1. Chapter 9 Hypothesis Testing Understanding Basic Statistics Fifth Edition By Brase and Brase Prepared by Jon Booze

2. 9 | 2 Copyright © Cengage Learning. All rights reserved. Methods for Drawing Inferences We can draw inferences on a population parameter in two ways: 1)Estimation (Chapter 8) 2)Hypothesis Testing (Chapter 9)

3. 9 | 3 Copyright © Cengage Learning. All rights reserved. Hypothesis Testing Hypothesis testing is the process of making decisions about the value of a population parameter.

4. 9 | 4 Copyright © Cengage Learning. All rights reserved. Establishing the Hypotheses Null Hypothesis: A hypothesis about a parameter that often denotes a theoretical value, a historical value, or a production specification. –Denoted as H 0 Alternate Hypothesis: A hypothesis that differs from the null hypothesis, such that if we reject the null hypothesis, we will accept the alternate hypothesis. –Denoted as H 1 (in other sources H A ).

5. 9 | 5 Copyright © Cengage Learning. All rights reserved. Hypotheses Restated

6. 9 | 6 Copyright © Cengage Learning. All rights reserved. Statistical Hypotheses The null hypothesis is always a statement of equality. –H 0 : μ = k, where k is a specified value The alternate hypothesis states that the parameter (μ or p) is less than, greater than, or not equal to a specified value.

7. 9 | 7 Copyright © Cengage Learning. All rights reserved. Statistical Hypotheses Which of the following is an acceptable null hypothesis? a). H 0 :   1.2b). H 0 :  > 1.2 c). H 0 :  = 1.2 d). H 0 :   1.2

8. 9 | 8 Copyright © Cengage Learning. All rights reserved. Statistical Hypotheses Which of the following is an acceptable null hypothesis? a). H 0 :   1.2b). H 0 :  > 1.2 c). H 0 :  = 1.2 d). H 0 :   1.2

9. 9 | 9 Copyright © Cengage Learning. All rights reserved. Types of Tests Left-Tailed Tests: H 1 : μ < k H 1 : p < k Right-Tailed Tests:H 1 : μ > k H 1 : p > k Two-Tailed Tests:H 1 : μ ≠ k H 1 : p ≠ k

10. 9 | 10 Copyright © Cengage Learning. All rights reserved. Types of Tests A production manager believes that a particular machine averages 150 or more parts produced per day. What would be the appropriate hypotheses for testing this claim? a). H 0 :   150; H 1 :  > 150 b). H 0 :  > 150; H 1 :  = 150 c). H 0 :  = 150; H 1 :   150 d). H 0 :  = 150; H 1 :  > 150

11. 9 | 11 Copyright © Cengage Learning. All rights reserved. Types of Tests A production manager believes that a particular machine averages 150 or more parts produced per day. What would be the appropriate hypotheses for testing this claim? a). H 0 :   150; H 1 :  > 150 b). H 0 :  > 150; H 1 :  = 150 c). H 0 :  = 150; H 1 :   150 d). H 0 :  = 150; H 1 :  > 150

12. 9 | 12 Copyright © Cengage Learning. All rights reserved. Hypothesis Testing Procedure 1)Select appropriate hypotheses. 2)Draw a random sample. 3)Calculate the test statistic. 4)Assess the compatibility of the test statistic with H 0. 5)Make a conclusion in the context of the problem.

13. 9 | 13 Copyright © Cengage Learning. All rights reserved. Hypothesis Test of μ x is Normal, σ is Known

14. 9 | 14 Copyright © Cengage Learning. All rights reserved. P-Value P-values are sometimes called the probability of chance. Low P-values are a good indication that your test results are not due to chance.

15. 9 | 15 Copyright © Cengage Learning. All rights reserved. P-Value for Left-Tailed Test

16. 9 | 16 Copyright © Cengage Learning. All rights reserved. P-Value for Right-Tailed Test

17. 9 | 17 Copyright © Cengage Learning. All rights reserved. P-Value for Two-Tailed Test

18. 9 | 18 Copyright © Cengage Learning. All rights reserved. Types of Errors in Statistical Testing Since we are making decisions with incomplete information (sample data), we can make the wrong conclusion. –Type I Error: Rejecting the null hypothesis when the null hypothesis is true. –Type II Error: Accepting the null hypothesis when the null hypothesis is false.

