1. Statistical Inferences Based on Two Samples
Chapter 10
Statistical Inferences Based on Two Samples

2. Chapter Outline
10.1 Comparing Two Population Means by Using Independent Samples: Variances Known
10.2 Comparing Two Population Means by Using Independent Samples: Variances Unknown
10.3 Paired Difference Experiments
10.4 Comparing Two Population Proportions by Using Large, Independent Samples
10.5 Comparing Two Population Variances by Using Independent Samples

3. 10.1 Comparing Two Population Means by Using Independent Samples: Variances Known
Suppose a random sample has been taken from each of two different populations
Suppose that the populations are independent of each other
Then the random samples are independent of each other
Then the sampling distribution of the difference in sample means is normally distributed

4. Sampling Distribution of the Difference of Two Sample Means #1
Suppose population 1 has mean µ1 and variance σ12
From population 1, a random sample of size n1 is selected which has mean x1 and variance s12
Suppose population 2 has mean µ2 and variance σ22
From population 2, a random sample of size n2 is selected which has mean x2 and variance s22
Then the sample distribution of the difference of two sample means…

5. Sampling Distribution of the Difference of Two Sample Means #2
Is normal, if each of the sampled populations is normal
Approximately normal if the sample sizes n1 and n2 are large
Has mean µx1–x2 = µ1 – µ2
Has standard deviation

6. Sampling Distribution of the Difference of Two Sample Means #3
Figure 10.1

7. z-Based Confidence Interval for the Difference in Means (Variances Known)
A 100(1 – ) percent confidence interval for the difference in populations µ1–µ2 is

8. z-Based Test About the Difference in Means (Variances Known)
Test the null hypothesis about H0: µ1 – µ2 = D0
D0 = µ1 – µ2 is the claimed difference between the population means
D0 is a number whose value varies depending on the situation
Often D0 = 0, and the null means that there is no difference between the population means

9. z-Based Test About the Difference in Means (Variances Known)
Use the notation from the confidence interval statement on a prior slide
Assume that each sampled population is normal or that the samples sizes n1 and n2 are large

10. Test Statistic (Variances Known)
The test statistic is
The sampling distribution of this statistic is a standard normal distribution
If the populations are normal and the samples are independent ...

11. z-Based Test About the Difference in Means (Variances Known)
Reject H0: µ1 – µ2 = D0 in favor of a particular alternative hypothesis at a level of significance if the appropriate rejection point rule holds or if the corresponding p-value is less than 
Rules are on the next slide…

12. z-Based Test About the Difference in Means (Variances Known) Continued

13. Example 10.2: The Bank Customer Waiting Time Case

14. 10.2 Comparing Two Population Means by Using Independent Samples: Variances Unknown
Generally, the true values of the population variances σ12 and σ22 are not known
They have to be estimated from the sample variances s12 and s22, respectively

15. Comparing Two Population Means Continued
Also need to estimate the standard deviation of the sampling distribution of the difference between sample means
Two approaches:
If it can be assumed that σ12 = σ22 = σ2, then calculate the “pooled estimate” of σ2
If σ12 ≠ σ22, then use approximate methods

16. Pooled Estimate of σ2

17. t-Based Confidence Interval for the Difference in Means (Variances Unknown)

18. Example 10.3: The Catalyst Comparison Case

19. t-Based Test About the Difference in Means: Variances Equal

20. Example 10.4: The Catalyst Comparison Case

21. t-Based Confidence Intervals and Tests for Differences with Unequal Variances

22. 10.3 Paired Difference Experiments
Before, drew random samples from two different populations
Now, have two different processes (or methods)
Draw one random sample of units and use those units to obtain the results of each process

23. Paired Difference Experiments Continued
For instance, use the same individuals for the results from one process vs. the results from the other process
E.g., use the same individuals to compare “before” and “after” treatments
Using the same individuals, eliminates any differences in the individuals themselves and just comparing the results from the two processes

24. Paired Difference Experiments #3
Let µd be the mean of population of paired differences
µd = µ1 – µ2, where µ1 is the mean of population 1 and µ2 is the mean of population 2
Let d̄ and sd be the mean and standard deviation of a sample of paired differences that has been randomly selected from the population
d̄ is the mean of the differences between pairs of values from both samples

25. t-Based Confidence Interval for Paired Differences in Means

26. Paired Differences Testing Rules

27. Example 10.6 and 10.7: The Repair Cost Comparison Case

28. 10.4 Comparing Two Population Proportions by Using Large, Independent Samples
Select a random sample of size n1 from a population, and let p̂1 denote the proportion of units in this sample that fall into the category of interest
Select a random sample of size n2 from another population, and let p̂2 denote the proportion of units in this sample that fall into the same category of interest
Suppose that n1 and n2 are large enough
n1·p1 ≥ 5, n1·(1 - p1) ≥ 5, n2·p2 ≥ 5, and n2·(1 – p2) ≥ 5

29. Comparing Two Population Proportions Continued
Then the population of all possible values of p̂1 - p̂2
Has approximately a normal distribution if each of the sample sizes n1 and n2 is large
Has mean µp̂1 - p̂2 = p1 – p2
Has standard deviation

30. Difference of Two Population Proportions

31. Example 10.9 and 10.10: The Advertising Media Case

32. 10.5 Comparing Two Population Variances Using Independent Samples
Population 1 has variance σ12 and population 2 has variance σ22
The null hypothesis H0 is that the variances are the same
H0: σ12 = σ22
The alternative is that one is smaller than the other
That population has less variable measurements
Suppose σ12 > σ22
More usual to normalize
Test H0: σ12/σ22 = 1 vs. σ12/σ22 > 1

33. Comparing Two Population Variances Using Independent Samples Continued
Reject H0 in favor of Ha if s12/s22 is significantly greater than 1
s12 is the variance of a random of size n1 from a population with variance σ12
s22 is the variance of a random of size n2 from a population with variance σ22
To decide how large s12/s22 must be to reject H0, describe the sampling distribution of s12/s22
The sampling distribution of s12/s22 is the F distribution

34. F Distribution
Figure 10.13

35. F Distribution
The F point F is the point on the horizontal axis under the curve of the F distribution that gives a right-hand tail area equal to 
The value of F depends on a (the size of the right-hand tail area) and df1 and df2
Different F tables for different values of 
Tables A.5 for  = 0.10
Tables A.6 for  = 0.05
Tables A.7 for  = 0.025
Tables A.8 for  = 0.01

36. Example 10.11: The Catalyst Comparison Case