1. Chapter 24
Comparing Means

2. Comparing Two Means
An educator believes that new reading activities for elementary school children will improve reading comprehension scores. She randomly assigns her third-grade students to one of two groups. The first group will use a traditional reading program and the second group will use the new reading activities. At the end of the experiment, both groups take a reading comprehension exam. Are the scores for the new reading activities group higher than for the traditional group?

3. Comparing Two Means
Look at boxplot of each group’s scores.

4. Comparing Two Means What do you see?
____________________________________

5. Comparing Two Means
Does the new reading program produce better average scores?
For this particular class _________________________________
For population of all third-graders
_________________________________

6. Comparing Two Means μ1 = _____________________________
μ2 = _____________________________
Interested in quantity μ1 - μ2.

7. Comparing Two Means μ1 and μ2 are parameters (unknown).
________________________________
Estimate μ1 - μ2 with

8. Sampling Distribution
Assumptions:
Random Samples
Samples are Independent
Nearly Normal Population Distributions

9. Sampling Distribution
If Assumptions hold, sampling distribution is

10. Sampling Distribution
σ1 and σ2 are parameters (unknown).
____________________________

11. Sampling Distribution

12. Degrees of Freedom? t really doesn’t have a t distribution.
The true distribution of t is ________________________________
when you use this formula for the degrees of freedom.

13. Degrees of Freedom?

14. Degrees of Freedom? Problem:
____________________________________
Can use simpler, more conservative method.

15. Inference for μ1 - μ2 C% confidence interval for μ1 - μ2
t* is critical value from t distribution table.
d.f. = n1 – 1 or n2 – 1, whichever is smaller.

16. Example #1
A statistics student designed an experiment to test the battery life of two brands of batteries. For the sample of 6 generic batteries, the mean amount of time the batteries lasted was minutes with a standard deviation of 10.3 minutes. For the sample of 6 name brand batteries, the mean amount of time the batteries lasted was minutes with a standard deviation of 14.6 minutes. Calculate a 90% confidence interval for the difference in battery life between the generic and name brand batteries.

17. Example #1 (cont.) Assumptions: Random samples Independent samplesOK
Independent samples
different batteries for each sample.
Nearly Normal
data shows no real outliers.

18. Example #1 (cont.) d.f. = 5 μ1 = ______________________________
μ2 = ______________________________

19. Example #1 (cont.)

20. Example #1 (cont.)

21. Example #2
The Core Plus Mathematics Project was designed to help students improve their mathematical reasoning skills. At the end of 3 years, students in both the CPMP program and students in a traditional math program took an algebra test (without calculators). The 312 CPMP students had a mean score of 29.0 and a standard deviation of 18.8 while the 265 traditional students had a mean score of 38.4 with a standard deviation of Calculate a 95% confidence interval for the mean difference in scores between the two groups.

22. Example #2 (cont.) Assumptions: Random samples Independent samples
no reason to think non-random
Independent samples
different students in each group
Nearly Normal
n1 and n2 are large, so not important.

23. Example #2 (cont.) d.f. = smaller of 311 and 264 = 264
μ1 = _____________________________
μ2 = _____________________________

24. Example #2 (cont.)

25. Example #2 (cont.)

26. Hypothesis Test for μ1 - μ2
HO: __________________________
HA: Three possibilities
HA: ______________________________

27. Hypothesis Test for μ1 - μ2
Assumptions
Random samples.
Independent samples.
Nearly Normal Population Distributions.

28. Hypothesis Test for μ1 - μ2
Test statistic:
d.f. = smaller of n1 – 1 and n2 – 1

29. P-value for Ha:__________________
P-value = P(t d.f. > t)

30. P-value for Ha: _________________
P-value = P(t d.f. < t)

31. P-value for Ha:__________________
2*P(t d.f. > |t|)

32. Hypothesis Test for μ1 - μ2
Small p-value
_____________________________________
Large p-value

33. Decision If p-value < α __________________________________

34. Hypothesis Test for μ1 - μ2
Conclusion: Statement about value of μ1 - μ2. Always stated in terms of problem.

35. Example #1
Back to the reading example. The educator takes a random sample of all third graders in a large school district and divides them into the two groups. The mean score of the 18 students in the new activities group was with a standard deviation of The mean score of the 20 students in the traditional group was 41.8 with a standard deviation of Is this evidence that the students in the new activities group have a higher mean reading score? Use α = 0.1.

36. Example #1 (cont.) μ1 = ______________________________
μ2 = ______________________________

37. Example #1 (cont.) HO: ____________ HA: ____________ Assumptions:
Random Samples OK
Independent Samples
Different set of students in each group.
Nearly Normal
boxplots look symmetric

38. Example #1 (cont.)

39. Example #1 (cont.)
d.f. = smaller of 17 or 19 = 17
P-value

40. Example #1 (cont.)
Decision:

41. Example #1 (cont.)
Conclusion:

42. Example #2
In June 2002, the Journal of Applied Psychology reported on a study that examined whether the content of TV shows influenced the ability of viewers to recall brand names of items featured in commercials. Researchers randomly assigned volunteers to watch either a program with violent content or a program with neutral content. Both programs featured the same 9 commercials. After the shows ended, subjects were asked to recall the brands in the commercials. Is there evidence that viewer memory for ads differs between programs? Use α = 0.05

43. Example #2 (cont.) μ1 = _____________________________
μ2 = _____________________________

44. Example #2 (cont.) HO: ______________ HA: ______________ Assumptions:
Random Samples
no reason to think not random
Independent Samples
Different people in each group.
Nearly Normal
n1 and n2 are large so not important

45. Example #2 (cont.)

46. Example #2 (cont.)
d.f. = smaller of 107 or 107 = 107
P-value

47. Example #2 (cont.)
Decision:

48. Example #2 (cont.)
Conclusion: