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# Copyright © 2014 Pearson Education, Inc. All rights reserved Chapter 8 Hypothesis Testing for Population Proportions.

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Copyright © 2014 Pearson Education, Inc. All rights reserved Chapter 8 Hypothesis Testing for Population Proportions

8 - 2 Copyright © 2014 Pearson Education, Inc. All rights reserved Learning Objectives  Know how to test hypotheses concerning a population proportion and hypotheses concerning the comparison of two population proportions.  Understand the meaning of p-value and how it is used.  Understand the meaning of significance level and how it is used.  Know the conditions required for calculating a p-value and significance level.

Copyright © 2014 Pearson Education, Inc. All rights reserved 8.1 The Main Ingredients of Hypothesis Testing

8 - 4 Copyright © 2014 Pearson Education, Inc. All rights reserved Hypothesis Testing  How do you determine if someone is cheating when they toss a coin?  If the toss is fair P(Heads) = p = 0.5  Otherwise P(Heads) = p ≠ 0.5  If only 15 Heads come up out of 40 tosses, is the person cheating?

8 - 5 Copyright © 2014 Pearson Education, Inc. All rights reserved Null and Alternative Hypotheses  H 0 : p = 0.5  “H naught” or “the Null Hypothesis”  The status quo, no cheating, no surprises, no change, no effect  H a : p ≠ 0.5  “The Alternative Hypothesis”  This is what we hope or guess is true.  Note that p is the population proportion. Hypothesis testing is always for the population parameter, never the statistic.

8 - 6 Copyright © 2014 Pearson Education, Inc. All rights reserved Choosing the Null and Alternative Hypothesis  First come up with an idea or hypothesis.  The dealer is palming aces.  Babies are born less often on Tuesdays.  Write down the null hypothesis.  H 0 : p = 1/4  H 0 : p = 1/7  Write down the alternative hypothesis.  H a : p > 1/4  H a : p < 1/7

8 - 7 Copyright © 2014 Pearson Education, Inc. All rights reserved Level of Significance  The Level of Significance, , is the probability of rejecting H 0 when H 0 is true.  Concluding that the coin tossing is not fair when it is fair.  Saying that the defendant is guilty but he did not commit the crime.  Concluding that the person has psychic abilities when she was just guessing.

8 - 8 Copyright © 2014 Pearson Education, Inc. All rights reserved Typical Levels of Significance  Since the level of significance is the probability of making an error, it should be small.   = 0.05 or 5% is the most typical.   = 0.01 or 1% is used when making this error has very bad repercussions.   = 0.10 or 10% is used when an error is less an issue than making no conclusion when H a is true.

8 - 9 Copyright © 2014 Pearson Education, Inc. All rights reserved The Test Statistic  In conducting a hypothesis test one takes a sample and finds.  The Central Limit Theorem for proportions:  The Test Statistic is the z-score:

8 - 10 Copyright © 2014 Pearson Education, Inc. All rights reserved The Meaning of the Test Statistic  The test statistic tells us how unlikely that sample proportion could have happened by random chance had the null hypothesis been true.  If the null hypothesis is true, then the test statistic should be close to 0. Therefore, the farther the test statistic is from 0, the more the null hypothesis is discredited.

8 - 11 Copyright © 2014 Pearson Education, Inc. All rights reserved The p-Value  The p-value is a probability. Assuming the null hypothesis is true, the p-value is the probability that if the experiment were repeated, you would get a test statistic as extreme as or more extreme than the one you actually got. A small p-value suggests that a surprising outcome has occurred and discredits the null hypothesis.

8 - 12 Copyright © 2014 Pearson Education, Inc. All rights reserved p-Value Example  A hypothesis test was conducted to see if 10% of the time that the weather report states a 10% chance of rain that there is rain. Of 300 randomly selected days with a 10% chance of rain, 6% had rain. The p-value was found to be 0.02. Interpret this p-value.  If another 300 days with 10% chance of rain were randomly selected, then there is only a 2% chance that the number of days of rain would be less than 18 (6%) or more than 42 (14%).

