Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 20- 1

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 20 Testing Hypotheses About Proportions

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Hypothesis Testing Motivating Example People are concerned about the national high school dropout rate: many people believe that the US is producing less and less high school graduates. The US Department of Ed reported that the national high school dropout rate for the year 2000 was 10.9%. A random sample of 1782 high school students across the nation showed that 210 had dropped out in the year Is there evidence to conclude that the dropout rate has increased?

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Hypothesis Testing Motivating Example 2 A marketing firm has been hired to evaluate the viability of the market for a new external computer hard drive. According to the firm’s research, 76% of consumers expressed interest in the old model. The firm obtains a random sample of n = 2,500 potential consumers and finds that the 82% of the sampled consumers are interested in the new model. Is the interest in the new hard drive greater than interest in the old hard drive?

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Hypothesis Testing Motivating Example 3 Reliance Electric produces industrial grade motors used in drilling and construction applications. They recently spent roughly a million dollars on upgrading the equipment on all of the production lines at their Athens, GA plant. The quality control manager collected data before and after the upgrade. Immediately before the upgrade, the plant was finding that 7.6% of the motors did not meet minimum standards. After the upgrade, a random sample of n = 120 motors showed that 6.2% of motors did not meet the standards. Has the proportion of bad motors decreased?

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Hypothesis Testing Motivating Example 4 Registered Nurses (RN’s) are in extremely high demand at present across the nation. As a result, many nursing schools receive more applications than students they can admit for training. In 2005, the Berry Nursing School accepted 57% of their applicants. Last year, the school accepted 20 applicants from a random sample of n = 33 of their total applicants. Is there statistical evidence to conclude that the acceptance rate has changed?

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Hypothesis Testing How can we make a scientific decision concerning a possible change in the national high school dropout rate when we only have a small sample (N=1782) of the large population of high school students across the nation? ANSWER: Use inferential statistics and our knowledge of sampling distributions to measure or quantify the amount of evidence for whether the dropout rate has increased. We could build a confidence interval (CI), but remember, CI’s are especially well-suited for estimating population parameters. Instead, what is often done is to perform a hypothesis test.

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Hypotheses In Statistics, a hypothesis proposes a model for the world. Then we look at the data. If the data are consistent with that model, we have no reason to disbelieve the hypothesis. Data consistent with the model lend support to the hypothesis, but do not prove it. But if the facts are inconsistent with the model, we need to make a choice as to whether they are inconsistent enough to disbelieve the model. If they are inconsistent enough, we can reject the model.

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Hypotheses (cont.) 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 © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Hypotheses (cont.) 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.

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Hypotheses (cont.) 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.

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Testing Hypotheses The null hypothesis, which we denote H 0, specifies a population model parameter of interest and proposes a value for that parameter. We might have, for example, H 0 : p = 0.20, as in the chapter example. 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 © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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 accept if we reject the null.

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Writing hypothesis statements There are three possible sets of hypothesis statements: Ho: parameter >= hypothesized value H A : parameter < hypothesized value Ho: parameter = hypothesized value H A : parameter ≠ hypothesized value Ho: parameter <= hypothesized value H A : parameter > hypothesized value

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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.

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide One-Proportion z-Test The conditions for the one-proportion z-test are the same as for the one proportion z-interval. 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.

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide The Reasoning of Hypothesis Testing (cont.) 3.Mechanics The ultimate goal of the calculation is to make a decision about the Ho using 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 © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Statistical decision making using p-values If the p-value is small, reject the Ho. “If the p-value is low, the null (Ho) must go.” If the p-value is large, fail to reject the Ho.

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Statistical decision making using p-values If p <= α, reject the Ho. If p > α, fail to reject the Ho.

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide What is α? α (alpha) = the significance level – this is the criterion that we will use to decide whether the p- value is small or large. α = (1-CL) Example, if CL = 98%, then α = (1-.98) =.02. Therefore, if the p-value is equal to or less than.02, we will consider the p-value to be small, AND reject Ho.

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide More about statistical decision making If the p-value <= α, the results are unlikely to happen simply due to sampling error. If it is unlikely that the results are due to sampling error, then we conclude that they must be due to some change in the true population proportion. These results are called “statistically significant”.

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Interpreting the p-value There is a (p-value) chance of obtaining the calculated z-test value or one more extreme simply due to sampling error given that the Ho is true.

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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 © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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 © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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 (in other words, we are equally interested in increases or decreases from the amount specified in the Ho. For two-sided alternatives, the p-value is the probability of deviating in either direction from the null hypothesis value.

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Alternative Alternatives (cont.) A one-sided alternative focuses on deviations from the null hypothesis value in only one direction (an increase OR a decrease, but NOT both). 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 © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Summarizing Conclusions We fail to reject the Ho. There is insufficient evidence to conclude (claim); z=x.xx, p=.xxx. OR We reject the Ho. There is sufficient evidence to conclude (claim); z=x.xx, p=.xxx.

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Components of your conclusion A decision about the Ho. A statement about the amount of evidence for the claim that explicitly re-states the context of the problem. The calculated z-test and corresponding p- value.