1. P-value Method 2 means, sigmas unknown

2. Sodium levels are measured in millimoles per liter (mmol/L) and a score between 136 and 145 is considered normal. A student wants to test the claim that the average sodium level is different for men and women. A sample of 9 men had an average sodium level of 143.3, with a standard deviation of 5.3. A sample of 10 women had an average sodium level of 139.3 with a standard deviation of 6.1 Source: selected entries from the Data Bank in Bluman, Elementary Statistics, eighth edition

3. Suppose that sodium levels are normally distributed in both men and women and that the standard deviations for the two populations are different. Evaluate the claim using the P-value method with α =.05.

4. If you want to work through this on your own, and just check your answer, click on the student to the left. Otherwise, click away from him and we’ll work through this together.

5. Set-up This is a test about 2 means, the mean sodium level for men and the mean sodium level for women. The hypotheses will be about these population means.

6. Step 1: State the hypotheses and identify the claim. The claim is that: The average sodium level for men is different from the average sodium level for women.

7. I don’t see an equals sign. This must be the Alternate Hypothesis.

8. 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 will compare the same quantities that show up in the Alternate. Equality Now!

9. If we subtract, we’ll be able to see what number will be at the center of our distribution!

10. It says here if we subtract “bigger minus smaller” we can ensure we get to work with positive numbers later on. Since the sample mean for men is bigger, let’s subtract “men minus women.”

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

12. Step (*): Draw the picture and mark off the observed value. Remember: Never draw the picture without first verifying that you have a normal distribution.

13. In this case, we can keep going because we were told to suppose that sodium levels are normally distributed in both men and women.

14. Step (*) First, draw the picture. Top level: Area Middle level: Standard Units (t) 0 Since we don’t know the population standard deviations, we will approximate them with the sample standard deviations. Whenever we do this, we compensate by using t-values instead of z-values. Click this person if you want further explanation about t-values.

15. Step (*) First, draw the picture. Top level: Area Middle level: Standard Units (t) 0 0 is always at the center in standard units.

16. Step (*) First, draw the picture. Top level: Area Middle level: Standard Units (t) 0 Bottom level: Actual units (mmol/L) We were told at the beginning that millimoles per liter are the units for sodium levels.

17. Step (*) First, draw the picture. Top level: Area Middle level: Standard Units (t) 0 Bottom level: Actual units (mmol/L) 0 The number from the Null always goes in the center here.

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

19. Step (*) continued Once you’ve drawn the picture, start at the bottom level and mark off the observed difference. Standard Units (t) 0 Actual units (mmol/L)0 Bottom level Always subtract sample values in the same order as you subtracted population values in the hypotheses---in this case, “men minus women.” = 143.3 – 139.3 = 4

20. Step (*) continued Once you’ve drawn the picture, start at the bottom level and mark off the observed difference. Standard Units (t) 0 Actual units (mmol/L)0 Bottom level 4 4 > 0 so it goes on the right side

21. Step (*) continued Standard Units (t) 0 Actual units (mmol/L)0 Bottom level 4 4 > 0 so it goes on the right side Mark off the tail that has 4 as its boundary---this is the right tail. Since this is a two-tailed test, remember to mark off the left tail as well.

22. 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 (mmol/L)0 4 Middle Level The observed value converted to standard units is called the test value. It goes here.

23. hypothesized difference in means Test value

24. Finishing up Step 2…. Standard Units (t) 0 Actual units (mmol/L)0 4 Middle Level The test value goes here!

25. Finishing up Step 2…. Standard Units (t) 0 Actual units (mmol/L)0 4 Middle Level The test value goes here! 1.529

26. Step 3: Move up to the top level and find the area in the two tails; this is the P-value. Standard Units (t) 0 Actual units (mmol/L)0 4 1.529 Top Level (area) P = total area in both tails.

27. Since our standard units are t-values, we will locate the area on Table F. We need to look for our test value, 1.529 in the row that corresponds to the correct degrees of freedom.

30. The degrees of freedom is the smaller of 8 and 9; the monkey is excited to tell us that that’s 8.

31. Look for 1.529 in the row for d.f. = 8. Since 1.529 is between 1.397 and 1.860, it will fall between these two columns.

32. Moving up to the top of the chart, we look in the row for a two-tailed test. The α-values here are the areas of the tails with the corresponding t-values as their cut-offs. We can see that our t-value puts us in between an area of.2 and.1.

33. .1 < P <.2 I don’t have to know exactly what P is; this should be enough information for me to make my decision.

34. Step 4: Decide whether or not to reject the Null. In the picture to the left, an unknown artist captures the Null Hypothesis begging for mercy.

36. .1 < P <.2 α =.05 Where does this fit in?.05 < Since α =.05 is less than.1, it is less than P.

37. P > α This means the probability we would get the result we did (or one further from 0) is greater than α. So it’s not so unlikely that we should reject it. Do not reject the Null.

38. Step 5: Answer the question. Talk about the claim. Since the claim is the Alternate Hypothesis, switch to the language of “support”. We did not reject the Null, so we don’t support the claim. There is not enough evidence to support the claim that men and women have different sodium levels.

39. I wouldn’t mind seeing all that summarized.

40. Each click will give you one step. Step (*) is broken into two clicks. Step (*) 0 Standard Units (t) Actual units (mmol/L)0 4 Step 2 1.529 Step 3 P = area is between.1 and.2 Step 4: Don’t reject the Null. Step 5: There’s not enough evidence to support the claim.

41. And there was much rejoicing.

42. Press the escape key to exit the slide show. If you continue to click through the show, you’ll see an informal explanation of why we use t-values when we don’t know both σ’s.

43. 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

44. 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 someone stomp all over our lovely normal distribution!

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

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

47. 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.

48. 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 person 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 person smush the curve---it gets really smushed and is very different from the standard normal curve.

49. Of course, there’s a rigorous explanation for why the t- distribution works. (Sadly, it doesn’t involve any people jumping on curves and smushing them.) 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.

50. Follow us! 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!