8 - 13 Copyright © 2014 Pearson Education, Inc. All rights reserved Hypothesis Testing Summary 1. Hypothesize  Come up with an idea or hypothesis.  Write down H 0 and H a. 2. Prepare  Choose the level of significance .  Select the test statistic  Check Sampling Distributions’ Conditions

8 - 14 Copyright © 2014 Pearson Education, Inc. All rights reserved Hypothesis Testing Summary (Continued) 3. Compute to Compare  Compute the test statistic.  Compute the p-Value. 4. Interpret  If p-Value < , reject H 0.  If p-Value > , fail to reject H 0.  State the conclusion in the context of the study.

Copyright © 2014 Pearson Education, Inc. All rights reserved 8.2 Characterizing p-values

8 - 16 Copyright © 2014 Pearson Education, Inc. All rights reserved Three Types of Hypothesis Tests  Left Tailed Hypothesis  H 0 : p = p 0  H a : p < p 0  Right Tailed Hypothesis  H 0 : p = p 0  H a : p > p 0  Two Tailed Hypothesis  H 0 : p = p 0  H a : p ≠ p 0

8 - 17 Copyright © 2014 Pearson Education, Inc. All rights reserved Left Tailed Hypothesis  H 0 : p = p 0  H a : p < p 0  The p-value represents the probability that if p = p 0 and another random sample is taken with the same sample size then the new sample proportion will be less than observed sample proportion.

8 - 18 Copyright © 2014 Pearson Education, Inc. All rights reserved Right Tailed Hypothesis  H 0 : p = p 0  H a : p > p 0  The p-value represents the probability that if p = p 0 and another random sample is taken with the same sample size then the new sample proportion will be greater than observed sample proportion.

8 - 19 Copyright © 2014 Pearson Education, Inc. All rights reserved Two Tailed Hypothesis  H 0 : p = p 0  H a : p ≠ p 0  The p-value represents the probability that if p = p 0 and another random sample is taken with the same sample size then the new sample proportion will be farther from p 0 than observed sample proportion.

8 - 20 Copyright © 2014 Pearson Education, Inc. All rights reserved p-Value Example  45% of Americans have type “O” blood. A study was done to see if the percent differs for college students. 47% of the 1000 randomly selected students had type O blood. The p-Value was 0.2.  Two tailed test: “differs”  If 45% of college students do have type “O” blood and if another 1000 randomly selected students are tested, then there is a 20% chance that either fewer than 43% or greater than 47% of this new group of students will have type “O” blood.

8 - 21 Copyright © 2014 Pearson Education, Inc. All rights reserved Requirements for a Valid p-Value 1. Random sample: The sample is collected randomly. 2. Large enough sample size: The sample size, n, is large enough that the sample has at least 10 expected successes and 10 expected failures np 0 ≥10 and n(1 - p 0 ) ≥ 10 3. Without replacement: If the sample is collected without replacement, then the population size is at least 10 times bigger than the sample size. 4. Independence: Each observation or measurement must have no influence on any others. 5. Null hypothesis : The null hypothesis is true.

8 - 22 Copyright © 2014 Pearson Education, Inc. All rights reserved StatCrunch and the p-Value  Have more than 70% of all dogs had their rabies shots? 592 of the 800 dogs examined had their rabies shots.  H 0 : p = 0.7H a : p > 0.7  Stat→Proportions→One sample→with summary

8 - 23 Copyright © 2014 Pearson Education, Inc. All rights reserved StatCrunch and the p-Value  Have more than 70% of all dogs had their rabies shots? 592 of the 800 dogs examined had their rabies shots.  H 0 : p = 0.7H a : p > 0.7  If 70% of all dogs have had their rabies shots and if another 800 randomly selected dogs are examined, then there is a 0.68% chance that more than 592 of them will have had their rabies shots.

8 - 24 Copyright © 2014 Pearson Education, Inc. All rights reserved Summary of the p-Value  Tells us how surprising the sample data is if the null hypothesis is true.  A very small p-Value, less than 0.05 for example, indicates that the results that were obtained would be very surprising.  A larger p-Value, greater than 0.05 for example, indicates that the results that were obtained would not be very surprising.

Copyright © 2014 Pearson Education, Inc. All rights reserved 8.3 Hypothesis Testing in Four Steps

8 - 26 Copyright © 2014 Pearson Education, Inc. All rights reserved Four Steps to Hypothesis Testing 1. Hypothesize: State your hypothesis about the population parameter. 2. Prepare: Choose a significance level and test statistic, check requirements, and state assumptions. 3. Compute to Compare: Compute the test statistic and find the p-value to measure the surprise. 4. Interpret: Reject or fail to reject the null hypothesis. State the conclusions in the context of the study.

