1. Statistics
Chapter 9: Inferences Based on Two Samples: Confidence Intervals and Tests of Hypotheses

2. Where We’ve Been
Made inferences based on confidence intervals and tests of hypotheses
Studied confidence intervals and hypothesis tests for µ and p
Selected the necessary sample size for a given margin of error
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

3. Where We’re Going
Learn to use confidence intervals and hypothesis tests to compare two populations
Learn how to use these tools to compare two population means, proportions and variances
Select the necessary sample size for a given margin of error when comparing parameters from two populations
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

4. 9.1: Identifying the Target Parameter
µ1 - µ2
Mean difference; difference in averages
Quantitative Data
p1 - p2
Difference between proportions, percentages, fractions or rates; compare proportions
Qualitative Data
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

5. 9.2: Comparing Two Population Means: Independent Sampling
Point Estimators
 → µ
1 - 2 → µ1 - µ2
To construct a confidence interval or conduct a
hypothesis test, we need the standard deviation:
Singe sample
Two samples
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

6. 9.2: Comparing Two Population Means: Independent Sampling
The Sampling Distribution for (1 - 2)
The mean of the sampling distribution is (µ1-µ2).
If the two samples are independent, the standard deviation of the sampling distribution (the standard error) is
The sampling distribution for (1 - 2) is approximately normal for large samples.
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

7. 9.2: Comparing Two Population Means: Independent Sampling
The Sampling Distribution for (1 - 2)
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

8. 9.2: Comparing Two Population Means: Independent Sampling
Large Sample Confidence Interval for (µ1 - µ2 )
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

9. 9.2: Comparing Two Population Means: Independent Sampling
Two samples concerning retention scores for first-year students at private and
public institutions were obtained from the Department of Education’s data
base to see if there was a significant difference in the two types of colleges.
Private Colleges
Public Universities
n: 71
Mean: 78.17
Standard Deviation: 9.55
Variance: 91.17
n: 32
Mean: 84
Standard Deviation: 9.88
Variance: 97.64
What does a 95% confidence interval tell us about retention scores?
Source: National Center for Education Statistics
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

10. 9.2: Comparing Two Population Means: Independent Sampling
Private Colleges
Public Universities
n: 71
Mean: 78.17
Standard Deviation: 9.55
Variance: 91.17
n: 32
Mean: 84
Standard Deviation: 9.88
Variance: 97.64
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

11. 9.2: Comparing Two Population Means: Independent Sampling
Private Colleges
Public Universities
n: 71
Mean: 78.17
Standard Deviation: 9.55
Variance: 91.17
n: 32
Mean: 84
Standard Deviation: 9.88
Variance: 97.64
Since 0 is not in the confidence interval, the difference in the sample means appears to indicate a real difference in retention.
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

12. 9.2: Comparing Two Population Means: Independent Sampling
One-Tailed Test
Two-Tailed Test
H0: (µ1 - µ2) = D0 Ha: (µ1 - µ2) < D0 (> D0) Rejection region: z < -z (> z ) Test Statistic: where
H0: (µ1 - µ2) = D0 Ha: (µ1 - µ2) ≠ D0 Rejection region: |z| > z /2
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

13. 9.2: Comparing Two Population Means: Independent Sampling
Conditions Required for Valid Large-Sample Inferences about (µ1 - µ2)
1. The two samples are randomly and independently selected from the target populations.
2. The sample sizes are both ≥ 30.
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

14. 9.2: Comparing Two Population Means: Independent Sampling
Let’s go back to the retention data and test the hypothesis that there is no significant difference in retention at privates and publics.
Private Colleges
Public Universities
n: 71
Mean: 78.17
Standard Deviation: 9.55
Variance: 91.17
n: 32
Mean: 84
Standard Deviation: 9.88
Variance: 97.64
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

15. 9.2: Comparing Two Population Means: Independent Sampling
Test statistic:
Reject the null hypothesis:
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

16. 9.2: Comparing Two Population Means: Independent Sampling
For small samples, the t-distribution can be used with a pooled sample estimator of 2, sp2
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

17. 9.2: Comparing Two Population Means: Independent Sampling
Small Sample Confidence Interval for (µ1 - µ2 )
The value of t is based on (n1 + n2 -2) degrees of freedom.
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

18. 9.2: Comparing Two Population Means: Independent Sampling
One-Tailed Test
Two-Tailed Test
H0: (µ1 - µ2) = D0 Ha: (µ1 - µ2) < D0 (> D0) Rejection region: t < -t (> t ) Test Statistic:
H0: (µ1 - µ2) = D0 Ha: (µ1 - µ2) ≠ D0 Rejection region: |t| > t /2
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

19. 9.2: Comparing Two Population Means: Independent Sampling
Conditions Required for Valid Small-Sample Inferences about (µ1 - µ2)
1. The two samples are randomly and independently selected from the target populations.
2. Both sampled populations have distributions that are approximately normal.
3. The population variances are equal.
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

