1. Slide Slide 1 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Overview

2. Slide Slide 2 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Definitions In statistics, a hypothesis is a claim or statement about a property of a population. A hypothesis test (or test of significance) is a standard procedure for testing a claim about a property of a population.

3. Slide Slide 3 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Rare Event Rule for Inferential Statistics If, under a given assumption, the probability of a particular observed event is exceptionally small, we conclude that the assumption is probably not correct.

4. Slide Slide 4 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Key Concept This section presents individual components of a hypothesis test, and the following sections use those components in comprehensive procedures. The role of the following should be understood:  null hypothesis  alternative hypothesis  test statistic  critical region  significance level  critical value  P-value  Type I and II error

5. Slide Slide 5 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Components of a Formal Hypothesis Test

6. Slide Slide 6 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Null Hypothesis: H 0  The null hypothesis (denoted by H 0 ) is a statement that the value of a population parameter (such as proportion, mean, or standard deviation) is equal to some claimed value.  We test the null hypothesis directly.  Either reject H 0 or fail to reject H 0.

7. Slide Slide 7 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Alternative Hypothesis: H 1  The alternative hypothesis (denoted by H 1 or H a or H A ) is the statement that the parameter has a value that somehow differs from the null hypothesis.  The symbolic form of the alternative hypothesis must use one of these symbols: ,.

8. Slide Slide 8 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Example: Use the given claims to express the corresponding null and alternative hypotheses in symbolic form. a)The proportion of drivers who admit to running red lights is greater than 0.5. b) The mean height of professional basketball players is at most 7 ft. c) The standard deviation of IQ scores of actors is equal to 15.

9. Slide Slide 9 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. a)The proportion of drivers who admit to running red lights is greater than 0.5. In Step 1, we express the given claim as p > 0.5. In Step 2, we see that if p > 0.5 is false, then p  0.5 must be true. In Step 3, we see that the expression p > 0.5 does not contain equality, so we let the alternative hypothesis H 1 be p > 0.5, and we let H 0 be p = 0.5.

10. Slide Slide 10 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. b) The mean height of professional basketball players is at most 7 ft. In Step 1, we express “a mean of at most 7 ft” in symbols as   7. In Step 2, we see that if   7 is false, then µ > 7 must be true. In Step 3, we see that the expression µ > 7 does not contain equality, so we let the alternative hypothesis H 1 be µ > 7, and we let H 0 be µ = 7.

11. Slide Slide 11 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. c) The standard deviation of IQ scores of actors is equal to 15. In Step 1, we express the given claim as  = 15. In Step 2, we see that if  = 15 is false, then   15 must be true. In Step 3, we let the alternative hypothesis H 1 be   15, and we let H 0 be  = 15.

12. Slide Slide 12 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

13. Slide Slide 13 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. The test statistic is a value used in making a decision about the null hypothesis, and is found by converting the sample statistic to a score with the assumption that the null hypothesis is true. Test Statistic

14. Slide Slide 14 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Test Statistic - Formulas z =z = x - µ x  n Test statistic for mean z = p - p  pq n  Test statistic for proportions 2 =2 = (n – 1)s 2    Test statistic for standard deviation

15. Slide Slide 15 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Example: A survey of n = 880 randomly selected adult drivers showed that 56% (or p = 0.56) of those respondents admitted to running red lights. Find the value of the test statistic for the claim that the majority of all adult drivers admit to running red lights.

16. Slide Slide 16 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Solution: The preceding example showed that the given claim results in the following null and alternative hypotheses: H 0 : p = 0.5 and H 1 : p > 0.5. we work under the assumption that the null hypothesis is true with p = 0.5, we get the following test statistic: n pq  z = p – p  = 0.56 - 0.5 ( 0.5)(0.5)  880 = 3.56

17. Slide Slide 17 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Interpretation: We know from previous chapters that a z score of 3.56 is exceptionally large. It appears that in addition to being “more than half,” the sample result of 56% is significantly more than 50%. See figure following.

18. Slide Slide 18 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Critical Region, Critical Value, Test Statistic

19. Slide Slide 19 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Critical Region The critical region (or rejection region) is the set of all values of the test statistic that cause us to reject the null hypothesis. For example, see the red-shaded region in the previous figure.

20. Slide Slide 20 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Significance Level The significance level (denoted by  ) is the probability that the test statistic will fall in the critical region when the null hypothesis is actually true. Common choices for  are 0.05, 0.01, and 0.10.

21. Slide Slide 21 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Two-tailed, Right-tailed, Left-tailed Tests The tails in a distribution are the extreme regions bounded by critical values.

