1. P-value method dependent samples

2. A group of friends wants to compare two energy drinks. They agree to meet on consecutive Saturdays to run a mile. One week, they drink “GoGetUm” Energy Drink prior to the run and record their times. The following week, the same group drinks “YippetyDoo” Energy Drink and again records their times. The results (in minutes) are recorded in the table below. RunnerABCDEF GoGetUm time 6.28.010.112.37.59.2 YippetyDoo time 6.18.810.212.17.58.0

3. Suppose that the time it takes to run a mile is approximately normally distributed and that both runs were on an indoor track to ensure comparable conditions. Is there enough evidence to support the claim that the two times were different? Use the P-value method with α=.05.

4. If you want to try this problem on your own and just check your answer, click on the math professor to the right. Otherwise, click away from the professor (or just hit the space bar) and we’ll work through this together.

5. Set-up With dependent samples, we subtract first, and then calculate the mean difference. This is very different from what we do with independent samples. Then, we calculate the means first and then subtract to calculate the difference between the two means.

6. RunnerABCDEF GoGetUm time 6.28.010.112.37.59.2 YippetyDoo time 6.18.810.212.17.58.0 Difference: GoGetUm - YippetyDoo Calculate the difference in the two times. We can subtract in either order, but we must be consistent once we’ve chosen the order of subtraction. As shown above, I’ve chosen to subtract the “YippetyDoo” time from the “GoGetUm” time..1.201.2

7. RunnerABCDEF GoGetUm time 6.28.010.112.37.59.2 YippetyDoo time 6.18.810.212.17.58.0 Difference: GoGetUm - YippetyDoo.1-.8-.1.201.2 Enter these values into your calculator.

8. Make sure that’s a sample standard deviation! And don’t forget to keep it stored in your calculator so we can call it up later and don’t have to use a rounded value!

9. Step 1: State the hypotheses and identify the claim That tells us what both hypotheses will be!

11. Step (*) Draw the picture and mark off the observed value. Do we know we have a bell-shaped distribution?

12. Yes! We were told to suppose the time it takes to run a mile is approximately normally distributed.

13. Step (*): First, draw the picture Top level: Area Middle Level: Standard Units (t) Since we don’t know σ, we will approximate it with s, but we have to compensate for this by using t-values.

14. Step (*): First, draw the picture Top level: Area Middle Level: Standard Units (t) 0 The center is always 0 in standard units. Label this whenever you draw the picture.

15. Step (*): First, draw the picture Top level: Area Middle Level: Standard Units (t) 0 Bottom level: Actual Units (min) In this case, the actual units are minutes.

16. Step (*): First, draw the picture Top level: Area Middle Level: Standard Units (t) 0 Bottom level: Actual Units (min) 0

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

18. Step (*): Standard Units (t) 0 Actual Units (min) 0 Bottom level.1.1 > 0 so it goes to the right if center. The observed value is always on the bottom level (actual units).

19. Standard Units (t) 0 Actual Units (min) 0 Bottom level.1.1 > 0 so it goes to the right if center. The observed value is always on the bottom level (actual units). The boundary of the right tail is directly above the observed value,.1. The left tail is drawn so that the picture is symmetric.

20. Step 2: Move up to the middle level and mark off the boundary of the right tail in standard units. To do this, convert the observed value to standard units. The result is called the test value. Standard Units (t) 0 Actual Units (min) 0.1 Middle level Put the test value here!

21. Remember to call up the value that is stored in your calculator so that we don’t round until after we’ve calculated t.

23. Let’s add it to the picture!! Standard Units (t) 0 Actual Units (min) 0.1 Middle level Put the test value here!.380

24. Step 3: Move up to the top level and calculate the area in the two tails. This will be the P-value. Standard Units (t) 0 Actual Units (min) 0.1.380 Top Level: Area

25. Since our standard units are t-values, we’ll need Table F. To know which row to look in, we need to calculate the degrees of freedom (d.f.): d.f. = n-1 = 6-1 = 5

26. Look in the row for d.f. = 5. Find where the t-value.380 would go.

27. .380 is smaller than all the t-values in this row, so it would be to the left of all of them. (Note that the t-values increase as we move left to right, so the smallest value will be furthest left.)

28. Follow this arrow up to the top of the table to see what this tells us about area. Since this is a two-tailed test, look in the row titled “Two tails, α.”

29. Remember, the α values in the table are the areas that correspond to the t-values below them. Since our t-value is to the left of all the t- values in the table, our P-value, which is area corresponding to that t- value, will also be to the left of all the areas in the table. Notice that area DECREASES as we move to the right, so that the area furthest left will be the BIGGEST area. Thus, P >.2

30. At this point, we don’t know exactly what P is, but we have enough information about it to proceed to the next step.

31. Compare P to α.

32. P >.2 Since α =.05, which is less than.2, we can add α to this inequality. So P > α

33. And P is “BIG.” (Well, at least it’s bigger than α.)

34. Having considered these arguments…

35. Step 5: Answer the question. There is not enough evidence to support the claim that the two energy drinks produced different results.

36. Could we see all that one more time?

37. Each click will give you one step. Step (*) is broken up into two clicks. Step 1. Step (*) standard units (t) actual units (min) 0000.1 Step 2.380 Step 3 P >.2 Step 5: There’s not enough evidence to support the claim.

38. And there was much rejoicing.