19. 9 | 19 Copyright © Cengage Learning. All rights reserved. Type I and Type II Errors

20. 9 | 20 Copyright © Cengage Learning. All rights reserved. Errors in Statistical Testing Unfortunately, we usually will not know when we have made an error. We can only talk about the probability of making an error. Decreasing the probability of making a type I error will increase the probability of making a type II error (and vice versa). We can only decrease the probability of both types of errors by increasing the sample size (obtaining more information), but this may not be feasible in practice.

21. 9 | 21 Copyright © Cengage Learning. All rights reserved. Level of Significance Good practice requires us to specify in advance the risk level of type I error we are willing to accept. The probability of type I error is the level of significance for the test, denoted by α (alpha).

22. 9 | 22 Copyright © Cengage Learning. All rights reserved. Type II Error The probability of making a type II error is denoted by β (Beta). 1 – β is called the power of the test. –1 – β is the probability of rejecting H 0 when H 0 is false (a correct decision).

23. 9 | 23 Copyright © Cengage Learning. All rights reserved. Type II Error The probability of making a type II error is denoted by β (Beta). 1 – β is called the power of the test. –1 – β is the probability of rejecting H 0 when H 0 is false (a correct decision).

24. 9 | 24 Copyright © Cengage Learning. All rights reserved. The Probabilities Associated with Testing

25. 9 | 25 Copyright © Cengage Learning. All rights reserved. Concluding a Statistical Test For our purposes, significant is defined as follows: At our predetermined level of risk α, the evidence against H 0 is sufficient to reject H 0. Thus we adopt H 1.

26. 9 | 26 Copyright © Cengage Learning. All rights reserved. Concluding a Statistical Test For a particular experiment, P = 0.17 and  = 0.05. What is the appropriate conclusion? a). Reject the null hypothesis. b). Do not reject the null hypothesis. c). Reject both the null hypothesis and the alternative hypothesis. d). Accept both the null hypothesis and the alternative hypothesis.

27. 9 | 27 Copyright © Cengage Learning. All rights reserved. Concluding a Statistical Test For a particular experiment, P = 0.17 and  = 0.05. What is the appropriate conclusion? a). Reject the null hypothesis. b). Do not reject the null hypothesis. c). Reject both the null hypothesis and the alternative hypothesis. d). Accept both the null hypothesis and the alternative hypothesis.

28. 9 | 28 Copyright © Cengage Learning. All rights reserved. Statistical Testing Comments Frequently, the significance is set at α = 0.05 or α = 0.01. When we “accept” the null hypothesis, we are not proving the null hypothesis to be true. We are only saying that the sample evidence is not strong enough to justify the rejection of H 0. –Some statisticians prefer to say “fail to reject H 0 ” rather than “accept H 0.”

29. 9 | 29 Copyright © Cengage Learning. All rights reserved. Interpretation of Testing Terms

30. 9 | 30 Copyright © Cengage Learning. All rights reserved. Testing µ When σ is Known 1)State the null hypothesis, alternate hypothesis, and level of significance. 2)If x is normally distributed, any sample size will suffice. If not, n ≥ 30 is required. Calculate:

31. 9 | 31 Copyright © Cengage Learning. All rights reserved. Testing µ When σ is Known 3)Use the standard normal table and the type of test (one or two-tailed) to determine the P- value. 4)Make a statistical conclusion: If P-value ≤ α, reject H 0. If P-value > α, do not reject H 0. 5)Make a context-specific conclusion.

32. 9 | 32 Copyright © Cengage Learning. All rights reserved. Testing µ When σ is Known Suppose that the test statistic z = 1.85 for a right- tailed test. Use Table 3 in the Appendix to find the corresponding P-value. a). 0.2514 b). 0.0322 c). 0.9678 d). 0.0161

33. 9 | 33 Copyright © Cengage Learning. All rights reserved. Testing µ When σ is Known Suppose that the test statistic z = 1.85 for a right- tailed test. Use Table 3 in the Appendix to find the corresponding P-value. a). 0.2514 b). 0.0322 c). 0.9678 d). 0.0161

34. 9 | 34 Copyright © Cengage Learning. All rights reserved. Testing µ When σ is Unknown 1)State the null hypothesis, alternate hypothesis, and level of significance. 2)If x is normally distributed (or mound-shaped), any sample size will suffice. If not, n ≥ 30 is required. Calculate:

35. 9 | 35 Copyright © Cengage Learning. All rights reserved. Testing µ When σ is Unknown 3)Use the Student’s t table and the type of test (one or two-tailed) to determine (or estimate) the P-value. 4)Make a statistical conclusion: If P-value ≤ α, reject H 0. If P-value > α, do not reject H 0. 5)Make a context-specific conclusion.

