1. Comparing Two Means
BPS 7e Chapter 21
© 2015 W. H. Freeman and Company

2. Two-Sample Problems
To study the change in attitude toward statistics over the course of the semester, a professor selected a simple random sample of students. She administered a questionnaire to the students at the beginning and again at the end of the semester.
What type of problem is this?
matched pairs
two independent samples

3. Two-Sample Problems (answer)
To study the change in attitude toward statistics over the course of the semester, a professor selected a simple random sample of students. She administered a questionnaire to the students at the beginning and again at the end of the semester.
What type of problem is this?
matched pairs
two independent samples

4. Two-Sample Problems
The National Park Service compared the average amount of money spent by visitors in two different national parks. They sampled visitors on the same day in each of the two parks.
What type of problem is this?
matched pairs
two independent samples

5. Two-Sample Problems (answer)
The National Park Service compared the average amount of money spent by visitors in two different national parks. They sampled visitors on the same day in each of the two parks.
What type of problem is this?
matched pairs
two independent samples

6. Two-Sample Problems
Researchers at a pharmaceutical company tested a new sunscreen formula. They applied the new formula to one arm and the old formula to the other arm of each of a set of individuals, randomly choosing the arms for each formula.
What type of problem is this?
matched pairs
two independent samples

7. Two-Sample Problems (answer)
Researchers at a pharmaceutical company tested a new sunscreen formula. They applied the new formula to one arm and the old formula to the other arm of each of a set of individuals, randomly choosing the arms for each formula.
What type of problem is this?
matched pairs
two independent samples

8. Two-Sample Problems
A dairy scientist compared milk production of cows fed two diets. He randomly divided a set of 10 cows into two groups. One group was fed diet 1 for a month, and the other was fed diet 2 for a month.
What type of problem is this?
matched pairs
two independent samples

9. Two-Sample Problems (answer)
A dairy scientist compared milk production of cows fed two diets. He randomly divided a set of 10 cows into two groups. One group was fed diet 1 for a month, and the other was fed diet 2 for a month.
What type of problem is this?
matched pairs
two independent samples

10. Two-Sample Problems
A dairy scientist compared milk production of cows fed two diets. He randomly assigned 10 cows to be fed one diet for a month and then the other diet for a month. The order of diets was determined randomly for each cow. Milk production during each of the two periods was recorded.
What type of problem is this?
matched pairs
two independent samples

11. Two-Sample Problems (answer)
A dairy scientist compared milk production of cows fed two diets. He randomly assigned 10 cows to be fed one diet for a month and then the other diet for a month. The order of diets was determined randomly for each cow. Milk production during each of the two periods was recorded.
What type of problem is this?
matched pairs
two independent samples

12. Comparing Two Means
Researchers compared the balance of a sample of elderly patients with that of a separate sample of younger patients. They exposed members of each age group to unpredictable noises and measured the amount of forward and backward sway.
What feature of this study suggests that two-sample inferences for the means of the groups might be appropriate?
Both groups were exposed to the same noise.
The noise was unpredictable.
Both were groups of patients.
The two groups were sampled independently.

13. Comparing Two Means (answer)
Researchers compared the balance of a sample of elderly patients with that of a separate sample of younger patients. They exposed members of each age group to unpredictable noises and measured the amount of forward and backward sway.
What feature of this study suggests that two-sample inferences for the means of the groups might be appropriate?
Both groups were exposed to the same noise.
The noise was unpredictable.
Both were groups of patients.
The two groups were sampled independently.

14. Comparing Two Means
Researchers compared the balance of a sample of elderly patients with that of a separate sample of younger patients. They exposed members of each age group to unpredictable noises and measured the amount of forward and backward sway.
In order for two-sample inferences for the means of the groups to be appropriate, what else must be true?
The data suggest that the populations are approximately Normal.
The data for the groups have skewed distributions.
The data have outliers.
The two samples are the same size.

15. Comparing Two Means (answer)
Researchers compared the balance of a sample of elderly patients with that of a separate sample of younger patients. They exposed members of each age group to unpredictable noises and measured the amount of forward and backward sway.
In order for two-sample inferences for the means of the groups to be appropriate, what else must be true?
The data suggest that the populations are approximately Normal.
The data for the groups have skewed distributions.
The data have outliers.
The two samples are the same size.

