S TATISTICS

1. STATISTICS Part IIIB. Hypothesis Testing

2. The observed : the P-value • S’pose HO : μ≤140 HA: μ> 140 α= .05 And sample results yield a Z value of 3.1. What is the associated P-value and what does it mean in plain English? If the null hypothesis is true, the probability of getting the sample result that we got or something further to the right is one in a thousand. What, in your mind, is really the risk—after looking at this sample information—of a Type I error?

3. The observed : the P-value • Before we look at sample information, we set the tails of rejection and thereby define our risk of rejecting a true null (α). • After looking at sample information, the risk of rejecting a true null seems more like the p value than the α value. • Hence the p-value is often called the “observed α ,” which means the observed risk of rejecting a true null.

4. Relationship between C.I.s and Hypothesis Testing • FOR TWO-TAILED TESTS ONLY …. One can assess the results of a hypothesis test at α by examining a (1 - α) confidence interval on the same data. We simply observe whether or not the hypothesized value for μ is in the C.I. Example:

5. HO : μ = 63.25 HA: μ 63.25 We’ll evaluate at anαof .06. S’pose a 94% confidence interval estimate based upon sample information yielded A. P(62.1 ≤≤ 68.1) = .94 B. P(64.1 ≤≤ 70.1) = .94

6. Example with MINITAB • I want to test these hypotheses at a .05 level of significance to see whether or not the typical household’s annual contribution to charity in Utah is 5% of household income: • HO : μ = 5.0 HA: μ 5.0 I survey 24 people and get these results: -For a hypothesis test, I get p = .45 -For a 95% C.I., I get 2.85 to 5.99 If I only had the results of the C.I., what would I know about the results of a hypothesis test at .05?

7. Try this one • HO : μ= 68.5 HA: μ 68.5 at an α of .08 Use MINITAB data to obtain a 92% C.I. Estimate of μ The C.I. Is between 64.57 and 67.22. What does this tell us about the null hypothesis at the .08 level? Note the MINITAB result of a hypothesis test: The P-value is .003. We reject the null. What if we built a C.I. at the .997 level of confidence?

8. HYPOTHESES TESTS ON TWO POPULATION PARAMETERS • Comparing two means • Independent samples • Paired samples • Comparing two population proportions

9. Two-Mean Hypotheses • The opposing hypotheses about the difference between two population means appear in one of three forms Form I: Two tailed Ho: A  B HA: A  B Form II: Left tailed Ho: A  B HA: A  B Form III: Right tailed Ho: A  B HA: A  B

10. Independent Sample Example • The Utah State Legislature wants to see if there is statistical evidence to support the strong belief that presidents of large banks (population A) receive larger salaries than presidents of comparably sized credit unions (population B). • What is the alternative hypothesis? • HA: A> B • MINITAB example . . . • P-Value is .33. We cannot accept the alternative hypothesis even though the sample mean salary for Bank Presidents is greater than the sample mean salary for Credit Union presidents.

11. Paired Sample Example • I wonder if the price of running shoes reflects performance. I have eight track students at Bonneville High School run the mile in expensive running shoe A on Thursday. Then I have these same eight individuals run the mile on the same track under the same conditions on Friday using bargain shoe B. I record their running times in seconds. • What is the alternative hypothesis? • HA: A  B or A - B  0 • MINITAB Commands • P-Value is .054. These results tend not to be statistically significant, or we could say we would accept the alternative only at alpha levels above .054.

12. Paired Sample Example • I want to see if I can use statistics to validate the claim by a training firm that a productivity course will increase workers’ productivity in my plant by more than 10 production units an hour. I pick 27 workers for the training and I measure their productivity before the training (Population B) and after the training (Population A) • What is the alternative hypothesis? • HA: A - B > 10 • MINITAB Commands • P-Value is .969! Obviously we could not possibly accept the alternative because the sample data are entirely consistent with the null!

13. Comparing two population proportions • I believe that the proportion of full-time students who have no gainful employment while attending school is less at Weber State (Population A) than it is at the University of Utah (Population B). • What is the alternative hypothesis? • HA: A< B or A- B < 0 Survey Results: WSU: 16 out of 80 surveyed don’t work UofU: 24 out of 77 surveyed don’t work • MINITAB Commands • We would tend to not reject the null.

14. Summary of Part IIIB • After doing the test, the risk of a Type I error is the P-value (the “observed alpha”). • For two-tailed tests, we reviewed the relationship between Confidence Intervals at 1 -  and hypothesis tests at . • We reviewed how MINITAB is employed to compare two population means (independent samples and paired samples) and two population proportions.

15. Final Comment on Hypothesis Testing Where is the burden of proof? Ho: Defendant is innocent HA: Defendant is guilty

16. Final Comment on Hypothesis Testing Where is the burden of proof? Ho: Defendant is innocent HA: Defendant is guilty Ho: π = .5 HA: π ≠ .5

17. Final Comment on Hypothesis Testing Where is the burden of proof? Ho: Defendant is innocent HA: Defendant is guilty Ho: π = .5 HA: π ≠ .5 The burden of proof is on the ALTERNATIVE HYPOTHESIS!