Testing Hypotheses about a Population ProportionDiscovering Characteristics of a PopulationSlide 3ExamplesExampleSlide 6Slide 7Two Approaches for Hypothesis TestingSlide 9Classical ApproachSlide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17p-Value ApproachSlide 19Slide 20Slide 21Slide 22Slide 23Slide 24Slide 25The Steps of Testing a Hypothesis (p-Value Approach)Slide 27Step 1: State the Null and Alternative HypothesesState the Null and Alternative HypothesesSlide 30Explaining the DataStep 2: State the Significance LevelThe Sampling Distribution of p^Slide 34Step 3: The Test StatisticThe Test StatisticSlide 37Step 4: Compute the Test StatisticStep 5: Compute the p-valueCompute the p-valueSlide 41Step 6: Make a DecisionStep 7: State the ConclusionSummarySlide 45Slide 46Supporting DataTesting Hypotheses on the TI-83Slide 49Slide 50Exercise Helps Prevent ColdsSlide 52The Classical ApproachSlide 54Slide 55Example of the Classical ApproachSlide 57Slide 58Slide 59Slide 60Slide 61Testing Hypotheses about a Population ProportionLecture 31Sections 9.1 – 9.3Wed, Mar 21, 2007Discovering Characteristics of a PopulationAny question about a population must first be described in terms of a population parameter.We will work with the population mean and the population proportion p.Discovering Characteristics of a PopulationThen the question about that parameter generally falls into one of two categories.EstimationWhat is the value of the parameter?Hypothesis testingDoes the evidence support or refute a claim about the value of the parameter?ExamplesIf we want to learn about voters’ preferences, how do we phrase the question?What parameter do we use?Do we estimate a parameter or test a hypothesis?ExampleIf we want to learn about the effectiveness of a new drug, how do we phrase the question?What parameter do we use?Do we estimate a parameter or test a hypothesis?ExampleIf we want to find out whether a newborn child is more likely to be male than female, how do we phrase the question?What parameter do we use?Do we estimate a parameter or test a hypothesis?ExampleA standard assumption is that a newborn baby is as likely to be a boy as to be a girl. However, some people believe that boys are more likely. Suppose a random sample of 1000 live births shows that 520 are boys and 480 are girls.We will test the hypothesis that male births are as likely as female births, using these data.Two Approaches for Hypothesis TestingClassical approach.Specify .Determine the critical value and the rejection region.See whether the statistic falls in the rejection region.Report the decision.Two Approaches for Hypothesis Testingp-Value approach.Compute the p-value of the statistic.Report the p-value.If is specified, then report the decision.Classical ApproachH0Classical ApproachH0Classical Approach0zcH0Critical valueClassical Approach0zcRejection RegionAcceptance RegionH0Classical Approach0zcRejection RegionAcceptance RegionH0Classical Approach0zcRejection RegionAcceptance RegionRejectzH0Classical Approach0zcRejection RegionAcceptance RegionH0Classical Approach0zcRejection RegionAcceptance RegionAcceptzH0p-Value ApproachH0p-Value ApproachH0p-Value ApproachH0p-Value Approach0zH0zp-Value Approach0zRejectp-value < H0zp-Value Approach0zH0p-Value Approach0zH0zp-Value Approach0zp-value > H0AcceptzThe Steps of Testing a Hypothesis (p-Value Approach)The seven steps:1. State the null and alternative hypotheses.2. State the significance level.3. State the formula for the test statistic.4. Compute the value of the test statistic.5. Compute the p-value.6. Make a decision.7. State the conclusion.The Steps of Testing a Hypothesis (p-Value Approach)See page 566. (Our seven steps are modified from what is in the book.)Step 1: State the Null and Alternative HypothesesLet p = proportion of live births that are boys.The null and alternative hypotheses areH0: p = 0.50.H1: p > 0.50.State the Null and Alternative HypothesesThe null hypothesis should state a hypothetical value p0 for the population proportion.H0: p = p0.State the Null and Alternative HypothesesThe alternative hypothesis must contradict the null hypothesis in one of three ways:H1: p < p0. (Direction of extreme is left.)H1: p > p0. (Direction of extreme is right.)H1: p p0. (Direction of extreme is left and right.)Explaining the DataThe observation is 520 males out of 1000 births, or 52%. That is, p^ = 0.52.Since we observed 52%, not 50%, how do we explain the discrepancy?Chance, orThe true proportion is not 50%, but something larger, maybe 52%.Step 2: State the Significance LevelThe significance level should be given in the problem.If it isn’t, then use = 0.05.In this example, we will use = 0.05.The Sampling Distribution of p^To decide whether the sample evidence is significant, we will compare the p-value to . is the probability that the value that we observe is at least as extreme as the critical value(s), if the null hypothesis is true.Therefore, when we compute the p-value, we do it under the assumption that H0 is true, i.e., that p = p0.The Sampling Distribution of p^We know that the sampling distribution of p^ is normal with mean p and standard deviationThus, we assume that p^ has mean p0 and standard deviation: nppp1ˆ nppp00ˆ1Step 3: The Test StatisticTest statistic – The z-score of p^, under the assumption that H0 is true.Thus, npppppZpp000ˆˆ1ˆˆThe Test StatisticIn our example, we computeTherefore, the test statistic is .01581.0100050.1)50(.ˆp01581.050.0ˆpZThe Test StatisticNow, to find the value of the test statistic, all we need to do is to collect the sample data and substitute the value of p^.Step 4: Compute the Test StatisticIn the sample, p^ = 0.52.Thus,265.101581.050.052.0ZStep 5: Compute the p-valueTo compute the p-value, we must first check whether it is a one-tailed or a two-tailed test.We will compute the probability that Z would be at least as extreme as the value of our test statistic.If the test is two-tailed, then we must take into account both tails of the distribution to get the p-value. (Double the value in
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