1. Exercises #8.74, 8.78 on page 403 #8.110, 8.112 on page 416
Problems from supplementary exercises:
# 8.125, on page 419
# 8.138, on page 421

2. Inferences About Two Means
In the previous chapter we used one sample to make inferences about a single population. Very often we are interested in comparing two populations.
1) Is the average midterm grade in Stat higher than the average midterm grade in Stat ?
2) Is the average grade in Quiz #1 higher than Quiz #2 in this section of Introductory Statistics?

3. Inferences about Two Means
Each sample is an example of testing a claim between two populations. However, there is a fundamental difference between 1) and 2).
In # 2) the samples are not independent where as in # 1), they are.
Why? 1. Different people in each class Same people writing different test.

4. Definition. Independent and Dependent Samples
Two samples are independent if the sample selected from one population is not related to the sample selected from the other population.
If one sample is related to the other, the samples are dependent. With dependent samples we get two values for each person, sometimes called paired-samples.

5. Notation for Two Dependent Samples

6. The (1-a)*100% confidence interval for the mean difference md is
Confidence Interval for the Mean Difference (Dependent Samples: Paired Data )
The (1-a)*100% confidence interval for the mean difference md is

7. Test Statistic for the Mean Difference (Dependent Samples)
For n<30 the appropriate test statistic for testing the mean difference between paired samples is 
with n-1 degrees of freedom.
For n>30 then we use ‘z’ 

8. Testing Claims about the Mean Difference (Independent Samples)
When making claims about the mean difference between independent samples a different procedure is used than that for dependent/paired samples.
Again there are different procedures for large (n>30) samples and small samples (n<30).
In the small sample case, we must assume that both populations are normal and have equal variances.

9. Example
Suppose we wish to compare two brands of 9-volt batteries, Brand 1 and Brand 2. Specifically, we would like to compare the mean life for the population of batteries of Brand 1, 1, and the mean life for the population of batteries of Brand 2, 2. To obtain a meaningful comparison we shall estimate the difference of the two population means by picking samples from the two populations.

10. For Brand 1 a sample of size 64 was chosen.
From the data a point estimate for 1, would be From the data a point estimate for 2 would be 7.78.
It would therefore be natural for us to take as a point estimate for (1-2) to be hours.

11. Point Estimator (Independent Samples)
The estimate is the best point estimator of (1-2).
Having found a point estimate, our next goal is to determine a confidence interval for it.

12. Point Estimator (Independent Samples)
To construct a confidence interval for (1-2) we need to know the distribution of its point estimator.
The distribution of is normal with mean (1-2) and standard deviation
where n1 is the size of sample 1, n2 is the size of sample 2.

13. Confidence Interval for Difference in two Means
Confidence Interval for Difference in two Means (Large samples or known variance)

14. Example: Life span of Batteries
Let a = .05 so we are looking for the 95% confidence interval for the mean difference.

15. Test Statistic for Two Means: Independent and large samples

16. Example: Life span of Batteries
Hypothesis Testing. I claim that the two brands of batteries have the same life span. Using a 5% level of significance, test this claim.

17. Example: Life span of Batteries
Hypothesis
Sample Data
Test Statistic

18. Example: Life span of Batteries
Critical Region
Decision The test statistic lies in the critical region, therefore we reject H0. The samples provide sufficient evidence to claim that the Batteries do indeed have different life spans.

19. Problems #9.36, 9.35 on page 465 #9.42 on page 466 #9.6 on page 448

20. Overview Comparing Two Populations: Mean (Small Dependent Samples)
Mean (Large Independent Samples).