8 - 27 Copyright © 2014 Pearson Education, Inc. All rights reserved Hypothesize  Each year 5% of the world tiger population is killed by poachers. Now that there is a campaign to educate people on the problem, has this percent gone down? 300 tigers were observed and 18 were killed by poachers.  H 0 : p = 0.05  H a : p < 0.05

8 - 28 Copyright © 2014 Pearson Education, Inc. All rights reserved Hypothesize  90% of all restaurants fail after one year. Is that number different for Chinese restaurants. Of the 108 new Chinese restaurants 87 had failed after one year.  H 0 : p = 0.90  H a : p ≠ 0.90

8 - 29 Copyright © 2014 Pearson Education, Inc. All rights reserved Prepare  Each year 5% of the world tiger population is killed by poachers. Now that there is a campaign to educate people on the problem, has this percent gone down? 300 tigers were observed and 18 were killed by poachers.  Use the standard  = 0.05.  np 0 = (300)(0.05) = 15 ≥ 10  n(1-p 0 ) = (300)(0.95) = 285 ≥ 10  Can use the z-statistic.  Assume that the 300 tigers were randomly selected and that there are more than 3000 tigers in the world.

8 - 30 Copyright © 2014 Pearson Education, Inc. All rights reserved Prepare  90% of all restaurants fail after one year. Is that number different for Chinese restaurants. Of the 108 new Chinese restaurants 87 had failed after one year.  Use the standard  = 0.05.  np 0 = (108)(0.9) = 97.2 ≥ 10  n(1-p 0 ) = (108)(0.1) = 10.8 ≥ 10  Can use the z-statistic.  Assume that the 108 Chinese restaurants were randomly selected and that there are more than 1080 Chinese restaurants in existence.

8 - 31 Copyright © 2014 Pearson Education, Inc. All rights reserved Compute to Compare  Each year 5% of the world tiger population is killed by poachers. Now that there is a campaign to educate people on the problem, has this percent gone down? 300 tigers were observed and 18 were killed by poachers.  z ≈ 0.795  p-value ≈ 0.79   If 5% of the tigers are killed by poaching and if another sample is observed, there is a 79% chance of getting a sample proportion less than 6%. 79% is very high.

8 - 32 Copyright © 2014 Pearson Education, Inc. All rights reserved Compute to Compare  90% of all restaurants fail after one year. Is that number different for Chinese restaurants. Of the 108 new Chinese restaurants 87 had failed after one year.  z ≈ -3.27  p-value ≈ 0.001   If 90% of all Chinese restaurants fail after one year and if another sample is observed, there is a 0.1% chance of getting a sample proportion less than 81%. 0.1% is a very small chance.

8 - 33 Copyright © 2014 Pearson Education, Inc. All rights reserved Rules to Interpret 1. If p-value < , reject H 0 and accept H a and state the conclusion. 2. If p-value > , fail to reject H 0 and just state that we do not have statistically significant evidence to make a conclusion.

8 - 34 Copyright © 2014 Pearson Education, Inc. All rights reserved Interpret  Each year 5% of the world tiger population is killed by poachers. Now that there is a campaign to educate people on the problem, has this percent gone down? 300 tigers were observed and 18 were killed by poachers.  z ≈ 0.795, p-value ≈ 0.79,  Since p-value ≈ 0.79 > 0.05 = , fail to reject H 0. There is statistically insignificant evidence to support the claim that since the campaign began, less than 5% of the world tiger population is killed by poachers.

8 - 35 Copyright © 2014 Pearson Education, Inc. All rights reserved Interpret  90% of all restaurants fail after one year. Is that number different for Chinese restaurants. Of the 108 new Chinese restaurants 87 had failed after one year.  z ≈ -3.27, p-value ≈ 0.001,  Since p-value ≈ 0.001 < 0.05 = , reject H 0 and accept H a. There is statistically significant evidence to support the claim that the percent of Chinese restaurants that fail after one year is different from 90%.