16. Conditions for Inference
For observations from a population with moderate skewness and no outliers, the sum of the two sample sizes should be at least:
10.
15.
20.
40.

17. Conditions for Inference (answer)
For observations from a population with moderate skewness and no outliers, the sum of the two sample sizes should be at least:
10.
15.
20.
40.

18. Conditions for Inference
For observations from a population with strong skewness and no outliers, the sum of the two sample sizes should be at least:
10.
15.
20.
40.

19. Conditions for Inference (answer)
For observations from a population with strong skewness and no outliers, the sum of the two sample sizes should be at least:
10.
15.
20.
40.

20. Conditions for Inference
True of False: Unlike the t procedures for means, the F test for standard deviations is extremely sensitive to non-Normal distributions.
True
False

21. Conditions for Inference (answer)
True of False: Unlike the t procedures for means, the F test for standard deviations is extremely sensitive to non-Normal distributions.
True
False

22. Comparing Two Means
Researchers compared the balance of a sample of elderly patients with that of a separate sample of younger patients. They exposed members of each age group to unpredictable noises and measured the amount of forward and backward sway.
To compare the means, they should give a confidence interval for:
melderly × myounger.
melderly ÷ myounger.
melderly ─ myounger.
melderly + myounger.

23. Comparing Two Means (answer)
Researchers compared the balance of a sample of elderly patients to that of a separate sample of younger patients. They exposed members of each age group to unpredictable noises and measured the amount of forward and backward sway.
To compare the means, they should give a confidence interval for:
melderly × myounger.
melderly ÷ myounger.
melderly ─ myounger.
melderly + myounger.

24. Comparing Two Means
Researchers compared the balance of a sample of elderly patients with that of a separate sample of younger patients. They exposed members of each age group to unpredictable noises and measured the amount of forward and backward sway.
To compare the means, they should test the hypotheses ___________ and ____________.
H0: melderly = myounger; Ha: melderly < myounger
H0: x̅elderly = x̅younger; Ha: x̅elderly < x̅younger
H0: melderly ≠ myounger; Ha: melderly = myounger
H0: x̅elderly ≠ x̅younger; Ha: x̅elderly = x̅younger

25. Comparing Two Means (answer)
Researchers compared the balance of a sample of elderly patients with that of a separate sample of younger patients. They exposed members of each age group to unpredictable noises and measured the amount of forward and backward sway.
To compare the means, they should test the hypotheses ___________ and ____________.
H0: melderly = myounger; Ha: melderly < myounger
H0: x̅elderly = x̅younger; Ha: x̅elderly < x̅younger
H0: melderly ≠ myounger; Ha: melderly = myounger
H0: x̅elderly ≠ x̅younger; Ha: x̅elderly = x̅younger

26. Two-Sample t Procedures
A dairy scientist compared milk production of cows fed two diets. He randomly divided a set of 10 cows into two groups. One group was fed diet 1 for a month, and the other was fed diet 2 for a month. Here are the milk-production results (lb/week):
What is the standard error of x̅1 ─ x̅2?
group n x̅ s
diet 1 10
385.7
25.7
diet 2
398.2
43.1

27. Two-Sample t Procedures (answer)
A dairy scientist compared milk production of cows fed two diets. He randomly divided a set of 10 cows into two groups. One group was fed diet 1 for a month, and the other was fed diet 2 for a month. Here are the milk-production results (lb/week):
What is the standard error of x̅1 ─ x̅2?
group n x̅ s
diet 1 10
385.7
25.7
diet 2
398.2
43.1

28. Two-Sample t Procedures
A dairy scientist compared milk production of cows fed two diets. He randomly divided a set of 10 cows into two groups. One group was fed diet 1 for a month, and the other was fed diet 2 for a month. Here are the milk-production results (lb/week):
Using Option 2, the two-sample t statistic for this situation has an approximate t distribution with how many degrees of freedom? 9 10 18 20
group n x̅ s
diet 1 10
385.7
25.7
diet 2
398.2
43.1
Option 1:
Option 2:

