Copyright © 2009 Pearson Education, Inc. Chapter 20 Testing Hypotheses About Proportions

Copyright © 2009 Pearson Education, Inc. Slide 1- 3 Objectives: The student will be able to: Perform a one-proportion z-test, to include: writing appropriate hypotheses, checking the necessary assumptions, drawing an appropriate diagram, computing the P-value, making a decision, and interpreting the results in the context of the problem.

Copyright © 2009 Pearson Education, Inc. Motivating Example A 1996 report from the U.S. Consumer Product Safety Commission claimed that at least 90% of all American homes have at least one smoke detector. A city’s fire department has been running a public safety campaign about smoke detectors consisting of posters, billboards, and ads on radio, TV, and in the newspaper. The city wonders if this concerted effort has raised the local level above the 90% national rate. Building inspectors visit 400 randomly selected homes and find that 376 have smoke detectors. Is this strong evidence that the local rate is higher than the national rate? Slide 1- 4

Copyright © 2009 Pearson Education, Inc. Slide 1- 5 Hypotheses Hypotheses are working models that we adopt temporarily. Our starting hypothesis is called the null hypothesis. The null hypothesis, that we denote by H 0, specifies a population model parameter of interest and proposes a value for that parameter. We usually write down the null hypothesis in the form H 0 : parameter = hypothesized value. The alternative hypothesis, which we denote by H A, contains the values of the parameter that we consider plausible when we reject the null hypothesis.

Copyright © 2009 Pearson Education, Inc. Slide 1- 6 Testing Hypotheses The null hypothesis, specifies a population model parameter of interest and proposes a value for that parameter. In our example, H 0 : p = In other words, the population parameter for the city is the same as the national level of 90%. We want to compare our data to what we would expect given that H 0 is true. We can do this by finding out how many standard deviations away from the proposed value we are. We then ask how likely it is to get results like we did if the null hypothesis were true.

Copyright © 2009 Pearson Education, Inc. Slide 1- 7 A Trial as a Hypothesis Test Think about the logic of jury trials: To prove someone is guilty, we start by assuming they are innocent. We retain that hypothesis until the facts make it unlikely beyond a reasonable doubt. Then, and only then, we reject the hypothesis of innocence and declare the person guilty.

Copyright © 2009 Pearson Education, Inc. Slide 1- 8 The same logic used in jury trials is used in statistical tests of hypotheses: We begin by assuming that a hypothesis is true. Next we consider whether the data are consistent with the hypothesis. If they are, all we can do is retain the hypothesis we started with. If they are not, then like a jury, we ask whether they are unlikely beyond a reasonable doubt. A Trial as a Hypothesis Test (cont.)

Copyright © 2009 Pearson Education, Inc. Slide 1- 9 The statistical twist is that we can quantify our level of doubt. We can use the model proposed by our hypothesis to calculate the probability that the event we’ve witnessed could happen. That’s just the probability we’re looking for—it quantifies exactly how surprised we are to see our results. This probability is called a P-value. P-Values

Copyright © 2009 Pearson Education, Inc. Slide When the data are consistent with the model from the null hypothesis, the P-value is high and we are unable to reject the null hypothesis. In that case, we have to “retain” the null hypothesis we started with. We can’t claim to have proved it; instead we “fail to reject the null hypothesis” when the data are consistent with the null hypothesis model and in line with what we would expect from natural sampling variability. If the P-value is low enough, we’ll “reject the null hypothesis,” since what we observed would be very unlikely were the null model true. P-Values (cont.)

Copyright © 2009 Pearson Education, Inc. Slide What to Do with an “Innocent” Defendant If the evidence is not strong enough to reject the presumption of innocent, the jury returns with a verdict of “not guilty.” The jury does not say that the defendant is innocent. All it says is that there is not enough evidence to convict, to reject innocence. The defendant may, in fact, be innocent, but the jury has no way to be sure.

Copyright © 2009 Pearson Education, Inc. Slide What to Do with an “Innocent” Defendant (cont.) Said statistically, we will fail to reject the null hypothesis. We never declare the null hypothesis to be true, because we simply do not know whether it’s true or not. Sometimes in this case we say that the null hypothesis has been retained.