Copyright © 2014 Pearson Education, Inc. All rights reserved 8.4 Comparing Proportions from Two Populations

8 - 37 Copyright © 2014 Pearson Education, Inc. All rights reserved Comparing Two Populations  Comparing two groups, men vs. women, blacks vs. whites, with treatment vs. without treatment, etc.  Instead of the sample size, n, there will be two sample sizes, n 1 and n 2.  p 1 and p 2 instead of a single population proportion, p.  and instead of the sample proportion,

8 - 38 Copyright © 2014 Pearson Education, Inc. All rights reserved Hypothesis Tests  The Null Hypothesis will be:  H 0 : p 1 = p 2  Three possibilities for the Alternative Hypothesis:  H a : p 1 < p 2 (Left Tailed Test)  H a : p 1 > p 2 (Right Tailed Test)  H a : p 1 ≠ p 2 (Two Tailed Test)

8 - 39 Copyright © 2014 Pearson Education, Inc. All rights reserved The Test Statistic  The estimator is just the difference between the sample proportions:  The null value is typically 0.  StatCrunch is easier to use than the formula.

8 - 40 Copyright © 2014 Pearson Education, Inc. All rights reserved Checking Conditions  Large Sample: Note the use of the pooled proportion.  Random Sample or at least close to random.  The two samples are independent of each other.  Independent Within Samples: The observations within each sample must be independent of one another.  The Null Hypothesis is true.

8 - 41 Copyright © 2014 Pearson Education, Inc. All rights reserved Using StatCrunch  Stat→Proportions→Two sample→with summary

8 - 42 Copyright © 2014 Pearson Education, Inc. All rights reserved Caffeine Therapy For Premature Infants  Apnea of prematurity occurs when premature babies have shallow breathing or stop breathing for more than 20 seconds.  Treatment group received caffeine therapy while the other group received a placebo.

8 - 43 Copyright © 2014 Pearson Education, Inc. All rights reserved Does caffeine therapy lower the rate of bad events? Of the 937 infants given the therapy, 377 suffered from death or disability. The placebo group had 932 infants, and of these, 431 suffered from death or disability. 1. Hypothesize  H 0 : p 1 = p 2, H a : p 1 < p 2 2. Prepare  Use  = 0.05.  z-statistic  937 x 0.43, 937 x 0.57, 932 x 0.43, 932 x 0.57 are all greater than or equal to 10. Assume that the infants were randomly and independently selected.

8 - 44 Copyright © 2014 Pearson Education, Inc. All rights reserved Does caffeine therapy lower the rate of bad events? Of the 937 infants given the therapy, 377 suffered from death or disability. The placebo group had 932 infants, and of these, 431 suffered from death or disability. 3. Compute to Compare  StatCrunch → z = -2.62, p-value = 0.004 4. Interpret  p-value = 0.004 < 0.05 =   Reject H 0, Accept H a. There is statistically significant evidence to conclude that a lower proportions of babies will die or suffer with this therapy than without this therapy.

8 - 45 Copyright © 2014 Pearson Education, Inc. All rights reserved Is there a difference between the proportions of men and women who play guitar? 82 of the 400 men and 54 of the 300 women surveyed played guitar.  Hypothesize  H 0 : p 1 = p 2, H a : p 1 ≠ p 2  Prepare  Use  = 0.05.  Large Sample? 400 x 0.19, 400 x 0.81, 300 x 0.19, 300 x 0.81 are all greater than or equal to 10.  Assume that the men and women were randomly and independently selected.

8 - 46 Copyright © 2014 Pearson Education, Inc. All rights reserved Is there a difference between the proportions of men and women who play guitar? 82 of the 400 men and 54 of the 300 women surveyed played guitar.  Compute to Compare  StatCrunch → z = 0.83, p-value = 0.41  Interpret  p-value = 0.41 > 0.05 =   Fail to Reject H 0. There is statistically insufficient evidence to make a conclusion about there being a difference between the proportions of men and women who own a guitar.

Copyright © 2014 Pearson Education, Inc. All rights reserved 8.5 Understanding Hypothesis Testing

8 - 48 Copyright © 2014 Pearson Education, Inc. All rights reserved What to do if Conditions Fail  The sample size is too small  Redo the study with a larger sample size.  Use an advanced test, e.g. Fisher’s Exact Test.  The samples are not independent  Take an advance statistics class or  Consult a statistician.  Samples not random  State the conclusion for the sample only.

