1. Traditional Method 1 mean, sigma unknown

2. In a national phone survey conducted in May 2012, adults were asked: Thinking about social issues, would you say your views on social issues are: 1.Very conservative 2.Conservative 3.Moderate 4.Liberal 5.Very liberal Source: The Gallup Poll, accessed online at http://brain.gallup.com/documents/questionnaire.aspx?STUDY=P1205006

3. very conservative very liberal The mean response was 3.10. I’m Average!

4. The mayor of a small town believes that the average town resident would also score him/herself at a 3.10 on this scale. She asks 45 randomly selected residents the same question and finds that their average response is 3.27 with a standard deviation of.52. Evaluate the mayors claim using the Traditional Method with α =.05. 3.10 Mayor’s claim 3.27 Survey results

5. If you want to try this problem on your own, click on the scale to the right when you’re ready to see the solution and check your answer. (If you want, you can click on the number that best describes you, but no one will be tabulating your response.) Otherwise, click away from the scale (or just hit the spacebar) and we’ll work through this together. very conservative very liberal

6. Set-up Let’s summarize what we know: Population The population in question is the residents of the town. We don’t know anything about this population, but the hypotheses will be about μ, the average response of all (adult) town residents.

7. Step 1: State the hypotheses and identify the claim. That’s the claim!

8. Translating this into symbols: μ = 3.10 (claim) Average resident score

9. μ = 3.10 (claim) The equals sign means this is the Null Hypothesis.

10. Step (*) Draw the picture and mark off the area in the critical region.

11. Uh-oh! The curve drawers have gone on strike! They refuse to draw any bell-shaped curves until they are assured that we do have an (approximately) bell-shaped distribution!

13. We start by drawing our picture… Top level: Area Middle level: standard units (t)0 Standard units are t-values because we don’t know σ and have to approximate it with s.

14. We start by drawing our picture… Top level: Area Middle level: standard units (t)0 Standard units are t-values because we don’t know σ and have to approximate it with s. If figuring out whether to use t-values or z-values makes your head spin, click on the person below for an informal explanation. Otherwise, click away from the person (or just hit the space bar) to keep going.

15. We start by drawing our picture… Top level: Area Middle level: standard units (t)0 Standard units are t-values because we don’t know σ and have to approximate it with s. 0 is always the center in standard units.

16. We start by drawing our picture… Top level: Area Middle level: standard units (t)0 Bottom level: actual units (points)3.10 These are points on the 1-5 scale of possible responses. The number from the Null always goes here.

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

18. Now start at the Top level and mark off the area in the critical region. standard units (t)0 actual units (points)3.10 Top level: Area

19. standard units (t)0 actual units (points)3.10 Top level: Area α =.05 = total area in both tails.025

20. Step 2 standard units (t)0 actual units (points)3.10.025 Move down to the middle level and mark off the critical values; these will be the boundaries of the tails in standard units. Middle level: Critical values go here

21. Since our standard units are t-values we’ll use Table F to get the critical values. We need to know which row to look in, so we need to calculate the degrees of freedom.

22. d.f. = n-1 = 45-1 =44 d.f. So we look for 44 in the column labeled “d.f.”. It ought to be here, but it isn’t!

23. d.f. That’s ok; the rule is when the value we want isn’t on the chart we choose the closest smaller value.

24. Missing Rows Note: the missing rows were deleted solely to make the part of the Table we need fit better on the slide.

25. Missing Rows Look in the row for d.f. = 40. At the top of the table, look for α =.05 in the row for two-tailed tests. Then look in the column below this. 2.021

26. Missing Rows 2.021 We get the absolute value from Table F. The plus/minus is because of the position of the critical values; in standard units anything to the left of center is negative and anything right of center is positive.

27. Adding the critical values to the picture: standard units (t)0 actual units (points)3.10.025 Middle level: Critical values go here -2.0212.021

28. Step 3 Move down to the bottom level and mark off the observed value standard units (t) 0 actual units (points) 3.10.025 2.021 -2.021 bottom level

29. standard units (t) 0 actual units (points) 3.10.025 2.021 -2.021 bottom level Argh!! I can see that 3.27 points is bigger than 3.10 points, but in order to see whether it falls in the critical region or not I need to know how it compares to 2.021 standard units! 3.27 Which one is right?

30. On order to see whether 3.27 is to the left or right of the critical value, we have to convert it to standard units. The result is called the test value. Observed value (in points) conversion formula Test value (in standard units)

31. Hypothesized value of μ 3.27 points conversion formula t = 2.193

32. Now we can add the test value and observed value to the picture! standard units (t) 0 actual units (points) 3.10.025 2.021 -2.021 2.193 2.193 > 2.021 so 2.193 goes to the right of 2.021 3.27 Line up 3.27 with 2.193

33. Step 4: Decide whether or not to reject the Null standard units (t) 0 actual units (points) 3.10.025 2.021 -2.021 2.193 3.27 The observed value is in the critical region; reject the Null.

34. Step 5: Answer the question in plain English I hate all this technical language!

35. Remember to talk about the claim. Since the claim is the Null, stick with the language of “rejection.” There is enough evidence to reject the claim that the town’s residents give themselves an average rating of 3.10.

36. Could we see a quick re-cap?

37. Each click will give you one step. Step (*) is broken into two clicks. Step 1. Step (*) standard units (t)0 actual units (points) 3.10.025 Step 2 2.021 -2.021 Step 3 2.193 3.27 Step 4: Reject the Null. Step 5: There’s enough evidence to reject the claim.

38. And there was much rejoicing.

39. Press the escape key (“esc”) to exit the slide show. If you keep clicking through, you’ll go to the informal explanation of t-distributions.

40. Ok, here’s a very informal explanation of t-distributions We use a t-distribution when we don’t know σ, the population standard deviation. If we did know σ, we’d go ahead and use a normal curve with the usual z-values as standard units. standard units (z) 0

41. 0 When we don’t know σ, we have to approximate it with s, the sample standard deviation. And while approximating σ with s is the best we can do, that doesn’t make it good. In fact, it’s kind of like letting a monster with very big feet stomp all over our lovely normal distribution!

42. The result is a smushed bell- shaped curve. It turns out, this “smushed normal curve” is our t- distribution.

43. The center is lower, so there’s less area in the middle! The tails are higher, so there’s more area in the tails!

44. This means we have to go farther from center (more standard units) to get a big area in the middle (for confidence intervals) or a small area in the tails (for hypothesis tests.) That’s why t-values are always bigger than z-values would be for the same area.

45. And remember, while approximating always has consequences, big samples lead to better approximations, and thus smaller consequences. Using a big sample is like letting a small monster smush the curve--- the curve still changes, but only a little, so it’s much closer to the standard normal curve. Using a small sample is like letting a really big monster smush the curve---it gets really smushed and is very different from the standard normal curve.

46. Of course, there’s a rigorous mathematical explanation for t-distributions. (Sadly, it doesn’t involve any monsters!) But the gist of it is this: Approximating things always has consequences. The consequence of approximating σ with s is that we use the t-distribution instead of the standard normal curve.

47. Follow me! Click anywhere on this slide to return to the main problem. Don’t just hit the space bar or you’ll exit the slide show!