Chapter 20 Chapter20 Presentation 1213 Testing Hypotheses About Proportions Copyright 2009 Pearson Education Inc 1 A 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 1213 Copyright 2009 Pearson Education Inc 2 Criminal Trial vs Statistical Hypothesis Testing Criminal Trial Initial Hypothesis Challenge the Hypothesis If given the evidence data the Initial Hypothesis seems unlikely to be true If given the evidence data the Initial Hypothesis seems plausible Chapter20 Presentation 1213 Statistical Hypothesis Testing The suspect is innocent The population proportion is 0 20 or The population mean is 100 5 mm or the population The prosecution presents evidence Data are collected the sample proportion or the sample mean or the sample is calculated Suspect is declared guilty The Initial Hypothesis is rejected Continue to believe the suspect is innocent Continue to believe that The population proportion is 0 20 or The population mean is 100 5 mm or Copyright 2009 Pearson Education Inc 3 Statistical 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 1213 Copyright 2009 Pearson Education Inc 4 Alternative Alternatives There are three possible alternative hypotheses HA parameter hypothesized value HA parameter hypothesized value HA parameter hypothesized value Chapter20 Presentation 1213 Copyright 2009 Pearson Education Inc 5 Testing Hypotheses Once we have both hypotheses we collect sample data We will compare our data to what we would expect given that H0 is 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 1213 Copyright 2009 Pearson Education Inc 6 P Values 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 Chapter20 Presentation 1213 Copyright 2009 Pearson Education Inc 7 P Values cont 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 Chapter20 Presentation 1213 Copyright 2009 Pearson Education Inc 8 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 Chapter20 Presentation 1213 Copyright 2009 Pearson Education Inc 9 The Reasoning of Hypothesis Testing cont 1 Hypotheses review The null hypothesis a statement about a parameter in a statistical model The alternative hypothesis the values of the parameter we consider plausible if we reject the null Chapter20 Presentation 1213 In general we have H0 parameter hypothesized value In general we have three options HA parameter hypothesized value HA parameter hypothesized value HA parameter hypothesized value Copyright 2009 Pearson Education Inc 10 The 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 oneproportion z test Chapter20 Presentation 1213 Copyright 2009 Pearson Education Inc 11 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 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 1213 Copyright 2009 Pearson Education Inc 12 The Reasoning of Hypothesis Testing cont 3 Mechanics Mechanics include the calculation of our test statistic from the data Chapter20 Presentation 1213 Different tests will have different formulas and different test statistics Copyright 2009 Pearson Education Inc 13 The Reasoning of Hypothesis Testing cont 3 Mechanics Mechanics also include the calculation of the P value Chapter20 Presentation 1213 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 Copyright 2009 Pearson Education Inc 14 The 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 reject or 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 1213 Copyright 2009 Pearson Education Inc 15 Alternative Hypotheses and P Values Recall there are three possible alternative hypotheses HA parameter hypothesized value HA parameter hypothesized value HA parameter hypothesized value Chapter20 Presentation 1213 Copyright 2009 Pearson Education Inc 16 Alternative Hypotheses and P Values Cont HA parameter value is known as a two sided alternative 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 Presentation 1213 Copyright 2009 Pearson Education Inc 17 Alternative Hypotheses and P Values Cont In the chapter example we have the following H0 p 0 20 HA p 0 20 Actual sample proportion p 0 17 A p just as unusual as 0 17 Actual p The p value is 2 0 067 0 134 Chapter20 Presentation 1213 Copyright 2009 Pearson Education Inc 18 Alternative Hypotheses and P Values Cont The other two alternative hypotheses are