PSU STAT 401 - Tests of Hypotheses Based on a Single Sample

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Slide 1Slide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Slide 20Slide 21Slide 22Slide 23Slide 24Slide 25Slide 26Slide 27Slide 28Slide 29Slide 30Slide 31Slide 32Slide 33Slide 34Slide 35Slide 36Slide 37Slide 38Slide 39Slide 40Slide 41Slide 42Slide 43Slide 44Slide 45Chapter 8Tests of Hypotheses Based on a Single Sample8.1Hypotheses and Test ProceduresHypothesesThe null hypothesis, denoted H0, is the claim that is initially assumed to be true. The alternative hypothesis, denoted by Ha, is the assertion that is contrary to H0. Possible conclusions from hypothesis-testing analysis are reject H0 or fail to reject H0.HypothesesH0 may usually be considered the skeptic’s hypothesis: Nothing new or interesting happening here! (And anything “interesting” observed is due to chance alone.)Ha may usually be considered the researcher’s hypothesis.Rules for HypothesesH0 is always stated as an equality claim involving parameters.Ha is an inequality claim that contradicts H0. It may be one-sided (using either > or <) or two-sided (using ≠).A Test of HypothesesA test of hypotheses is a method for using sample data to decide whether the null hypothesis should be rejected.Test ProcedureA test procedure is specified by1. A test statistic, a function of the sample data on which the decision is to be based.2. (Sometimes, not always!) A rejection region, the set of all test statistic values for which H0 will be rejectedErrors in Hypothesis TestingA type I error consists of rejecting the null hypothesis H0 when it was true. A type II error consists of not rejecting H0 when H0 is false. and a b are the probabilities of type I and type II error, respectively.Level TestaA test corresponding to the significance level is called a level test. A test with significance level is one for which the type I error probability is controlled at the specified level.aaSometimes, the experimenter will fix the value of , also known as the significance level.aRejection Region: and a bSuppose an experiment and a sample size are fixed, and a test statistic is chosen. Decreasing the size of the rejection region to obtain a smaller value of results in a larger value of for any particular parameter value consistent with Ha.ab8.2Tests About a Population MeanCase I: A Normal Population With Known sNull hypothesis:0 0:H m m=Test statistic value:0/xznms-=Case I: A Normal Population With Known sa 0:H m m>Alternative HypothesisRejection Region for Level Testa 0:H m m<a 0:H m m�az za�z za�-/ 2z za�/ 2z za�-orRecommended Steps in Hypothesis-Testing Analysis1. