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Notes 7: Hypothesis Testing for from a single sample (Z Test, T test, Bootstrap Test, and Power)Using Sample Information to Make Inferences about Population ParametersHypothesis Testing OutlineHypothesis Testing Step 1: Identify the Population Parameter of Interest and Describe it in the Context of the ProblemHypothesis Testing Step 2: State the Null Hypothesis and Alternative HypothesisExpected Errors in Hypothesis TestingHypothesis Testing Step 3: Check the Distributional AssumptionsHypothesis Testing Step 4: State and Compute the Appropriate Test Statistic Based on Distributional AssumptionsHypothesis Testing Step 5(a): Evaluate the Evidence Using the Rejection Region MethodHypothesis Testing Step 5(b): Evaluate the Evidence Using the P-value MethodOther Considerations when Conducting a Hypothesis TestPractical vs. Statistical SignificancePowerDetermining the Sample Size Required to Achieve Desired PowerThe Relationship Between Confidence Intervals and Hypothesis TestsBootstrap Hypothesis TestsMore T/Z/Bootstrap Hypothesis Test PracticeStatistics 371: Notes7 University of Wisconsin–MadisonNOTES 7: HYPOTHESIS TESTING FOR μ FROM A SINGLE SAMPLE (Z TEST, T TEST, BOOTSTRAP TEST, AND POWER)USING SAMPLE INFORMATION TO MAKE INFERENCES ABOUT POPULATION PARAMETERSWe often want to know what is true for a population of subjects. When it is impossible to obtain a census and summarize the population directly, we must rely on statistical inference. In this inference section we continue to apply the probabilityand simulation ideas we studied previously. We will be developing techniques to determine if a claim or conjecture about a parameter of our population of interest is supported or refuted by what is observed in our sample. We will also consider the limitations of our statistical inference. Statistical inference is the practice of making informed estimates of population parameters or relationships based on sample data.Tests of statistical hypotheses (hypothesis tests) examine whether sample data support or contradict a claim about the true value of a parameter or relationship. These tests begin with making a statement about a population parameter or relationship.Estimation of parameters is obtaining an estimate of the unknown parameter (this is typically accomplished with a confidence interval)HYPOTHESIS TESTING OUTLINESteps for testing a hypothesis about a parameter:(1) Identify the population parameter of interest in the context of the problem (we’ll start with μ in this note set)(2) State the null and alternative hypothesis(3) Check the distributional assumptions(4) State and compute the appropriate test statistic based on distributional assumptions(5) Evaluate the evidence based on the test statistica. Using the rejection region methodb. Using the p-value methodc. State the conclusion in the context of the problem1Statistics 371: Notes7 University of Wisconsin–MadisonHYPOTHESIS TESTING STEP 1: IDENTIFY THE POPULATION PARAMETER OF INTEREST AND DESCRIBE IT IN THE CONTEXT OF THE PROBLEMA statistical hypothesis is a claim or assertion about the value of a single population parameter, about the values of several population parameters, or about the form of an entire probability distribution.• May want to test the hypothesis that μ=5 where μ is the true population average• May want to test the hypothesis that p=.5 where p is the true population proportion• May want to test the hypothesis that μ1−μ2=0 where μ1 and μ2 are the true population averages for populations 1 and 2, respectivelyIn this note set we will focus on hypothesis tests about the value of the population mean (or average), μ .Paint Thickness example (a). The paint thicknesses (measured in mm) for 16 blocks randomly chosen over the course of the day were measured. The observed thicknesses are:2.25, 2.77, 2.54, 2.32, 2.03, 2.66, 2.99, 2.04, 2.53, 3.03, 2.74, 2.94, 3.03, 2.27, 3.22, 2.23Summary statistics: x=2.599 , s=0.3801Do these data give enough evidence that the population average paint thickness is different from 2.50 mm? If so, we may decide to recalibrate the machine. Step 1: The parameter of interest is the true population average paint thickness, μT2Statistics 371: Notes7 University of Wisconsin–MadisonHYPOTHESIS TESTING STEP 2: STATE THE NULL HYPOTHESIS AND ALTERNATIVE HYPOTHESISThe null hypothesis, H0, is the claim that is assumed to be true. -H0 is assumed to be true unless the sample provides sufficient evidence to the contrary -H0 statements regarding a single parameter always include the equality (e.g.,Ho: μG=30)- If the sample does not offer strong evidence against the null, we say we “fail to reject the null hypothesis”The alternative hypothesis, HA, is the claim that is contradictory to H0- Often a research hypothesis offers a specific range of values for a parameter (e.g., HA: μG>30 ¿- An alternative hypothesis can take one of three forms (where μ0 is referred toas the null value):oHA: μ ≠ μ0oHA: μ>μ0oHA: μ<μ0- If the sample offers strong evidence against the null hypothesis, we say we “reject the null hypothesis”Paint Thickness example (b). Suppose the device that applies paint to blocks has specifications that say the true average thickness of the paint is 2.50 mm. The device operator periodically evaluates a sample of paint thicknesses to determine if there is evidence that the average paint thickness is different from 2.50 mm, and if so, he will recalibrate the machine. State the null and alternative hypothesis. Parameter of interest: μT denotes the true average paint thickness of paint applied to all blocks.Null Hypothesis: H0: μT=2.50Alternative Hypothesis: HA: μT≠ 2.50(no recalibration necessary) (recalibration necessary)Pumpkin example (a). Tierra is a pumpkin farmer. The pumpkin seed company claims the average weight of their Bushkin pumpkins is 8 lbs. with standard deviation 0.75 lbs. Tierra thinks her pumpkins look larger than that and wonders if 3Statistics 371: Notes7 University of Wisconsin–Madisonthe mean weight of her Bushkin pumpkins is heavier than claimed by the seed company. Identify the population parameter of interest and write the null and alternative hypothesis for this problem.Parameter if interest: Let µ = the true average weight of all Buskin pumpkinsH0: µ = 8 versusHA: µ > 8 implicitH0: µ <= 8 lbs.Proportion of Cars example: A county official wonders if the proportion of cars in the county with modified emission

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