20. 9.2: Comparing Two Population Means: Independent Sampling
Does class time affect performance?
The test performance of students in two sections of international trade, meeting at different times, were compared.
8:00 a.m. Class
Mean: 78
Standard Deviation: 14
Variance: 196
n: 21
9:30 a.m. Class
Mean: 82
Standard Deviation: 17
Variance: 289
n: 21
With = .05, test H0 : µ1 = µ2
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

21. 9.2: Comparing Two Population Means: Independent Sampling
8:00 a.m. Class
Mean: 78
Variance: 196
n: 21
9:30 a.m. Class
Mean: 82
Variance: 289
n: 21
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

22. 9.2: Comparing Two Population Means: Independent Sampling
8:00 a.m. Class
Mean: 78
Variance: 196
n: 21
9:30 a.m. Class
Mean: 82
Variance: 289
n: 21
With df = – 2 = 40, t /2 = t.025 = Since out test statistic t = |t| < t.025.
Do not reject the null hypothesis
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

23. 9.2: Comparing Two Population Means: Independent Sampling
8:00 a.m. Class
Mean: 72
Variance: 154
n: 13
9:30 a.m. Class
Mean: 86
Variance: 163
n: 21
Your text book uses the following formula when the sample sizes are DIFFERENT, but you do NOT need to use this in Math 201. Just use the approach of the preceding slides even when
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

24. 9.3: Comparing Two Population Means: Paired Difference Experiments
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

25. 9.3: Comparing Two Population Means: Paired Difference Experiments
Suppose ten pairs of puppies were housetrained using two different methods: one puppy from each pair was paper-trained, with the paper gradually moved outside, and the other was taken out every three hours and twenty minutes after each meal. The number of days until the puppies were considered housetrained (three days straight without an accident) were compared. Nine of the ten paper-trained dogs took longer than the other paired dog to complete training, with the average difference equal to 4 days, with a standard deviation of 3 days.
What is a 90% confidence interval on the difference in successful training?
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

26. 9.3: Comparing Two Population Means: Paired Difference Experiments
Since 0 is not in the interval, one program does seem to work more effectively.
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

27. 9.3: Comparing Two Population Means: Paired Difference Experiments
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

28. 9.3: Comparing Two Population Means: Paired Difference Experiments
Suppose 150 items were priced at two online stores, “cport” and “warriorwoman.”
Mean difference: $1.75
Standard Deviation: $10.35
Test at the 5% level of significance that the difference in the two stores is zero.
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

29. 9.3: Comparing Two Population Means: Paired Difference Experiments
Suppose 150 items were priced at two online stores, “cport” and “warriorwoman.”
Mean difference: $1.75
Standard Deviation: $10.35
= .05
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

30. 9.3: Comparing Two Population Means: Paired Difference Experiments
Suppose 150 items were priced at two online stores, “cport” and “warriorwoman.”
Mean difference: $1.75
Standard Deviation: $10.35
= .05
The critical value z.025 is 1.96, so we would reject this null hypothesis.
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

31. 9.4: Comparing Two Population Proportions: Independent Sampling
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

32. 9.4: Comparing Two Population Proportions: Independent Sampling
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

33. 9.4: Comparing Two Population Proportions: Independent Sampling
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

34. 9.4: Comparing Two Population Proportions: Independent Sampling
A group of men and women were asked their opinions about an important issue (!):
Is the new comedy series (for TV) funny?
The results are as follow:
Men
Women
Yes
290
200 No 50 n
340
250
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

35. 9.4: Comparing Two Population Proportions: Independent Sampling
Calculate a 95% confidence interval on the difference in the opinions of men and women.
Men
Women p
.85
.80 q
.15
.20 n
340
250
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

36. 9.4: Comparing Two Population Proportions: Independent Sampling
Calculate a 95% confidence interval on the difference in the opinions of men and women.
Since 0 is in the confidence interval, we cannot rule out the possibility that both genders find the series equally funny.
Men
Women p
.85
.80 q
.15
.20 n
340
250
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

37. 9.4: Comparing Two Population Proportions: Independent Sampling
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

38. 9.4: Comparing Two Population Proportions: Independent Sampling
Randy Stinchfield of the University of Minnesota studied the gambling activities of public school students in 1992 and 1998 (Journal of Gambling Studies, Winter 2001). His results are reported below:
Do these results represent a statistically significant difference at the = .01 level?
1992
1998
Survey n
21,484
23,199
Number who gambled
4.684
5,313
Proportion who gambled
.218
.229
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

39. 9.4: Comparing Two Population Proportions: Independent Sampling
1992
1998
Survey n
21,484
23,199
Number who gambled
4.684
5,313
Proportion who gambled
.218
.229
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

40. 9.4: Comparing Two Population Proportions: Independent Sampling
1992
1998
Survey n
21,484
23,199
Number who gambled
4.684
5,313
Proportion who gambled
.218
.229
Since the computed value of z, , is of greater magnitude than the critical value, 2.576, we can reject the null hypothesis at the = .01 level.
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

41. 9.4: Comparing Two Population Proportions: Independent Sampling
For valid inferences
The two samples must be independent
The two sample sizes must be large:
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples