Chapter 11 Testing Hypotheses About Proportions Hypothesis testing method: uses data from a sample to judge whether or not a statement about a population may be true. Steps in Any Hypothesis Test 1. Determine the null and alternative hypotheses. 2. Verify necessary data conditions, and if met, 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 2006 Brooks/Cole, a division of Thomson Learning, Inc. Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc. 2 11.1 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. 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. Hypothesis 2: The population proportion favoring the new standard is a Copyright 2006 Brooks/Cole, a division majority. of Thomson Learning, Inc. 3 Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc. 4 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. Generally 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. 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 hypothesis examples: There is extrasensory perception. Men have lower mean pulse rates than women do. Increasing exercise intensity increases the resulting aerobic benefit. Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc. 5 Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc. 6 1
Example 11.1 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. The alternative is one-sided. Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc. 7 Example 11.2 Does a Majority Favor the Proposed Blood Alcohol Limit? Legislator s plan is to vote for the proposal if there is conclusive evidence that a majority of her constituents favor the proposal. H 0 : p.5 H a : p >.5 (not a majority) (a majority) Note: p = the proportion of her constituents that favors the proposal. The alternative is one-sided. Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc. 8 11.2 Logic of Hypothesis Testing What if the Null is True? 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 2006 Brooks/Cole, a division of Thomson Learning, Inc. 9 Example 11.3 Psychic Powers Cartoon: Two characters playing a coin-flipping game. Character 1: correctly guesses outcome of 100 flips. Character 2: just a coincidence Null: Alternative: Character 1 does not have Psychic Powers (is just guessing) Character 1 has Psychic Powers Q: If character only guessing, how likely is correctly guessing 100 consecutive fair coin tosses? A: (½) 100 => extraordinarily small. We reject the null hypothesis because the sample results are extremely inconsistent with it. We conclude character was using psychic powers. Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc. 10 11.3 Reaching a Conclusion About the Two Hypotheses Data summary used to evaluate the two hypotheses is called the test statistic. Likelihood of observing a test statistic as extreme as what we did, or something even more extreme, if the null hypothesis is true is called the p-value. Decision: reject H 0 if the p-value is smaller than a designated level of significance, denoted by α (usually 0.05, sometimes 0.10 or 0.01). In this case the result is statistically significant. Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc. 11 Stating the Two Possible Conclusions When the p-value is small, we reject the null hypothesis or, equivalently, we accept the alternative 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 2006 Brooks/Cole, a division of Thomson Learning, Inc. 12 2
11.4 Testing Hypotheses About a 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. Often H 0 for a one-sided test is written as H 0 : p = p 0. Remember a p-value is computed assuming H 0 is true, and p 0 is the value used for that computation. Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc. 13 The z-test for a Proportion Determine the sampling distribution of possible sample proportions when the true population proportion is p 0 (called the null value), the value specified in H 0. Using properties of this sampling distribution, calculate a standardized score (z-score) for the observed sample proportion. If the standardized score has a large magnitude, conclude that the sample proportion would be unlikely if the null value p 0 is true, and reject the null hypothesis. Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc. 14 Conditions for Conducting the z-test 1. The sample should be a random sample from the population. Not always practical most use test procedure as long as sample is representative of the population for the question of interest. 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 2006 Brooks/Cole, a division of Thomson Learning, Inc. 15 Example 11.6 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 = p 0 versus H a : p > p 0 (one-sided) Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc. 16 Step 2: Verify necessary data conditions, and if met, 1. The sample should be a random sample from the population. The sample is a convenience sample of students who were enrolled for a class. Does not seem this will bias results for this question, so will view the sample as a random sample. 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. Step 2: Verify necessary data conditions, and if met, 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, the standardized score for measuring the difference between the =.60, and the null hypothesis value, p 0 =.50. The z-statistic = 2.76 (formula comes later). Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc. 17 Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc. 18 3
Step 3: Assuming the null hypothesis is true, find the 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 (or larger)? Using computer (or reading from print-out): p-value = 0.003 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 0.05. The p-value = 0.003 so the results are statistically significant and we can reject the null hypothesis. Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc. 19 Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc. 20 Step 5: Report the conclusion in the context of the problem. Details for Calculating the z-statistic The z-statistic for the significance test is Statistical Conclusion = Reject the null hypothesis that p = 0.50 Context Conclusion = there is statistically significant evidence that the first letter presented is preferred. represents the sample estimate of the proportion p 0 represents the specific value in null hypothesis n is the sample size Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc. 21 Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc. 22 Example 11.1 Fewer than 20%? (cont). Clinical Trial of n = 400 patients. 68 patients experienced side effects. Can the company claim that fewer than 20% will experience side effects? Hypothesis testing steps: Step 1: Determine the null and alternative hypotheses Step 2: Verify necessary data conditions, and if met, Step 3: Assuming the null hypothesis is true, find the p-value. Step 4: Decide whether or not the result is statistically significant based on the p-value. Step 5: Report the conclusion in the context of the problem. Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc. 23 24 4
Using Minitab: To test hypotheses about a proportion, use Stat>Basic Statistics>1 Proportion. If the raw data are in a column of the worksheet, specify the column. If not, enter the summarized data. Click on Options, select confidence level, alternative, and check Use test and box. Check Perform hypothesis test and enter p 0. Click OK and read off results. Example 11.7 Left and Right Foot Lengths Sample: 112 of 215 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? 25 Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc. 26 11.5 Role of Sample Size in Statistical Significance Cautions about Sample Size and Statistical Significance If a small to moderate effect in the population, a small sample has little chance of being statistically significant. With a large sample, even a small and unimportant effect in the population may be statistically significance. Example 11.8 Same Sample Proportion Can Produce Different Conclusions Taste Test: Sample of people taste both drinks and record how many like taste of Drink A better than B. Let p = H 0 : p =.5 H a : p.5 proportion in population that would prefer Drink A. (no preference) (preference for one or other) Results based on two sample sizes: n = 60 and n = 960 and the sample proportion for both is 0.55. Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc. 27 Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc. 28 Example 11.8 Different Conclusions (cont) Results when n = 60 33 of the 60 preferred Drink A; Results when n = 960 528 or the 960 preferred Drink A; Why more significant for larger n? The z-value changes because the sample size affects the standard error. When n =60, the null standard error =.065. When n = 960, the null standard error =.016. Increasing n decreases null standard error => an absolute difference between the sample proportion and null value is more significant Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc. 29 Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc. 30 5
11.6 Real Importance versus Statistical Significance The p-value does not provide information about the magnitude of the effect. The magnitude of a statistically significant effect can be so small that the practical effect is not important. If sample size large enough, almost any null hypothesis can be rejected. Example 11.9 Birth Month and Height Headline: Spring Birthday Confers Height Advantage Austrian study of heights of 507,125 military recruits. Men born in spring were, on average, about 0.6 cm taller than men born in fall (Weber et al., Nature, 1998, 391:754 755). A small difference: 0.6 cm = about 1/4 inch. Sample size so large that even a very small difference was statistically significant. Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc. 31 Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc. 32 Case Study 11.1 Internet and Loneliness greater use of the Internet was associated with declines in participants communication with family members in the household, declines in size of their social circle, and increases in their depression and loneliness (Kraut et al., 1998, p. 1017) A closer look: actual effects were quite small. one hour a week on the Internet was associated, on average, with an increase of 0.03, or 1 percent on the depression scale (Harman, 30 August 1998, p. A3). 11.7 What Can Go Wrong? 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 2006 Brooks/Cole, a division of Thomson Learning, Inc. 33 Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc. 34 Example 11.10 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 2006 Brooks/Cole, a division of Thomson Learning, Inc. 35 6