Chapter 12 Testing Hypotheses About Proportions

Transcription

1 Chapter 12 Testing Hypotheses About Proportions Copyright 2011 Brooks/Cole, Cengage Learning

2 Hypothesis testing method: uses data from a sample to judge whether or not a statement about a population may be true An Overview of Hypothesis Testing Steps in Any Hypothesis Test 1. Determine the null and alternative hypotheses. 2. Verify necessary data conditions, and if met, summarize the data into an appropriate test statistic. 3. Assuming the null hypothesis is true, find the p-value. 4. Decide whether or not the result is statistically significant based on the p-value. 5. Report the conclusion in the context of the situation. Copyright 2011 Brooks/Cole, Cengage Learning 2

3 Lesson 1: Formulating Hypothesis Statements Does a majority of the population favor a new legal standard for the blood alcohol level that constitutes drunk driving? Hypothesis 1: The population proportion favoring the new standard is not a majority. Hypothesis 2: The population proportion favoring the new standard is a majority. Copyright 2011 Brooks/Cole, Cengage Learning 3

4 More on Formulating Hypotheses Do female students study, on average, more than male students do? Hypothesis 1: On average, women do not study more than men do. Hypothesis 2: On average, women do study more than men do. Copyright 2011 Brooks/Cole, Cengage Learning 4

5 Terminology for the Two Choices Null hypothesis: Represented by H 0, is a statement that there is nothing happening. Generally thought of as the status quo, or no relationship, or no difference. Usually the researcher hopes to disprove or reject the null hypothesis. Alternative hypothesis: Represented by H a, is a statement that something is happening. In most situations, it is what the researcher hopes to prove. It may be a statement that the assumed status quo is false, or that there is a relationship, or that there is a difference. Copyright 2011 Brooks/Cole, Cengage Learning 5

6 Examples of H 0 and H a Null hypothesis examples: There is no extrasensory perception. There is no difference between the mean pulse rates of men and women. There is no relationship between exercise intensity and the resulting aerobic benefit. Alternative hypotheses examples: There is extrasensory perception. Men have lower mean pulse rates than women do. Increasing exercise intensity increases the resulting aerobic benefit. Copyright 2011 Brooks/Cole, Cengage Learning 6

7 Example 12.2 Are Side Effects Experienced by Fewer than 20% of Patients? Pharmaceutical company wants to claim that the proportion of patients who experience side effects is less than 20%. Null: 20% (or more) of users will experience side effects. Alternative: Fewer than 20% of users will experience side effects. Notice that the claim that the company hopes to prove is used as the alternative hypothesis. H 0 : p =.20 (or p.20) H a : p <.20 Copyright 2011 Brooks/Cole, Cengage Learning 7

8 One-Sided and Two-Sided Hypothesis Tests A one-sided hypothesis test is one for which the alternative hypothesis specifies parameter values in a single direction from a specified null value. A one-sided test may also be called a one-tailed hypothesis test. A two-sided hypothesis test is one for which the alternative hypothesis specifies parameter values in both directions from the specified null value. A two-sided test may also be called a two-tailed hypothesis test. Copyright 2011 Brooks/Cole, Cengage Learning 8

9 Notation and Null Value H 0 : population parameter = null value where the null value is the specific number the parameter equals if the null hypothesis is true. Alternative hypothesis written in one of the three ways: Two-sided alternative hypothesis: H a : population parameter null value One-sided alternative hypothesis (choose one): H a : population parameter > null value H a : population parameter < null value Copyright 2011 Brooks/Cole, Cengage Learning 9

10 Lesson 2: Test Statistic, p-value, and Deciding between the Hypotheses Similar to presumed innocent until proven guilty logic. We assume the null hypothesis is a possible truth until the sample data conclusively demonstrate otherwise. The Probability Question on Which Hypothesis Testing is Based If the null hypothesis is true about the population, what is the probability of observing sample data like that observed? Copyright 2011 Brooks/Cole, Cengage Learning 10

11 Example 12.4 Stop Pain before It Starts Painkiller Study: Men randomly assigned to experimental group (began taking painkillers before operation) or to control group (began taking painkillers after the operation). But 9 1/2 weeks later... only 12 members of the 60 men in the experimental group were still feeling pain. Among the 30 control group members, 18 were still feeling pain. the likelihood of this difference being due to chance was only 1 in 500. Null: Effectiveness of Painkillers is the same whether taken before or after surgery. If null hypothesis is true, probability is only 1 in 500 that the observed difference could have been as large as it was or larger. Reasonable to reject the null hypothesis of equal effectiveness. Copyright 2011 Brooks/Cole, Cengage Learning 11

12 Test Statistic and p-value The test statistic for a hypothesis test is the data summary used to evaluate the null and alternative hypotheses. The p-value is computed by assuming that the null hypothesis is true and then determining the probability of a test statistic as extreme as or more extreme than the observed test statistic in the direction of the alternative hypothesis. Copyright 2011 Brooks/Cole, Cengage Learning 12