22. Slide Slide 22 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Two-tailed Test H 0 : = H 1 :   is divided equally between the two tails of the critical region Means less than or greater than

23. Slide Slide 23 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Right-tailed Test H 0 : = H 1 : > Points Right

24. Slide Slide 24 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Left-tailed Test H 0 : = H 1 : < Points Left

25. Slide Slide 25 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. P-Value The P-value (or p-value or probability value) is the probability of getting a value of the test statistic that is at least as extreme as the one representing the sample data, assuming that the null hypothesis is true. The null hypothesis is rejected if the P-value is very small, such as 0.05 or less.

26. Slide Slide 26 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Traditional method: Reject H 0 if the test statistic falls within the critical region. Fail to reject H 0 if the test statistic does not fall within the critical region. Decision Criterion

27. Slide Slide 27 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. P-value method: Reject H 0 if the P-value   (where  is the significance level, such as 0.05). Fail to reject H 0 if the P-value > . Decision Criterion - cont

28. Slide Slide 28 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Decision Criterion - cont Confidence Intervals: Because a confidence interval estimate of a population parameter contains the likely values of that parameter, reject a claim that the population parameter has a value that is not included in the confidence interval.

29. Slide Slide 29 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Procedure for Finding P-Values Figure 8.6

30. Slide Slide 30 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Procedure for Finding P-Values

31. Slide Slide 31 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Example: Finding P-values. First determine whether the given conditions result in a right- tailed test, a left-tailed test, or a two-tailed test, then find the P-values and state a conclusion about the null hypothesis. a) A significance level of  = 0.05 is used in testing the claim that p > 0.25, and the sample data result in a test statistic of z = 1.18. b) A significance level of  = 0.05 is used in testing the claim that p  0.25, and the sample data result in a test statistic of z = 2.34.

32. Slide Slide 32 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Example: Finding P-values. First determine whether the given conditions result in a right- tailed test, a left-tailed test, or a two-tailed test, then find the P-values and state a conclusion about the null hypothesis. a) With a claim of p > 0.25, the test is right-tailed. Because the test is right-tailed, Figure 8-6 shows that the P-value is the area to the right of the test statistic z = 1.18. We refer to Table A-2 and find that the area to the right of z = 1.18 is 0.1190. The P-value of 0.1190 is greater than the significance level  = 0.05, so we fail to reject the null hypothesis. The P-value of 0.1190 is relatively large, indicating that the sample results could easily occur by chance.

33. Slide Slide 33 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Example: Finding P-values. First determine whether the given conditions result in a right- tailed test, a left-tailed test, or a two-tailed test, then find the P-values and state a conclusion about the null hypothesis. b) With a claim of p  0.25, the test is two-tailed. Because the test is two-tailed, and because the test statistic of z = 2.34 is to the right of the center, Figure 8-6 shows that the P-value is twice the area to the right of z = 2.34. We refer to Table A-2 and find that the area to the right of z = 2.34 is 0.0096, so P-value = 2 x 0.0096 = 0.0192. The P-value of 0.0192 is less than or equal to the significance level, so we reject the null hypothesis. The small P-value o 0.0192 shows that the sample results are not likely to occur by chance.

34. Slide Slide 34 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Wording of Final Conclusion Figure 8-7

35. Slide Slide 35 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Accept Versus Fail to Reject  Some texts use “accept the null hypothesis”  We are not proving the null hypothesis.  The sample evidence is not strong enough to warrant rejection (such as not enough evidence to convict a suspect).

36. Slide Slide 36 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Type I Error  A Type I error is the mistake of rejecting the null hypothesis when it is true.  The symbol   (alpha) is used to represent the probability of a type I error.

37. Slide Slide 37 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Type II Error  A Type II error is the mistake of failing to reject the null hypothesis when it is false.  The symbol  (beta) is used to represent the probability of a type II error.

38. Slide Slide 38 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Suppose that we are interested in the burning rate of a solid propellant. Now burning rate is a random variable that can be described by a probability distribution. we are interested in deciding whether or not the mean burning rate is 50 centimeters per second.

39. Slide Slide 39 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. How to Compute Type I and II Errors

40. Slide Slide 40 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

41. Slide Slide 41 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

42. Slide Slide 42 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

43. Slide Slide 43 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Increasing n to 16: When n = 10:

44. Slide Slide 44 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Considering a mean value closer to the hypothetical mean (50.5)

45. Slide Slide 45 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Summary

46. Slide Slide 46 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Definition The power of a hypothesis test is the probability (1 -  ) of rejecting a false null hypothesis, which is computed by using a particular significance level  and a particular value of the population parameter that is an alternative to the value assumed true in the null hypothesis. That is, the power of the hypothesis test is the probability of supporting an alternative hypothesis that is true.

