2 Known
p-Value and Critical Value Approaches
1. Develop the hypotheses.
H0: 1 - 2 = 0 
Ha: 1 - 2  0
where:
1 = average spending on textbooks by students
in university A
2 = average spending on textbooks by students
in university B
2. Specify the level of significance.
a = .05

13. Hypothesis Tests About 1 - 2: s 1 and s 2 Known
p-Value and Critical Value Approaches
3. Compute the value of the test statistic:

14. Hypothesis Tests About 1 - 2: s 1 and s 2 Known
p-Value Approach
4. Compute the p-value:
For z=2.17, the area to the left of 2.17 is
= So, the p-value is 2(0.015)=0.03 for the
two-tailed test.
Determine whether to reject H0:
Because p-value < a = .05, we reject H0.
At the .05 level of significance, the sample data
indicate that the average spending on textbooks by students
in university A is NOT the same at that in university B.

15. Hypothesis Tests About 1 - 2: s 1 and s 2 Known
Critical Value Approach
4. Determine the critical value and rejection rule:
For  = .05, z0.05/2= z0.025 = 1.96.
Reject H0 if z > 1.96 or z < -1.96
5. Determine whether to reject H0:
Because the test statistic z = 2.17 > 1.96,
we reject H0.
Again, at the .05 level of significance, the sample data
indicate that the average spending on textbooks by students
in university A is NOT the same at that in university B.