Copyright © 2009 Pearson Education, Inc. Slide The Reasoning of Hypothesis Testing There are four basic parts to a hypothesis test: 1. Hypotheses 2. Model 3. Mechanics 4. Conclusion Let’s look at these parts in detail…

Copyright © 2009 Pearson Education, Inc. Slide The Reasoning of Hypothesis Testing (cont.) 1.Hypotheses The null hypothesis: To perform a hypothesis test, we must first translate our question of interest into a statement about model parameters. In general, we have H 0 : parameter = hypothesized value. The alternative hypothesis: The alternative hypothesis, H A, contains the values of the parameter we consider plausible if we reject the null.

Copyright © 2009 Pearson Education, Inc. Slide Alternative Alternatives There are three possible alternative hypotheses: H A : parameter < hypothesized value H A : parameter ≠ hypothesized value H A : parameter > hypothesized value

Copyright © 2009 Pearson Education, Inc. Slide Alternative Alternatives (cont.) H A : parameter ≠ value is known as a two-sided alternative because we are equally interested in deviations on either side of the null hypothesis value. For two-sided alternatives, the P-value is the probability of deviating in either direction from the null hypothesis value.

Copyright © 2009 Pearson Education, Inc. Slide Alternative Alternatives (cont.) The other two alternative hypotheses are called one-sided alternatives. A one-sided alternative focuses on deviations from the null hypothesis value in only one direction. Thus, the P-value for one-sided alternatives is the probability of deviating only in the direction of the alternative away from the null hypothesis value.

Copyright © 2009 Pearson Education, Inc. Slide The Reasoning of Hypothesis Testing (cont.) 2.Model To plan a statistical hypothesis test, specify the model you will use to test the null hypothesis and the parameter of interest. All models require assumptions, so state the assumptions and check any corresponding conditions. Your plan should end with a statement like Because the conditions are satisfied, I can model the sampling distribution of the proportion with a Normal model. Watch out, though. It might be the case that your model step ends with “Because the conditions are not satisfied, I can’t proceed with the test.” If that’s the case, stop and reconsider.

Copyright © 2009 Pearson Education, Inc. Slide The Reasoning of Hypothesis Testing (cont.) 2.Model Each test we discuss in the book has a name that you should include in your report. The test about proportions is called a one- proportion z-test. Right now this is an easy choice, but later on it will be harder to decide – ask yourself are you looking at means or proportions? Do you have one group or two? If two, do you have independent or matched samples? In our case we have one group and we are looking at proportions.

Copyright © 2009 Pearson Education, Inc. Slide One-Proportion z-Test The conditions for the one-proportion z-test are the same as for the one proportion z-interval. (Independence, Randomization Condition, 10% Condition, and Success/Failure Condition) We test the hypothesis H 0 : p = p 0 using the statistic where When the conditions are met and the null hypothesis is true, this statistic follows the standard Normal model, so we can use that model to obtain a P-value. Notice that here we are finding the SD(p^) not the SE(p^)

Copyright © 2009 Pearson Education, Inc. Motivating Example (again) A 1996 report from the U.S. Consumer Product Safety Commission claimed that at least 90% of all American homes have at least one smoke detector. A city’s fire department has been running a public safety campaign about smoke detectors consisting of posters, billboards, and ads on radio, TV, and in the newspaper. The city wonders if this concerted effort has raised the local level above the 90% national rate. Building inspectors visit 400 randomly selected homes and find that 376 have smoke detectors. Is this strong evidence that the local rate is higher than the national rate? Slide 1- 21

Copyright © 2009 Pearson Education, Inc. Example – Step 1: Hypothesis Step 2: Model H 0 : p = 0.90 H A : p>0.90 The one proportion z-test is needed, but first we need to check its conditions: Independence assumption (Is it reasonable to assume that the randomly selected homes are independent)? Randomization Condition (Is this a random sample)? 10% condition (Are the 400 homes inspected less than 10% of the total city population)? Success/Failure Condition: is np≥10 and is nq≥10? These conditions are satisfied so it is okay to use a Normal model and perform a one-proportion z-test. Slide 1- 22

Copyright © 2009 Pearson Education, Inc. Slide The Reasoning of Hypothesis Testing (cont.) 3.Mechanics Under “mechanics” we place the actual calculation of our test statistic from the data. Different tests will have different formulas and different test statistics. Usually, the mechanics are handled by a statistics program or calculator, but it’s good to know the formulas.