47. Slide Slide 47 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

48. Slide Slide 48 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Comprehensive Hypothesis Test – P-Value Method

49. Slide Slide 49 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Comprehensive Hypothesis Test – Traditional Method

50. Slide Slide 50 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Comprehensive Hypothesis Test - cont A confidence interval estimate of a population parameter contains the likely values of that parameter. We should therefore reject a claim that the population parameter has a value that is not included in the confidence interval.

51. Slide Slide 51 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Testing a Claim About a Proportion

52. Slide Slide 52 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. p – pp – p pq n z =z =  Test Statistic for Testing a Claim About a Proportion

53. Slide Slide 53 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. P-Value Method Use the same method as described in Section 8-2 and in Figure 8-8. Use the standard normal distribution (Table A-2). Traditional Method Use the same method as described in Section 8-2 and in Figure 8-9. Confidence Interval Method Use the same method as described in Section 8-2 and in Table 8-2.

54. Slide Slide 54 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Example: An article distributed by the Associated Press included these results from a nationwide survey: Of 880 randomly selected drivers, 56% admitted that they run red lights. The claim is that the majority of all Americans run red lights. That is, p > 0.5. The sample data are n = 880, and p = 0.56.  np = (880)(0.5) = 440  5 nq = (880)(0.5) = 440  5

55. Slide Slide 55 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Example: An article distributed by the Associated Press included these results from a nationwide survey: Of 880 randomly selected drivers, 56% admitted that they run red lights. The claim is that the majority of all Americans run red lights. That is, p > 0.5. The sample data are n = 880, and p = 0.56. We will use the P-value Method.  Referring to Table A-2, we see that for values of z = 3.50 and higher, we use 0.9999 for the cumulative area to the left of the test statistic. The P-value is 1 – 0.9999 = 0.0001. H 0 : p = 0.5 H 1 : p > 0.5  = 0.05 pq n p – p z =z =  0.56 – 0.5 (0.5)(0.5) 880 = = 3.56

56. Slide Slide 56 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Example: An article distributed by the Associated Press included these results from a nationwide survey: Of 880 randomly selected drivers, 56% admitted that they run red lights. The claim is that the majority of all Americans run red lights. That is, p > 0.5. The sample data are n = 880, and p = 0.56. We will use the P-value Method.  Since the P-value of 0.0001 is less than the significance level of  = 0.05, we reject the null hypothesis. There is sufficient evidence to support the claim. H 0 : p = 0.5 H 1 : p > 0.5  = 0.05 pq n p – p z =z =  0.56 – 0.5 (0.5)(0.5) 880 = = 3.56

57. Slide Slide 57 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Example: An article distributed by the Associated Press included these results from a nationwide survey: Of 880 randomly selected drivers, 56% admitted that they run red lights. The claim is that the majority of all Americans run red lights. That is, p > 0.5. The sample data are n = 880, and p = 0.56. We will use the P-value Method.  H 0 : p = 0.5 H 1 : p > 0.5  = 0.05 z = 3.56

58. Slide Slide 58 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Example: An article distributed by the Associated Press included these results from a nationwide survey: Of 880 randomly selected drivers, 56% admitted that they run red lights. The claim is that the majority of all Americans run red lights. That is, p > 0.5. The sample data are n = 880, and p = 0.56. We will use the Traditional Method.  H 0 : p = 0.5 H 1 : p > 0.5  = 0.05 pq n p – p z =z =  0.56 – 0.5 (0.5)(0.5) 880 = = 3.56 This is a right-tailed test, so the critical region is an area of 0.05. We find that z = 1.645 is the critical value of the critical region. We reject the null hypothesis. There is sufficient evidence to support the claim.

59. Slide Slide 59 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Example: An article distributed by the Associated Press included these results from a nationwide survey: Of 880 randomly selected drivers, 56% admitted that they run red lights. The claim is that the majority of all Americans run red lights. That is, p > 0.5. The sample data are n = 880, and p = 0.56. We will use the confidence interval method.  For a one-tailed hypothesis test with significance level , we will construct a confidence interval with a confidence level of 1 – 2 . We construct a 90% confidence interval. We obtain 0.533 0.5.

60. Slide Slide 60 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. CAUTION When testing claims about a population proportion, the traditional method and the P-value method are equivalent and will yield the same result since they use the same standard deviation based on the claimed proportion p. However, the confidence interval uses an estimated standard deviation based upon the sample proportion p. Consequently, it is possible that the traditional and P-value methods may yield a different conclusion than the confidence interval method. A good strategy is to use a confidence interval to estimate a population proportion, but use the P-value or traditional method for testing a hypothesis. 