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Hypothesis Testing Chapter 9 Understandable Statistics Ninth Edition By Brase and Brase Prepared by Yixun Shi Bloomsburg University of Pennsylvania.

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Presentation on theme: "Hypothesis Testing Chapter 9 Understandable Statistics Ninth Edition By Brase and Brase Prepared by Yixun Shi Bloomsburg University of Pennsylvania."— Presentation transcript:

1 Hypothesis Testing Chapter 9 Understandable Statistics Ninth Edition By Brase and Brase Prepared by Yixun Shi Bloomsburg University of Pennsylvania

2 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.9 | 2 Methods for Drawing Inference We can draw inference on a population parameter in two ways: 1)Estimation (Chapter 8) 2)Hypothesis Testing (Chapter 9)

3 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.9 | 3 Hypothesis Testing In essence, hypothesis testing is the process of making decisions about the value of a population parameter.

4 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.9 | 4 Establishing the Hypotheses Null Hypothesis: A hypothesis about the parameter in question that often denotes a theoretical value, an 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 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.9 | 5 Hypotheses Restated

6 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.9 | 6 Types of Tests 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, p) is less than, greater than, or not equal to a specified value.

7 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.9 | 7 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

8 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.9 | 8 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.

9 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.9 | 9 Hypothesis Test of μ x is Normal, σ is Known

10 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.9 | 10 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.

11 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.9 | 11 P-Value for Left-Tailed Test

12 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.9 | 12 P-Value for Right-Tailed Test

13 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.9 | 13 P-Value for Two-Tailed Test

14 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.9 | 14 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.

15 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.9 | 15 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 (obtain more information), but this may not be feasible in practice.

16 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.9 | 16 Type I and Type II Errors

17 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.9 | 17 Level of Significance Good practice is to specify in advance the level of type I error we are willing to risk. The probability of type I error is the level of significance for the test, denoted by α (alpha).

18 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.9 | 18 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).

19 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.9 | 19 The Probabilities Associated with Testing

20 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.9 | 20 Concluding a Statistical Test For our purposes, significant is defined as follows: At our predetermined α level of risk, the evidence against H 0 is sufficient to discredit H 0. Thus we adopt H 1.

21 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.9 | 21 Statistical Testing Comments In most statistical applications, α = 0.05 or α = 0.01 is used 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 the null –Some statisticians prefer to say “fail to reject the null” rather than “accept the null.”

22 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.9 | 22 Interpretation of Testing Terms

23 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.9 | 23 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:

24 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.9 | 24 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.

25 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.9 | 25 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:

26 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.9 | 26 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.

27 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.9 | 27 Using Table 6 to Estimate P-Values Suppose we calculate t = 2.22 for a one-tailed test from a sample size of 6. Thus, df = n – 1 = 5. We obtain: 0.025 < P-Value < 0.050

28 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.9 | 28 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.

29 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.9 | 29 Critical Regions for H 0 : µ = k

30 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.9 | 30 Critical Regions for H 0 : µ = k

31 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.9 | 31 Critical Regions for H 0 : µ = k

32 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.9 | 32 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:

33 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.9 | 33 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.

34 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.9 | 34 Left-Tailed Tests

35 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.9 | 35 Right-Tailed Tests

36 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.9 | 36 Two-Tailed Tests

37 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.9 | 37 Testing a Proportion p Test assumptions: r is a binomial variable n is the number of independent trials p is the probability of success on each trial np > 5 and n(1-p) > 5

38 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.9 | 38 Types of Proportion Tests

39 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.9 | 39 The Distribution of the Sample Proportion

40 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.9 | 40 Converting the Sample Proportion to z

41 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.9 | 41 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

42 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.9 | 42 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.

43 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.9 | 43 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.

44 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.9 | 44 Critical Thinking: Issues Related to Hypothesis Testing Central question – Is the value of sample test statistics too far away from the value of the population parameter proposed in H 0 to occur by chance alone? – P-value tells the probability for that to occur by chance alone.

45 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.9 | 45 Critical Thinking: Issues Related to Hypothesis Testing If the P-value is so 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.

46 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.9 | 46 Tests Involving Paired Differences Data pairs occur naturally in many settings: –Before and after measurements on the same observation after a treatment. –Be sure to have a definite and uniform method for creating pairs of data points.

47 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.9 | 47 Advantages to Using Paired Data Reduces the danger of extraneous or uncontrollable variables Theoretically reduces measurement variability. Increases the accuracy of statistical conclusions.

48 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.9 | 48 Testing for Differences We take the difference between each pair of data points. –Denoted by d We then test the average difference against the Student’s t distribution.

49 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.9 | 49 Hypotheses for Differences H 0 : µ d = 0

50 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.9 | 50 Sample Test Statistic for Differences with

51 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.9 | 51 Finding the P-Value Just as in the test for µ when σ is unknown, use Table 6 to estimate the P-Value of the test.

52 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.9 | 52 Testing d 1)State the null hypothesis, alternate hypothesis, and level of significance. 2)If you can assume d is normal (mound-shaped), any sample size will do. If not, make sure n ≥ 30. Calculate: df = n-1

53 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.9 | 53 Testing d 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.

54 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.9 | 54 Testing the Differences Between Independent Samples Many practical applications involve testing the difference between population means or population proportions.

55 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.9 | 55 Testing µ 1 - µ 2 when σ 1, σ 2 are Known

56 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.9 | 56 Hypotheses for Testing µ 1 - µ 2 when σ 1, σ 2 are Known

57 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.9 | 57 Testing µ 1 - µ 2 when σ 1, σ 2 are Known

58 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.9 | 58 Testing µ 1 - µ 2 when σ 1, σ 2 are Known

59 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.9 | 59 Testing µ 1 - µ 2 when σ 1, σ 2 are Known

60 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.9 | 60 Testing µ 1 - µ 2 when σ 1, σ 2 are Unknown Just as in the one-sample test for the mean, we will resort to the Student’s t distribution and proceed in a similar fashion. –Remark: in practice, the population standard deviation will be unknown in most cases.

61 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.9 | 61 Testing µ 1 - µ 2 when σ 1, σ 2 are Unknown

62 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.9 | 62 Testing µ 1 - µ 2 when σ 1, σ 2 are Unknown

63 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.9 | 63 Testing µ 1 - µ 2 when σ 1, σ 2 are Unknown

64 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.9 | 64 Deciding Which Test to Use for H 0 : µ 1 - µ 2

65 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.9 | 65 Testing for a Difference in Proportions, p 1 – p 2 Suppose we have two independent binomial experiments. We would like to test if the two population proportions are equal.

66 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.9 | 66 Testing for a Difference in Proportions

67 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.9 | 67 Testing for a Difference in Proportions

68 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.9 | 68 Testing for a Difference in Proportions The test statistic is as follows:

69 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.9 | 69 The Test Procedure

70 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.9 | 70 The Test Procedure

71 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.9 | 71 The Test Procedure

72 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.9 | 72 The Test Procedure

73 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.9 | 73 Critical Regions For Tests of Differences Recall, our emphasis is on the P-Value method. Most scientific studies use this technique. Also, for a fixed α-level test, the methods are equivalent and lead to identical results.


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