Copyright © 2009 Pearson Education, Inc. Slide The Reasoning of Hypothesis Testing (cont.) 3.Mechanics The ultimate goal of the calculation is to obtain a P-value. The P-value is the probability that the observed statistic value (or an even more extreme value) could occur if the null model were correct. If the P-value is small enough, we’ll reject the null hypothesis. Note: The P-value is a conditional probability—it’s the probability that the observed results could have happened if the null hypothesis is true.

Copyright © 2009 Pearson Education, Inc. Example: Step 3: Mechanics Mechanics: n=400, x=376. p^=376/400=0.94 p 0 =0.9 and q 0 =0.1 Draw a curve for the Normal model of p^ under the null hypothesis. Calculate the value of the z-test statistic, then find the associated P-value. The P-value is the probability that that the observed statistic p^ or a more extreme value would occur if the null model is correct. Use the normal table or your calculators distribution function Slide 1- 25

Copyright © 2009 Pearson Education, Inc. Example: Step 3: Mechanics Mechanics: n=400, x=376. p^=376/400=0.94 p 0 =0.9 and q 0 =0.1 Draw a curve for the Normal model of p^ under the null hypothesis. Calculate the value of the z-test statistic, then find the associated P-value. z = (0.94 – 0.90) / √( (0.9)(0.1) / 400 ) = 2.67 P = P(z>2.67) = Slide 1- 26

Copyright © 2009 Pearson Education, Inc. Using the TI-83+ Stat…Tests…1-PropZtest p0 is the hypothecated proportion.90 x: is the number of favorable observations from the sampling 376 n: is the number in your sample 400 prop: choose the sign of the alternative hypothesis: > Choose calculate Slide 1- 27

Copyright © 2009 Pearson Education, Inc. Slide The Reasoning of Hypothesis Testing (cont.) 4.Conclusion The conclusion in a hypothesis test is always a statement about the null hypothesis. The conclusion must state either that we reject or that we fail to reject the null hypothesis. And, as always, the conclusion should be stated in context.

Copyright © 2009 Pearson Education, Inc. Slide The Reasoning of Hypothesis Testing (cont.) 4.Conclusion Your conclusion about the null hypothesis should never be the end of a testing procedure. Often there are actions to take or policies to change.

Copyright © 2009 Pearson Education, Inc. Slide P-Values and Decisions: What to Tell About a Hypothesis Test How small should the P-value be in order for you to reject the null hypothesis? It turns out that our decision criterion is context- dependent. When we’re screening for a disease and want to be sure we treat all those who are sick, we may be willing to reject the null hypothesis of no disease with a fairly large P-value. A longstanding hypothesis, believed by many to be true, needs stronger evidence (and a correspondingly small P-value) to reject it. The threshold P-value for rejecting the null hypothesis is often denoted alpha α and is often 0.05 or 0.01 Another factor in choosing a P-value is the importance of the issue being tested.

Copyright © 2009 Pearson Education, Inc. Slide P-Values and Decisions (cont.) Your conclusion about any null hypothesis should be accompanied by the P-value of the test. If possible, it should also include a confidence interval for the parameter of interest. Don’t just declare the null hypothesis rejected or not rejected. Report the P-value to show the strength of the evidence against the hypothesis. This will let each reader decide whether or not to reject the null hypothesis.

Copyright © 2009 Pearson Education, Inc. Example – Step 4: Conclusion Conclusion: Because the P-value of is very low, it is unlikely that the observed results can be explained by sampling error. We reject the null hypothesis. There is strong evidence that in this city more than 90% of homes have smoke detectors Go a step further -> interpret P and give a confidence interval for the observed parameter. Use the SE(p^) as learned in chapter 19. Further, If in fact the rate in this city is 90% (the same as the national rate) that would mean that only 0.4% of samples (about 1 in 250) would find (as ours did) a proportion of 94% or more. While that could have happened, it is not the most logical explanation. In fact we are 95% confident that between 91.7% and 96.3% of the city’s homes have smoke detectors. Slide 1- 32

Copyright © 2009 Pearson Education, Inc. Hypothesis Writing Examples Write the null and alternative hypotheses and draw the appropriate curve. In the 1950s only about 40% of high school graduates went on to college. Has the percentage changed? 20% of the cars of a certain model have needed costly transmission work after being driven between 50,000 and 100,000 miles. The manufacturer hopes that the redesign of the transmission has solved the problem. We field test a new flavor of soft drink, planning to market it only if we are sure that at least 60% of the people like the flavor. Slide 1- 33

