1Chapter20 Presentation 1213Copyright © 2009 Pearson Education, Inc.Chapter 20 Testing Hypotheses About ProportionsChapter20 Presentation 1213Copyright © 2009 Pearson Education, Inc.2A Trial as a Hypothesis Test Think about the logic of criminal jury trials in the USA:  All suspects are presumed innocent until proven guilty “beyond a reasonable doubt”.Chapter20 Presentation 1213Copyright © 2009 Pearson Education, Inc.3Criminal Trial vs. Statistical Hypothesis TestingCriminal TrialStatistical Hypothesis TestingInitial HypothesisThe suspect is innocentThe population proportion is 0.20orThe population mean is 100.5 mmor the population…..Challenge the HypothesisThe prosecutionpresents evidenceData are collected, the sample proportion or the sample mean or the sample…. is calculatedIf, given the evidence (data), the Initial Hypothesis seems unlikely to be true:Suspect is declared guiltyThe Initial Hypothesis is “rejected”If, given the evidence (data), the InitialHypothesis seems plausible:Continue to believe the suspect is innocentContinue to believe thatThe population proportion is 0.20orThe population mean is 100.5 mmor…..Chapter20 Presentation 1213Copyright © 2009 Pearson Education, Inc.4Statistical Hypotheses Our “Initial Hypothesis” is called the null hypothesis.  Notation H0: population parameter = hypothesized value. The alternative hypothesis, which we denote by HA, contains the values of the parameter that we consider plausible when we reject the null hypothesis.Chapter20 Presentation 1213Copyright © 2009 Pearson Education, Inc.5Alternative Alternatives There are three possible alternative hypotheses: HA: parameter ≠ hypothesized value HA: parameter < hypothesized value HA: parameter > hypothesized valueChapter20 Presentation 1213Copyright © 2009 Pearson Education, Inc.6Testing Hypotheses Once we have both hypotheses, we collect sample data. We will compare our data to what we would expect given that H0is true.  We calculate the probability of getting the sample data we got (or results more unusual than that) if the null hypothesis were true. To calculate this probability, we start by calculate how many standard deviations the sample statistic is from the proposed population parameter.Chapter20 Presentation 1213Copyright © 2009 Pearson Education, Inc.7 We can use our understanding of sampling distributions to calculate the probability of our sample statistic being that many (or more) standard deviations from the proposed population parameter. This probability is called a P-value.P-ValuesChapter20 Presentation 1213Copyright © 2009 Pearson Education, Inc.8 When the P-value is large, say, above 0.05, we are unable to reject the null hypothesis. We can’t claim to have proved it; instead we say we “fail to reject the null hypothesis”. When the P-value is small, say, 0.05 or less, we say we “reject the null hypothesis”, since what we observed would be very unlikely were the null hypothesis true.P-Values (cont.)Chapter20 Presentation 1213Copyright © 2009 Pearson Education, Inc.9The Reasoning of Hypothesis Testing There are four basic parts to a hypothesis test:1. Hypotheses2. Model3. Mechanics4. Conclusion Let’s look at these parts in detail…Chapter20 Presentation 1213Copyright © 2009 Pearson Education, Inc.10The Reasoning of Hypothesis Testing (cont.)1. Hypotheses (review) The null hypothesis: a statement about a parameter in a statistical model.  In general, we have H0: parameter = hypothesized value. The alternative hypothesis: the values of the parameter we consider plausible if we reject the null. In general, we have three options: HA: parameter ≠ hypothesized valueHA: parameter > hypothesized valueHA: parameter < hypothesized valueChapter20 Presentation 1213Copyright © 2009 Pearson Education, Inc.11The Reasoning of Hypothesis Testing (cont.)2. Model Depending on what type of population parameter you are testing, a sampling distribution model is used to compare the sample statistic to the corresponding hypothesized population parameter. All models require assumptions, so state the assumptions and check any corresponding conditions. The test about proportions is called a one-proportion z-test.Chapter20 Presentation 1213Copyright © 2009 Pearson Education, Inc.12One-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 H0: p = p0using the “test statistic”where When the conditions are met and the null hypothesis is true, the sampling distribution of the sample proportion follows the standard Normal model, so we can use that model to obtain a P-value.Chapter20 Presentation 1213Copyright © 2009 Pearson Education, Inc.13The Reasoning of Hypothesis Testing (cont.)3. Mechanics Mechanics include the calculation of our test statistic from the data. Different tests will have different formulas and different test statistics.Chapter20 Presentation 1213Copyright © 2009 Pearson Education, Inc.14The Reasoning of Hypothesis Testing (cont.)3. Mechanics Mechanics also include the calculation of the P-value. The P-value is the probability that the observed test statistic value (or an even more extreme value) could occur if the null hypothesis were true.Chapter20 Presentation 1213Copyright © 2009 Pearson Education, Inc.15The Reasoning of Hypothesis Testing (cont.)4. Conclusion The conclusion always begins with a statement about the null hypothesis.  It must begin with a statement that we rejector that we fail to reject the null hypothesis. The remainder of the conclusion should be in easy to understand language, and in the context of the specific situation.Chapter20 Presentation 1213Copyright © 2009 Pearson Education, Inc.16Alternative Hypotheses and P-Values Recall, there are three possible alternative hypotheses: HA: parameter ≠ hypothesized value HA: parameter < hypothesized value HA: parameter > hypothesized valueChapter20 Presentation 1213Copyright © 2009 Pearson Education, Inc.17Alternative Hypotheses and P-Values (Cont.) HA: parameter ≠ value is known as a two-sidedalternative because we are equally interested in deviations on either side of the null hypothesized value.  For two-sided alternatives, the P-value is the probability of deviating in either direction from the null hypothesis value.Chapter20