13 Using p-value to Reach a Conclusion The level of significance, denoted by α (alpha), is a value chosen by the researcher to be the borderline between when a p-value is small enough to choose the alternative hypothesis over the null hypothesis, and when it is not. When the p-value is less than or equal to α, we reject the null hypothesis. When the p-value is larger than α, we cannot reject the null hypothesis. The level of significance may also be called the α-level of the test. Decision: reject H 0 if the p-value is smaller than α (usually 0.05, sometimes 0.10 or 0.01). In this case the result is statistically significant. Copyright 2011 Brooks/Cole, Cengage Learning 13

14 Stating the Two Possible Conclusions When the p-value is small, we reject the null hypothesis. Small is defined as a p-value α, where α = level of significance (usually 0.05). When the p-value is not small, we conclude that we cannot reject the null hypothesis or, equivalently, there is not enough evidence to reject the null hypothesis. Not small is defined as a p-value > α, where α = level of significance (usually 0.05). Copyright 2011 Brooks/Cole, Cengage Learning 14

15 Lesson 3: What Can Go Wrong? Example 12.7 Medical Analogy Null hypothesis: You do not have the disease. Alternative hypothesis: You do have the disease. Type 1 Error: You are told you have the disease, but you actually don t. The test result was a false positive. Consequence: You will be unnecessarily concerned about your health and you may receive unnecessary treatment. Type 2 Error : You are told that you do not have the disease, but you actually do. The test result was a false negative. Consequence: You do not receive treatment for a disease that you have. If this is a contagious disease, you may infect others. Copyright 2011 Brooks/Cole, Cengage Learning 15

16 Type 1 and Type 2 Errors A type 1 error can only occur when the null hypothesis is actually true. The error occurs by concluding that the alternative hypothesis is true. A type 2 error can only occur when the alternative hypothesis is actually true. The error occurs by concluding that the null hypothesis cannot be rejected. Copyright 2011 Brooks/Cole, Cengage Learning 16

17 Probability of a Type 1 Error and the Level of Significance When the null hypothesis is true, the probability of a type 1 error, the level of significance, and the α-level are all equivalent. When the null hypothesis is not true, a type 1 error cannot be made. Copyright 2011 Brooks/Cole, Cengage Learning 17

18 Type 2 Errors Two factors that affect probability of a type 2 error 1. Sample size; larger n reduces the probability of a type 2 error without affecting the probability of a type 1 error. 2. Level of significance; larger α reduces probability of a type 2 error by increasing the probability of a type 1 error. Copyright 2011 Brooks/Cole, Cengage Learning 18

19 12.2 Testing Hypotheses about a Proportion Example Does a Majority Favor a Lower BAC Limit? Legislator wants to know if there a majority of her constituents favor the lower limit. H 0 : p.5 (not a majority) H a : p >.5 (a majority) Note: p = the proportion of her constituents that favors the lower limit. The alternative is one-sided. Copyright 2004 Brooks/Cole, a division of Thomson Learning, Inc. 19

20 Null and Alternative Hypotheses for a Population Proportion Possible null and alternative hypotheses: 1. H 0 : p = p 0 versus H a : p p 0 (two-sided) 2. H 0 : p = p 0 versus H a : p < p 0 (one-sided) 3. H 0 : p = p 0 versus H a : p > p 0 (one-sided) p 0 = specific value called the null value. Remember a p-value is computed assuming H 0 is true, and p 0 is the value used for that computation. Copyright 2011 Brooks/Cole, Cengage Learning 20

21 Details for Calculating the z-statistic The z-statistic for the significance test is pˆ z = sample estimate null value = null standard error ( 1 p ) represents the sample estimate of the proportion p 0 represents the specific value in null hypothesis n is the sample size p pˆ 0 n p 0 0 Copyright 2011 Brooks/Cole, Cengage Learning 21

22 Computing the p-value for the z-test For H a less than, find probability the test statistic z could have been equal to or less than what it is. For H a greater than, find probability the test statistic z could have been equal to or greater than what it is. For H a two-sided, p-value includes the probability areas in both extremes of the distribution of the test statistic z. Copyright 2011 Brooks/Cole, Cengage Learning 22

23 Conditions for Conducting the z-test 1. The sample should be a random sample from the population. 2. The quantities np 0 and n(1 p 0 ) should both be at least 10. A sample size requirement. Some authors say at least 5 instead of our conservative 10. Copyright 2011 Brooks/Cole, Cengage Learning 23

24 Example The Importance of Order Survey of n = 190 college students. About half (92) asked: Randomly pick a letter - S or Q. Other half (98) asked: Randomly pick a letter - Q or S. Is there a preference for picking the first? Step 1: Determine the null and alternative hypotheses. Let p = proportion of population that would pick first letter. Null hypothesis: statement of nothing happening. If no general preference for either first or second letter, p =.5 Alternative hypothesis: researcher s belief or speculation. A preference for first letter p is greater than.5. H 0 : p =.5 versus H a : p >.5 (one-sided) Copyright 2011 Brooks/Cole, Cengage Learning 24

