1. Traditional Method 2 means, σ’s known

2. The makers of a standardized exam have two versions of the exam: version A and version B. They believe the two versions are comparable in difficulty, so that the average score on each version should be the same. They are administering both versions to several groups to test this claim before they release the exams.

3. At one school, 40 students take version A and 36 students to take version B. The average score on version A is 52 and the average score on version B is 54. If the population standard deviation for version A is 10.3 and the population standard deviation for version B is 9.1, evaluate the test-makers’ claim. Use the traditional method with α=.05.

4. If you want to try this problem on your own and just check your answer, click on the wizard to the right. Otherwise, click away from the wizard and we’ll work through this together.

5. Set-up This test is about two means, the mean score on version A and the mean score on version B. We know: Version A Population Sample Version B Population Sample These are what the hypotheses will be about!

6. Set-up: This test is about two means, the mean score on version A and the mean score on version B. We know: Version A Population Sample Version B Population Sample

7. Step 1: State the hypotheses and identify the claim. The claim is that the average scores on both versions should be the same.

8. The equals sign means this is the Null Hypothesis. The fly means it’s dinner time!

9. When the Null Hypothesis is the claim, the Alternate Hypothesis is that the two quantities aren’t equal.

10. If we subtract, we’ll get a number in the hypotheses; we want that, since this number will be at the center of our distribution. Ok, but which way do we subtract?

11. If we subtract “B-A”, then when we subtract the sample means, we’ll get a positive number! I like positive numbers! So I’ll subtract that way. But we could also subtract “A-B” if we didn’t mind the negative numbers. Either way works, as long as we’re consistent.

12. We get 0 whichever way we subtract here; the positive number will show up when we’re working with the observed values. I like to plan ahead!

13. Step (*) Draw the picture and label the area in the critical region. Do we know we have a normal distribution?

14. We have a normal distribution since both sample sizes are big enough---they are both at least 30.

15. Step (*) Since we have a normal distribution, draw a normal curve. Top level: Area Middle level: standard units(z) We always use z-values when we know both σ’s, the population standard deviations.

16. Step (*): Since we have a normal distribution, draw a normal curve. Top level: Area Middle Level: Standard Units (z) 0 The center is always 0 in standard units. Label this whenever you draw the picture.

17. Step (*): Since we have a normal distribution, draw a normal curve. Top level: Area Middle Level: Standard Units (z) 0 Bottom level: Actual Units (points) In this case, the actual units are points, since our hypotheses are about the average difference in scores.

18. Step (*): Since we have a normal distribution, draw a normal curve. Top level: Area Middle Level: Standard Units (z) 0 Bottom level: Actual Units (points) 0 The number from the Null Hypothesis always goes in the center of the bottom level; that’s because we’re drawing the picture as if the Null is true.

19. Then remember: The raditional Method T is op-down T

20. Step (*): (continued) Once you’ve drawn the picture, start at the Top level and label the area in the critical region. Standard Units (z) 0 Actual Units (points) 0 Top level: Area.025

21. Step (*): (continued) Standard Units (z) 0 Actual Units (points) 0 Top level: Area.025

22. Step (*): (continued) Top level: Area.025 α =.05 = total area in both tails; the area in each tail is half of this, or.025

23. Step 2: Move down to the middle level. Label the critical values, which are the boundaries between the critical and non-critical regions. Standard Units (z) 0 Actual Units (points) 0.025 Middle Level Put critical values here!

24. We can find the critical value using either Table E or Table F. Click on the table you want to use. Table E gives us the z-values associated with certain areas under the standard normal curve The bottom row of table F gives us the z- values associated with the area in the tail/s.

25. Our picture looks like this: (we know the area in both tails and want to know the critical values.) 0. 025 ? ? To use Table E, we’ll take advantage of the fact that the left side of our picture matches the one that goes with the left side of Table E. I see the resemblance. Me too!

26. If we look up.025 in the area section we will be able to find the left critical value. Look’s like we need to zoom in!

