1. P-value method 1 mean, σ unknown

2. A student claims that the average statistics textbook has fewer than 650 pages. In a sample of 52 statistics texts, she finds that the average number of pages is 603, with a standard deviation of 143.9. Evaluate the student’s claim using the P-value method with α=.05. Source: Data bank in Bluman, Elementary Statistics, eighth edition

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

4. Set-up This test is about the mean number of pages in a statistics textbook. Population (all statistics textbooks) μ= ? This is what the hypotheses will be about!

5. Set-up This test is about the mean number of pages in a statistics textbook. Population (all statistics textbooks) μ= ? How do we know this is the sample standard deviation?

6. Read the problem carefully. A student claims that the average statistics textbook has fewer than 650 pages. In a sample of 52 statistics texts, she finds that the average number of pages is 603, with a standard deviation of 143.9. The standard deviation is introduced in a sentence that was telling us about the sample.

7. Step 1: State the hypotheses and identify the claim. The student’s claim is that: The average number of pages in a statistics textbook is less than 650. µ < 650

9. Step 1 The Null Hypothesis has to have an equals sign, since the Null always claims there is no difference between things. The Null Hypothesis has to use the same number that shows up in the Alternate.

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

11. Do we know we have a bell- shaped distribution?

12. Yippee!! We do! That’s because our sample size (52) is big enough---it is at least 30.

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. Click this person if you share his confusion about t-values and want an explanation. Otherwise, click away from this person to keep going.

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 (pages) In this case, the actual units are pages, since our hypothesis is about the number of pages in a statistics textbook.

16. Step (*): First, draw the picture Top level: Area Middle Level: Standard Units (t) 0 Bottom level: Actual Units (pages) 650 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.

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

18. Step (*): (continued) Once you’ve drawn the picture, start at the Bottom level and mark off the observed value Standard Units (t) 0 Actual Units (pages) 650 603 603 is less than 650, so it goes to the left of the center mark.

19. Step 2: Move up to the middle level. Convert the observed value to standard units and mark this off. Standard Units (t) 0 Actual Units (pages) 650 603 Middle level: The observed value converted to standard units is called the test value. It goes here.

21. Let’s add it to the picture! Standard Units (t) 0 Actual Units (pages) 650 603 Middle level: -2.355

22. Step 3: Move up to the top level and calculate the area in the tail. This is our P-value. Standard Units (t) 0 Actual Units (pages) 650 603 -2.355 Top Level: Area

23. To find the area (P), we’ll need table F; use Table F whenever the standard units are t-values. Each row of table F corresponds to a different t- distribution. To know which row to use, we calculate the degrees of freedom (d.f.), which is 1 less than the sample size.

24. d.f. = n-1 = 52-1 = 51

25. d.f. 51 ought to be right here, but it’s not on the table!

26. That’s ok. The rule is: When the exact degrees of freedom don’t appear on the table, use the closest smaller value.

27. d.f. In this case, 50.

28. d.f. Note that we work with the absolute value of our (negative) t-value. Table F only has positive t-values, but that’s ok, because the normal curve is symmetric. In the row for d.f. = 50 we look for where a t-value of 2.355 would be.

29. d.f. a t-value of 2.355 would be somewhere in here, between 2.009 and 2.403

30. Following the arrow up, we see that it is between these two columns.

31. Since this is a one-tailed test, look at the row reading “One tail, α.” The numbers in this row are the areas corresponding to the t-values below. Our arrow puts us between an area of.025 and.01.

32. This means that the area in our tail---that is, the P-value, is between.025 and.01. Since.01 is the smaller of these two numbers, we can write this as an inequality:.01 < P <.025 Note that we don’t know exactly what P is, but we do know enough about it to make our decision.

35. .01 < P <.025 Figure out where α=.05 belongs in relation to these numbers.

36. P < α This means:

38. Step 5: Answer the question. There is enough evidence to support the claim that the average statistics textbook has fewer than 650 pages.

39. A quick summary…

40. Each click will give you a new step; step (*) is broken into two clicks. Step 1: Step (*) Standard units (t) 0 Actual units (pages)650 603 Step 2 -2.355 Step 3.01<P<.025 Step 5: There is enough evidence to support the claim.

41. And there was much rejoicing.

42. Press the “Escape” key to exit; if you just hit the space bar or click the mouse you’ll go through the slides explaining about t-values.

43. Ok, so what’s up with this t-distribution? When we don’t know σ, the population standard deviation, we approximate it with s, the sample standard deviation. We look more alike than they do!

44. Of course, any time we use approximations, we can expect our results to be less accurate. In fact, using s to approximate σ actually skews the shape the shape of the distribution of sample means, so that there are fewer values near the center and more values in the tails. This is demonstrated in the picture below. Acknowledgement: The picture was found on Google images and is attributed to the website http://www.myoops.org/twocw/tufts/courses/1/content/D193325/C207686.jpg

45. Like much of statistics, you should be able to understand the problem: using approximations messes up the accuracy of our results. But it’s beyond the scope of the course to understand the solution, which is to use the t-distribution rather than the standard normal distribution. We’ll accept that as a gift from The Mathematicians Who Have Gone Before. In this case, this mathematician (William Sealy Gosset) who developed his theory while working for this brewing company. Acknowledgements: Picture of William Sealy Gosset: http://www.swlearning.com/quant/kohler/stat/biographical_sketches/bio12.1.html Picture of a pint of guinness: http://reviews.productwiki.com/guinness/http://reviews.productwiki.com/guinness/

46. With less area in the middle and more in the tails, if we were to go the same number of standard units on the t- distribution as the standard normal curve (z-distribution), we would, in fact, get less area in the middle and more in the tail. We have to go more standard units on the t-distribution to get the same areas we would get on the standard normal curve. This is why the t-values associated with any given area are always bigger than the z-values associated with the same area.

47. Finally, we know that large samples give us much better approximations than small samples. This means that the “problem” we create by approximating σ is a much smaller problem when the sample size is big. This is why there isn’t just one t-distribution. The t-distributions are actually a family of distributions---one for each sample size. (The family is usually categorized not by sample size, but by degrees of freedom, which is just one less than the sample size. )

48. Follow me back to the main problem! (Just put the mouse anywhere on this slide and click. DON’T just hit the space bar or you’ll go to the end of the slide show!)