25 Example The Importance of Order Step 2: Verify necessary data conditions, and if met, summarize the data into an appropriate test statistic. 1. The sample should be a random sample from the population. 2. The quantities np 0 and n(1 p 0 ) should both be at least 10. With n = 190 and p 0 =.5, both n p 0 and n(1 p 0 ) equal 95, a quantity larger than 10, so the sample size condition is met. Copyright 2011 Brooks/Cole, Cengage Learning 25

26 Example The Importance of Order Step 2: Verify necessary data conditions, and if met, summarize the data into an appropriate test statistic. Of 92 students asked S or Q, 61 picked S, the first choice. Of 98 students asked Q or S, 53 picked Q, the first choice. Overall: 114 students picked first choice 114/190 =.60. The sample proportion,.60, is used to compute the z-test statistic. z = pˆ p p0 n = = ( 1 p ).5( 1.5) Copyright 2011 Brooks/Cole, Cengage Learning 26

27 Example The Importance of Order Step 3: Assuming null hypothesis true, find p-value. If the true p is.5, what is the probability that, for a sample of 190 people, the sample proportion could be as large as.60 (or larger)? or equivalently If the null hypothesis is true, what is the probability that the z-statistic could be as large as 2.76? p-value = = Copyright 2011 Brooks/Cole, Cengage Learning 27

28 Example The Importance of Order Step 4: Decide whether or not the result is statistically significant based on the p-value. Convention used by most researchers is to declare statistical significance when the p-value is smaller than The p-value = so the results are statistically significant and we can reject the null hypothesis. Copyright 2011 Brooks/Cole, Cengage Learning 28

29 Example The Importance of Order Step 5: Report the conclusion in the context of the problem. Statistical Conclusion = Reject the null hypothesis that p = 0.50 Context Conclusion = there is statistically significant evidence that the first letter presented is preferred. Copyright 2011 Brooks/Cole, Cengage Learning 29

30 Example Fewer than 20%? Clinical Trial of n = 400 patients. 68 patients experienced side effects. Can the company claim that fewer than 20% will experience side effects? Step 1: Determine the null and alternative hypotheses. H 0 : p.20 (company s claim is not true) H a : p <.20 (company s claim is true) Copyright 2011 Brooks/Cole, Cengage Learning 30

31 Example Fewer than 20%? Step 2: Verify necessary data conditions, and if met, summarize the data into an appropriate test statistic. 1. A random sample from the population reasonable. 2. The quantities np 0 and n(1 p 0 ) should both be at least 10. With n = 400 and p 0 =.2, the sample size condition is met. Out of 400 patients, 68 experienced side effects. Sample proportion = 68/400 =.17. z = pˆ p = = ( 1 p ).20( 1.20) n p Copyright 2011 Brooks/Cole, Cengage Learning 31

32 Example Fewer than 20%? Step 3: Assuming the null hypothesis is true, find the p-value. The area to the left of z = -1.5 is So p-value = Copyright 2011 Brooks/Cole, Cengage Learning 32

33 Example Fewer than 20%? Step 4: Decide whether or not the result is statistically significant based on the p-value. The p-value = so the results are not statistically significant and we cannot reject the null hypothesis. Step 5: Report the conclusion in the context of the problem. There is not sufficient evidence to conclude that the population proportion who would experience side effects is less than.20 Copyright 2011 Brooks/Cole, Cengage Learning 33

34 Example If Your Feet Don t Match Sample: n = 112 college students with unequal right and left foot measurements. Let p = population proportion with a longer right foot. Are Left and Right Foot Lengths Equal or Different? Step 1: Determine the null and alternative hypotheses. H 0 : p =.5 versus H a : p.5 Step 2: Verify necessary data conditions, and if met, summarize the data into an appropriate test statistic. Sample proportion with longer right foot = 63/112 =.5625 pˆ p z = = = 1.32 p0( 1 p0 ).5( 1.5) n 112 Copyright 2011 Brooks/Cole, Cengage Learning 34

35 Example If Your Feet Don t Match Step 3: Assuming the null hypothesis is true, find the p-value. The area to the left of z = is So p-value = 2(0.093) = Copyright 2011 Brooks/Cole, Cengage Learning 35

36 Example If Your Feet Don t Match Step 4: Decide whether or not the result is statistically significant based on the p-value. The p-value = so the results are not statistically significant and we cannot reject the null hypothesis. Step 5: Report the conclusion in the context of the problem. Although was a tendency toward a longer right foot in sample, there is insufficient evidence to conclude the proportion in the population with a longer right foot is different from the proportion with a longer left foot. Copyright 2011 Brooks/Cole, Cengage Learning 36