36. 9 | 36 Copyright © Cengage Learning. All rights reserved. Using Table 4 to Estimate P-values 0.025 < P-value < 0.050 Suppose we calculate t = 2.22 for a one-tailed test from a sample size of 6. df = n – 1 = 5.

37. 9 | 37 Copyright © Cengage Learning. All rights reserved. Testing µ Using the Critical Value Method The values of that will result in the rejection of the null hypothesis are called the critical region of the distribution. When we use a predetermined significance level α, the Critical Value Method and the P- Value Method are logically equivalent.

38. 9 | 38 Copyright © Cengage Learning. All rights reserved. Critical Regions for H 0 : µ = k

39. 9 | 39 Copyright © Cengage Learning. All rights reserved. Critical Regions for H 0 : µ = k

40. 9 | 40 Copyright © Cengage Learning. All rights reserved. Critical Regions for H 0 : µ = k

41. 9 | 41 Copyright © Cengage Learning. All rights reserved. Testing µ When σ is Known (Critical Region Method) 1)State the null hypothesis, alternate hypothesis, and level of significance. 2)If x is normally distributed, any sample size will suffice. If not, n ≥ 30 is required. Calculate:

42. 9 | 42 Copyright © Cengage Learning. All rights reserved. Testing µ When σ is Known (Critical Region Method) 3)Show the critical region and critical value(s) on a graph (determined by the alternate hypothesis and α). 4)Conclude in favor of the alternate hypothesis if z is in the critical region. 5)State a conclusion within the context of the problem.

43. 9 | 43 Copyright © Cengage Learning. All rights reserved. Left-Tailed Tests

44. 9 | 44 Copyright © Cengage Learning. All rights reserved. Right-Tailed Tests

45. 9 | 45 Copyright © Cengage Learning. All rights reserved. Two-Tailed Tests

46. 9 | 46 Copyright © Cengage Learning. All rights reserved. Testing a Proportion p Binomial Experiments: r (# of successes) is a binomial variable n is the number of independent trials p is the probability of success on each trial Test Assumption: np > 5 and n(1 – p) > 5

47. 9 | 47 Copyright © Cengage Learning. All rights reserved. Testing a Proportion p Test Assumption: np > 5 and n(1 – p) > 5 The values of n and p for several experiments are shown below. Which experiment should not be tested using the normal distribution? a). n = 48, p = 0.39 b). n = 843, p = 0.09 c). n = 52, p = 0.93 d). n = 12, p = 0.51

48. 9 | 48 Copyright © Cengage Learning. All rights reserved. Testing a Proportion p Test Assumption: np > 5 and n(1 – p) > 5 The values of n and p for several experiments are shown below. Which experiment should not be tested using the normal distribution? a). n = 48, p = 0.39 b). n = 843, p = 0.09 c). n = 52, p = 0.93 d). n = 12, p = 0.51

49. 9 | 49 Copyright © Cengage Learning. All rights reserved. Types of Proportion Tests

50. 9 | 50 Copyright © Cengage Learning. All rights reserved. The Distribution of the Sample Proportion

51. 9 | 51 Copyright © Cengage Learning. All rights reserved. Converting the Sample Proportion to z

52. 9 | 52 Copyright © Cengage Learning. All rights reserved. Testing p 1)State the null hypothesis, alternate hypothesis, and level of significance. 2)Check np > 5 and nq > 5 (recall q = 1 – p). Compute: p = the specified value in H 0

53. 9 | 53 Copyright © Cengage Learning. All rights reserved. Testing p 3)Use the standard normal table and the type of test (one or two-tailed) to determine the P- value. 4)Make a statistical conclusion: If P-value ≤ α, reject H 0. If P-value > α, do not reject H 0. 5)Make a context-specific conclusion.

54. 9 | 54 Copyright © Cengage Learning. All rights reserved. Using the Critical Value Method for p As when testing for means, we can use the critical value method when testing for p. Use the critical value graphs exactly as when testing µ.

55. 9 | 55 Copyright © Cengage Learning. All rights reserved. Critical Thinking: Issues Related to Hypothesis Testing Central question – Is the value of test statistic too different from zero for the difference to be due to chance alone? The P-value gives the probability that the test statistic’s value is due to chance alone.

56. 9 | 56 Copyright © Cengage Learning. All rights reserved. Critical Thinking: Issues Related to Hypothesis Testing If the P-value is close to α, then we might attempt to clarify the results by - increasing the sample size - controlling the experiment to reduce the standard deviation How reliable is the study and the measurements in the sample? – Consider the source of the data and the reliability of the organization doing the study.