8 - 49 Copyright © 2014 Pearson Education, Inc. All rights reserved Power  The Power of a hypothesis test is the probability of rejecting the null hypothesis when the null hypothesis is false.  Example: If the alternative hypothesis is that the person cheated, then the power is the probability that if the person is cheating then we will conclude correctly that cheating occurred.  Always strive for a large power.

8 - 50 Copyright © 2014 Pearson Education, Inc. All rights reserved Hypotheses, Power, and Level of Significance  Before the data is collected you can choose to make it more likely to reject H 0 or more likely to fail to reject H 0, but not both.  Both choices lead to an increased chance of one of the “Bad” outcomes.  Only increasing the sample size decreases the chance of both “Bad” outcomes.

8 - 51 Copyright © 2014 Pearson Education, Inc. All rights reserved Power vs. Level of Significance  Increasing the level of significance decreases the power.  If you want a better chance of rejecting H 0 when H 0 is false, you can decrease the level of significance.  The only way to have a large power and a large level of significance is to have a large sample size.

8 - 52 Copyright © 2014 Pearson Education, Inc. All rights reserved Statistical Significance vs. Practical Significance  The term Statistical Significance is used when the null hypothesis is rejected. There is a small probability that if H 0 is correct then results as extreme as were obtained would happen randomly.  The term Practical Significance means that the results obtained are clearly far from the hypothesized value.

8 - 53 Copyright © 2014 Pearson Education, Inc. All rights reserved Statistical Significance and Practical Significance  A poll to see if the incumbent will get more than 50% of the votes results resulted in a sample proportion 58%.  = 0.05 and the p-value = 0.06.  This is practically significant but statistically insignificant.  A study to see if there is a difference in pass rates for male vs. female statistics students resulted in a sample proportion of men: 72.1% and women: 72.3%.  = 0.05 and the p-value = 0.04.  This is practically insignificant but statistically significant.

8 - 54 Copyright © 2014 Pearson Education, Inc. All rights reserved Hypotheses Should Never Be Changed After the Data is Analyzed  The null and alternative hypotheses must be written down before the data is analyzed.  Never adjust H 0 and H a to fit your results.  If you change your mind based on the data, you must collect a new data set in order to support the adjusted hypotheses.

8 - 55 Copyright © 2014 Pearson Education, Inc. All rights reserved Using Proper Language  If the p-value is less than the level of significance, , then state that we “reject H 0 ” and “accept H a ”. Do not state that the alternative hypothesis is proven or true.  If the p-value is greater than  then state that we “fail to reject H 0 ” and that no conclusion can be made. Do not state that H 0 is accepted or true. It is possible that the power is too small and that H 0 is still false.

8 - 56 Copyright © 2014 Pearson Education, Inc. All rights reserved Conclude Only the Inequality  A two tailed test was used to see whether there is a difference between the proportion of genetically modified seeds that germinate and natural seeds that germinate. The sample proportions were 0.68 and 0.43 and H 0 was rejected.  State that there is evidence for a difference in the population proportions. Do not state that for all seeds, the ones that are genetically modified are 25% more likely to germinate.

Copyright © 2014 Pearson Education, Inc. All rights reserved Chapter 8 Case Study

8 - 58 Copyright © 2014 Pearson Education, Inc. All rights reserved TV Violence Leads to Actual Violence?  A study was done to see if children who view violent TV shows are more likely to become violent towards their spouses when they grow up. 1. Hypothesize  H 0 : p 1 = p 2  H a : p 1 > p 2 2. Prepare   = 0.05

8 - 59 Copyright © 2014 Pearson Education, Inc. All rights reserved TV Violence Leads to Actual Violence?

8 - 60 Copyright © 2014 Pearson Education, Inc. All rights reserved Checking Conditions  We will assume that the participants were selected randomly and are independent from each other.  The population sizes are certainly large compared to the sample sizes. 

8 - 61 Copyright © 2014 Pearson Education, Inc. All rights reserved 3.Compute to Compare

8 - 62 Copyright © 2014 Pearson Education, Inc. All rights reserved 4. Interpret  Since p-value = 0.0033 < 0.05 = , we reject the null hypothesis and accept the alternative hypotheses.  There is statistically significant evidence to conclude that children who grow up watching violent TV are more likely to become violent towards their spouses when they grow up.  We cannot conclude based solely on the data that watching violent TV as children causes adults to become violent towards their spouses.

