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 Review #1 The following graphics summarize hypothetical results from a well designed experiment. For each of the graphics below, identify the name of the graphic, describe the distributions, and identify sources of variation. Be sure to use proper statistical vocabulary.

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

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Review #2: Statistical Notation Consider the following situation: To compare the effectiveness of two types of psychotherapy in the treatment of depression, two groups of 30 individuals were selected from a large group of persons known to be suffering from depression. Subjects were randomly assigned to two groups: cognitive-behavioral therapy (CBT) or psychonanalysis treatments. Each individual completed a set regimen of therapy and then was given a depression test that produced a depression score from (a) What does x-bar CBT refer to? (b) What does μ CBT refer to?

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Review #3: Statistical Notation Consider the following situation: Scientists want to know how many cars running on regular gasoline meet emissions standards compared to those running on premium. They take 200 cars randomly selected from all makes & models. They then run 100 of them on regular gasoline and the other 100 on premium for 6 weeks and measure emissions. Based on these emission levels, they then tally the number of cars meeting emissions in both groups. (a) What does p-hat regular refer to? (b) What does p premium refer to?

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 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 Assumptions of the One-Proportion z-Test 1) Independence of Observations Plausible Independence Condition Randomization Condition 10% Condition 2) Sample Size np 0 >= 10 nq 0 >= 10

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-value <= α, reject the Ho. If p-value > 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)*100%] 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 Interpreting the p-value For p-value =.0489 There is a 4.89% 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.) For 2-sided alternatives, we divide the significance level, α, by 2 and place α/2 in each tail. IMPORTANT: 2-sided tests have an extra step at the end: After finding the z-test and the associated p-value, you MUST multiply the p-value you find by 2 to compare it to α.

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Alternative Alternatives (cont.) The regions shaded in red below are called the rejection regions, because, if the z-test statistic that you compute falls in either of these regions, we reject the H 0. Find the critical value of a z-test of one proportion by finding the z-score that delimits the left-tailed rejection region.

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 Alternative Alternatives (cont.) Find the critical value of a 1-sided z-test of one proportion by finding the z-score that delimits the rejection region. If the rejection region falls in the right tail, take the absolute value of the z-score that delimits the region.

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide More About Statistical decision making If the z-test statistic is more extreme (more positive or more negative) than the critical value for the test, reject the Ho. If |z p-hat | >= z*, reject the Ho. If |z p-hat | < z*,, fail to reject the Ho.

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 comment on whether or not the results are statistically significant. 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.