Copyright © 2009 Pearson Education, Inc. Hypothesis Writing Examples Write the null and alternative hypotheses and draw the appropriate curve. The drug Lipitor is meant to lower cholesterol. Is there evidence to support the claim that over 1.9% of the users experience flu like symptoms as a side effect? According to the US dept. of Health, 16.3% of Americans did not have health insurance coverage in A politician claims that this percentage has decreased since During the past 40 years, the monthly rate of return for a particular item has been 4.2 percent. A store analyst claims that it is different. Slide 1- 34

Copyright © 2009 Pearson Education, Inc. Practice using the TI-83+ It is believed that the percent of convicted felons who have a history of juvenile delinquency is 70%. Is there evidence to support the claim the the actual percentage is more than the 70% if out of 200 convicted felons, we find that 154 have a history of juvenile delinquency? Alpha =.05 Stat…Tests…1-PropZtest p0 is the hypothecated proportion.70 x: is the number of favorable observations from the sampling 154 n: is the number in your sample 200 prop: choose the sign of the alternative hypothesis- > Choose calculate You should get a p value of.015 therefore, reject the null hypothesis which means that we reject the null hypothesis. The claim that more than 70% of the felons have a history of juvenile delinquency is supported by the data. Slide 1- 35

Copyright © 2009 Pearson Education, Inc. More examples There are supposed to be 20% orange M&Ms in a bag. Suppose a bag of 122 has only 21 orange ones. Does this contradict the company’s 20% claim? Note that this is a 2-tailed test… so H 0 : p=0.20 and H A : p≠0.20. Also, the P-value will now be 2xP(z<z-test statistic). Check the model assumptions/conditions! Slide 1- 36

Copyright © 2009 Pearson Education, Inc. Practice In the 1980s it was generally believed that autism affected about 6% of the nation’s children. Some people believe that the increase in the number of chemicals in the environment has led to an increase in the incidence of autism. A recent study examined 384 children and found that 46 of them showed signs of some form of autism. Is there strong evidence that the level of autism has increased? (Use an alpha of.05) Write the hypotheses, check the assumptions, draw the curve, find the pertinent statistics and critical values, find the p value, state your conclusion, etc. H0: p=0.06 HA: p>0.06 Test statistic = z = 4.93 P-value = (i.e. a really small #) Conclusion: Since the P-value is less than alpha, we can reject the hypothesis that the true population proportion of kids with autism is 6%. The statistical evidence indicates that the true rate of autism has probably increased. Slide 1- 37

Copyright © 2009 Pearson Education, Inc. 14. During the 2000 season, the home team won 138 of the 240 regular season games. Is this strong evidence of a home field advantage? (Let alpha=0.05) H0: p=0.50 HA: p>0.50 Test statistic = z = 2.32 P-value = 0.01 Conclusion: Since the P-value is less than alpha, we can reject the hypothesis that there is no home field advantage. Therefore, the statistical evidence suggests that there IS a home field advantage. Slide 1- 38

Copyright © 2009 Pearson Education, Inc. 15. A garden center wants to store leftover packets of vegetable seeds for sale the following spring but the center is concerned that the seeds may not germinate at the same rate a year later. The manager finds a packet from the previous year and plants the seeds as a test. Although the package claims a rate of at least 92% only 171 of the 200 seeds sprout. Is this evidence that the seeds have lost their viability during the year in storage? (Let alpha=0.05) H0: p=0.92 HA: p<0.92 Test statistic = z = P-value = Conclusion: Since the P-value is smaller than alpha, we reject the hypothesis that the true percentage of germinated seeds is 92%. The statistical evidence suggests that the seeds lose their effectiveness after a year. Slide 1- 39

Copyright © 2009 Pearson Education, Inc. More examples There are supposed to be 20% orange M&Ms in a bag. Suppose a bag of 122 has only 21 orange ones. Does this contradict the company’s 20% claim? In groups, repeat this example, but assume that there are supposed to be 5%, 10%, 15%, or 22% orange M&Ms. Write Hypotheses Check Model Mechanics Conclusion What is the 95% confidence interval on p 0 ? Slide 1- 40