28. Finishing up step 2: Standard Units (z) 0 Actual Units (points) 0.025 -1.96 The left critical value goes here. 1.96 The right critical value is +1.96. It goes here.

29. Step 3: Standard Units (z) 0 Actual Units (points) 0.025 1.96 Move down to the bottom level. Bottom level Remember: always subtract in the same order as that used in the hypotheses. We chose to subtract “B minus A” in order to get a positive result here. Hmmmm. 2 points is bigger than 0 points, but I don’t know how it compares to 1.96 standard units. Should it go here or there?

30. We want to compare 2 points to 1.96 standard units, but as long as they’re measured in different units, that’s difficult. So we convert 2 (our observed difference) to standard units. The result is called the test value, and it’ll show us where 2 belongs on the picture.

31. hypothesized difference

32. Standard units (z) 0 Actual units (points) 0.025 -1.96 1.96.90 < 1.96, so it goes somewhere to the left of 1.96..90 2 Line up the observed difference with the test value; note that it is not in the critical region.

33. Step 4: Decide whether or not to reject the Null.

34. Standard units (z) 0 Actual units (points) 0.025 -1.96 1.96.90 2 Since 2 is not in the critical region, don’t reject the Null.

35. Step 5: Answer the question. Talk about the claim. Since the claim is the Null Hypothesis, stick with the language of “rejection.” We did not reject the Null, so we don’t reject the claim. There is not enough evidence to reject the claim that the two tests will have the same mean score.

36. Let’s look at a quick re-cap.

37. Each click will give you one step. Step (*) is broken into two clicks. Step (*) Standard units (z) 0 Actual units (points) 0.025 Step 2 -1.961.96 Step 3.90 2 Step 4: Don’t reject the Null. Step 5: There’s not enough evidence to reject the claim.

38. And there was much rejoicing.

39. Press the Escape Key to exit the slide show. If you continue to click through the slideshow, you’ll see how to calculate the critical value using Table F.

40. Look at the top of Table F to determine which column will contain our critical value. Since this is a two-tailed test, look for α =.05 in this row.

41. Since our standard units are z-values, be sure to go all the way to the bottom row of Table F, the one labeled “z.”

42. z = 1.960. Note that Table F always gives us the critical value that is the boundary of the right tail; the boundary of the left tail is always its opposite.

43. Adding the critical values to our picture: Standard Units (z) 0 Actual Units (points) 0.025 -1.960 The left critical value is -1.960. It goes here. 1.960 The right critical value goes here.

44. Step 3: Standard Units (z) 0 Actual Units (points) 0.025 1.960 Move down to the bottom level. Bottom level Remember: always subtract in the same order as that used in the hypotheses. We chose to subtract “B minus A” in order to get a positive result here. Hmmmm. 2 points is bigger than 0 points, but I don’t know how it compares to 1.960 standard units. Should it go here or there?

45. We want to compare 2 points to 1.960 standard units, but as long as they’re measured in different units, that’s difficult. So we convert 2 (our observed difference) to standard units. The result is called the test value, and it’ll show us where 2 belongs on the picture.

46. hypothesized difference

47. Standard units (z) 0 Actual units (points) 0.025 -1.960 1.960.899 < 1.960, so it goes somewhere to the left of 1.960..899 2 Line up the observed difference with the test value; note that it is in the critical region.

48. Step 4: Decide whether or not to reject the Null.

49. Standard units (z) 0 Actual units (points) 0.025 -1.960 1.960.899 2 Since 2 is not in the critical region, don’t reject the Null.

50. Step 5: Answer the question. Talk about the claim. Since the claim is the Null Hypothesis, stick with the language of “rejection.” We did not reject the Null, so we don’t reject the claim. There is not enough evidence to reject the claim that the two tests will have the same mean score.

51. Let’s look at a quick re-cap.

52. Each click will give you one step. Step (*) is broken into two clicks. Step (*) Standard units (z) 0 Actual units (points) 0.025 Step 2 -1.960 1.960 Step 3.899 2 Step 4: Don’t reject the Null. Step 5: There’s not enough evidence to reject the claim.

53. And there was much rejoicing.