29. Two-Sample t Procedures (answer)
A dairy scientist compared milk production of cows fed two diets. He randomly divided a set of 10 cows into two groups. One group was fed diet 1 for a month, and the other was fed diet 2 for a month. Here are the milk-production results (lb/week):
Using Option 2, the two-sample t statistic for this situation has an approximate t distribution with how many degrees of freedom? 9 10 18 20
group n x̅ s
diet 1 10
385.7
25.7
diet 2
398.2
43.1
Option 1:
Option 2:

30. Two-Sample t Procedures
A dairy scientist compared milk production of cows fed two diets. He randomly divided a set of 10 cows into two groups. One group was fed diet 1 for a month, and the other was fed diet 2 for a month. Here are the milk-production results (lb/week):
What is the two-sample t statistic for testing H0: m1 = m2?
group n x̅ s
diet 1 10
385.7
25.7
diet 2
398.2
43.1

31. Two-Sample t Procedures (answer)
A dairy scientist compared milk production of cows fed two diets. He randomly divided a set of 10 cows into two groups. One group was fed diet 1 for a month, and the other was fed diet 2 for a month. Here are the milk-production results (lb/week):
What is the two-sample t statistic for testing H0: m1 = m2?
group n x̅ s
diet 1 10
385.7
25.7
diet 2
398.2
43.1

32. Two-Sample t Procedures
A dairy scientist compared milk production of cows fed two diets. He randomly divided a set of 10 cows into two groups. One group was fed diet 1 for a month, and the other was fed diet 2 for a month. Here are the milk-production results (lb/week):
The two-sample t statistic for testing H0: m1 = m2 is –0.79. If the test is one-sided, what is the P-value?
0.40 to 0.50
0.20 to 0.25
0.15 to 0.20
0.10 to 0.15
group n x̅ s
diet 1 10
385.7
25.7
diet 2
398.2
43.1
TABLE C t distribution critical values
d.f. 8
0.703
0.883
1.100
1.383 9
0.700
0.897
1.093
1.372 10
0.697
0.876
1.088
1.363
1-sided P
.25
.20
.15
.10

33. Two-Sample t Procedures (answer)
A dairy scientist compared milk production of cows fed two diets. He randomly divided a set of 10 cows into two groups. One group was fed diet 1 for a month, and the other was fed diet 2 for a month. Here are the milk-production results (lb/week):
The two-sample t statistic for testing H0: m1 = m2 is –0.79. If the test is one-sided, what is the P-value?
0.40 to 0.50
0.20 to 0.25
0.15 to 0.20
0.10 to 0.15
group n x̅ s
diet 1 10
385.7
25.7
diet 2
398.2
43.1
Option 1:
Option 2:
TABLE C t distribution critical values
d.f. 8
0.703
0.883
1.100
1.383 9
0.700
0.897
1.093
1.372 10
0.697
0.876
1.088
1.363
1-sided P
.25
.20
.15
.10

34. t Approximation
A dairy scientist compared milk production of cows fed two diets. He randomly divided a set of 10 cows into two groups. One group was fed diet 1 for a month, and the other was fed diet 2 for a month. Here are the milk-production results (lb/week):
Would the confidence interval for m1 – m2 be wider or narrower if the Option 1 degrees of freedom were used instead of the Option 2 degrees of freedom?
much wider
slightly wider
slightly narrower
narrower
group n x̅ s
diet 1 10
385.7
25.7
diet 2
398.2
43.1
Option 1:
Option 2:

35. t Approximation (answer)
A dairy scientist compared milk production of cows fed two diets. He randomly divided a set of 10 cows into two groups. One group was fed diet 1 for a month, and the other was fed diet 2 for a month. Here are the milk-production results (lb/week):
Would the confidence interval for m1 – m2 be wider or narrower if the Option 1 degrees of freedom were used instead of then Option 2 degrees of freedom?
much wider
slightly wider
slightly narrower
narrower
group n x̅ s
diet 1 10
385.7
25.7
diet 2
398.2
43.1

36. Confidence Interval The confidence interval for m1 – m2 is:
(x̅1 + x̅2) ± t *
(x̅1 ─ x̅2) ± t *
(x̅1 ─ x̅2) ± t *
(m1 – m2) ± t *

37. Confidence Interval (answer)
The confidence interval for m1 – m2 is:
(x̅1 + x̅2) ± t *
(x̅1 ─ x̅2) ± t *
(m1 – m2) ± t *