1. P-value Method One Mean, sigma known

2. The average length of a certain insect has been determined to be.52 cm with a standard deviation of.03 cm. A researcher studies a sample of this insect that live on a particular island. In a sample of 70 insects from the island, she finds that the average length is.51 cm. She claims that the average length of the island insects is the same as that of the overall population. Assuming that the standard deviation of the island insects is the same as that of the overall population, evaluate her claim using the P-value method with α=.01.

3. If you want to try this problem on your own and just check your answer, go ahead. Click on the picture of the biologist below when you’re ready to check your answer. Otherwise, click away from the biologist or just hit the space bar and we’ll work this problem together.

4. Set-up This is a test about 1 mean, the mean length of the island insects. This person is confused; he thought it was about 2 means, one for island insects and one for all insects. Click on him if you share his confusion. Otherwise, move the mouse away from him and click (or just hit the space bar) to keep going.

5. Population Note that “population” here refers to the population being studied---that is, the population of all island insects of this species. The researcher is comparing this sub-population to the larger population of all insects of this species. Set-up

6. Population μ= ? This is what the hypotheses will be about!

7. Set-up Population μ= ? σ=.03 We are told to assume that the standard deviation for the population of the island insects is the same as that of the overall population. In general, “assuming” things is something we try to avoid in statistical analysis, since false assumptions lead to unreliable conclusions. Here, we’ll accept that the researcher has a good evidence-based reason for her assumption. And we’ll breathe a sigh of relief knowing that we’ll soon learn how to deal with situations in which we know the sample standard deviation, s, but not the population standard deviation, σ.

8. S et-up Population μ= ? σ=.03

9. Step 1 State the hypotheses and identify the claim. The researcher’s claim is that: the length of the island insects is the same as the length of the insects generally μ =.52

12. Step (*) Draw the picture and mark off the observed value.

13. Do we know we have a normal Distribution?

14. Yes! Our sample size (70) is big enough---it is at least 30. Well done! We measured enough bugs!

15. Step (*): First, draw the picture Top level: Area Middle Level: Standard Units(z) We always use z-values when we know σ, the population standard deviation.

16. Step (*): First, draw the picture 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 (*): First, draw the picture Top level: Area Middle Level: Standard Units (z) 0 Bottom level: Actual Units (cm) In this case, the actual units are cm, since our hypothesis is about the length of the insects, which we measure in cm.

18. Step (*): First, draw the picture Top level: Area Middle Level: Standard Units (z) 0 Bottom level: Actual Units (cm).52 The number from the Null Hypothesis always goes in the center in standard units; that’s because we’re drawing the picture as if the Null is true.

19. Then remember: The -value Method P is ottom-up b

20. Step (*): (continued) Once you’ve drawn the picture, start at the Bottom level and mark off the observed value Standard Units (z) 0 Actual Units (cm).52 Bottom level.51.51 is less than.52, so it goes to the left of the center mark. Mark off the tail/s associated with the observed value; since this example involves a two-tailed test, be sure you remember to draw in the right tail too.

21. Step 2: Move up to the middle level. Convert the observed value to standard units and mark this off. Standard Units (z) 0 Actual Units (cm).52.51 Middle level: The observed value converted to standard units is called the test value. It goes here.

23. Let’s add it to the picture! Middle level: Standard Units (z) 0 Actual Units (cm).52.51 -2.79

24. Step 3: Move up to the top level and calculate the area in the tails. The total area is our P-value. Standard Units (z) 0 Actual Units (cm).52.51 -2.79 Top Level Area

25. We can use the left side of Table E to find the area in the left tail. Or we can use our calculator. Click on the picture of Table E if you want to use the table; click on the picture of the calculator if you prefer that option. (Note: the calculator used in this PowerPoint is the Casio fx-115 Msplus.)

26. Ok, we’ll use Table E. Let’s zoom in!

27. .0026 The area in the left tail is.0026.

28. Standard Units (z) 0 Actual Units (cm).52.51 -2.79.0026 By symmetry, the area in the right tail is also.0026. P = total area in both tails =.0026 +.0026 =.0052

31. Pα.0052.01 <

33. Step 5: Answer the question. There is enough evidence to reject the claim that the island insects have the same average length as the general population of insects.

34. Let’s review...

35. Each click will give you one step. Step (*) is broken into two clicks. Step 1: Step (*) Standard units (z) 0 Actual units (cm).52.51 Step 2 -2.97 Step 3.0026 Step 5: There is enough evidence to reject the claim..

36. And there was much rejoicing

37. Press the escape key to exit this slideshow. If you continue to click the mouse or hit the spacebar, you’ll go through the slides explaining why this hypothesis test was, in fact, about one mean, not two.

38. It’s true; we are comparing the mean length of the island insects to the mean length of the insects in general, but here’s the important difference between these means: We know the mean length of the insects in general. We don’t know the mean length of the island insects. Our hypotheses are about how this one unknown mean compares to the one we already know.

39. We say the hypothesis test is about one mean when there is just one mean that we don’t know, even if we are comparing it to a known mean. Click anywhere on this slide to return to the hypothesis test. Don’t just hit the space bar or you’ll go to the wrong place.

40. With the calculator, there’s no need to round the critical value, so be sure you’ve still got the calculated critical value displayed on your screen. Then hit the “shift” key followed by the “3” key.

41. You’ll see this menu. LEFT MIDDLE RIGHT

42. LEFT We want the area in the left tail, which is the area to the left of our critical value. So press the “1” key to select this option. You’ll see P( at the top of your screen.

43. If you didn’t clear your screen after calculating the critical value, you can just hit the “Ans” key to enter in this number. Otherwise, you’ll have to type ALL the digits of the critical value---remember, we don’t want to use the rounded value.

44. Our answer is so small, it is given in scientific notation! Move the decimal 3 places to the left to get an area of.00264.

45. Standard Units (z) 0 Actual Units (cm).52.51 -2.79.00264 By symmetry, the area in the right tail is also.00264. P = total area in both tails =.00264 +.00264 =.00528

48. Pα.00528.01 <

50. Step 5: Answer the question. There is enough evidence to reject the claim that the island insects have the same average length as the general population of insects.

51. Let’s review...

52. Each click will give you one step. Step (*) is broken into two clicks. Step 1: Step (*) Standard units (z) 0 Actual units (cm).52.51 Step 2 -2.97 Step 3.00264 Step 5: There is enough evidence to reject the claim..

53. And there was much rejoicing.