Copyright © 2014 Pearson Education, Inc. All rights reserved Chapter 8 Guided Exercise 1

8 - 64 Copyright © 2014 Pearson Education, Inc. All rights reserved Gun Control  Historically, the percentage of U.S. residents who support stricter gun control laws has been 52%. A recent Gallup Poll of 1011 people showed 495 in favor of stricter gun control laws. Assume the poll was given to a random sample of people.  QUESTION Test the claim that the proportion of those favoring stricter gun control has changed from 0.52. Perform a hypothesis test, using a significance level of 0.05.

8 - 65 Copyright © 2014 Pearson Education, Inc. All rights reserved Hypothesize  Test the claim that the proportion of those favoring stricter gun control has changed from 0.52.  H 0 : The population proportion that supports gun control is 0.52, p = 0.52  H a : p ≠ 0.52

8 - 66 Copyright © 2014 Pearson Education, Inc. All rights reserved Prepare  Choose the one-proportion z-test.  Random sample: yes  Sample size:  np 0 = 1011(0.52) ≈ 526 ≥ 10  n(1 - p 0 ) = 1011(0.48) ≈ 526 ≥ 10  Population size is more than 10 times 1011.

8 - 67 Copyright © 2014 Pearson Education, Inc. All rights reserved Compute and Compare

8 - 68 Copyright © 2014 Pearson Education, Inc. All rights reserved Interpret  Reject H 0 (if the p-value is 0.05 or less) or do not reject H 0 and choose one of the following conclusions: 1. The percentage is not significantly different from 52%. (A significant difference is one for which the p-value is less than or equal to 0.05.) 2. The percentage is significantly different from 52%.  Fail to reject H 0. There is insufficient evidence to make a conclusion about the percentages being different.

Copyright © 2014 Pearson Education, Inc. All rights reserved Chapter 8 Guided Exercise 2

8 - 70 Copyright © 2014 Pearson Education, Inc. All rights reserved  Perform a hypothesis test to test whether those who take Vioxx have a greater rate of heart attack than those who take a placebo. Use a level of significance of 0.05. Can we conclude that Vioxx causes an increased risk of heart attack?

8 - 71 Copyright © 2014 Pearson Education, Inc. All rights reserved The Study  In the fall of 2004, drug manufacturer Merck Pharmaceutical withdrew Vioxx, a drug that had been used for arthritis pain, from the market after a study revealed that its use was associated with an increase in the risk of heart attack. The experiment was placebo-controlled, randomized, and double- blind. Out of 1287 people taking Vioxx there were 45 heart attacks, and out of 1299 people taking the placebo there were 25 heart attacks (Source: Los Angeles Times, October 23, 2004).

8 - 72 Copyright © 2014 Pearson Education, Inc. All rights reserved Sample Proportions  Find the proportion of people in the sample taking Vioxx who had a heart attack and the proportion of people in the sample taking a placebo who had a heart attack. Compare these proportions.   There is a clear practical significance with 3.5% much higher than 1.9%.

8 - 73 Copyright © 2014 Pearson Education, Inc. All rights reserved Hypotheses  Let p V be the population proportion of those taking Vioxx who had a heart attack, and let p p be the population proportion of those taking the placebo who had a heart attack.  H 0 : p V = p p  H a : p V > p p

8 - 74 Copyright © 2014 Pearson Education, Inc. All rights reserved Prepare  Choose the two-proportion z -test. Although we don’t have a random sample, we have random assignment to groups. The pooled proportion of heart attacks is

8 - 75 Copyright © 2014 Pearson Education, Inc. All rights reserved Compute and Compare  z ≈ 2.46  p-value = 0.0069

8 - 76 Copyright © 2014 Pearson Education, Inc. All rights reserved Interpret  Reject or do not reject H 0 and choose i or ii:  Those taking Vioxx did not have a significantly different rate of heart attack than those taking the placebo.  Those who took Vioxx had a significantly higher heart attack rate than those who took the placebo.  Since p-value = 0.0069, we reject H 0 and accept H a. There is statistically significant evidence to support the claim that Vioxx users have a higher heart attack rate compared to non-Vioxx users.  Note that this does not mean that Vioxx causes heart attacks.

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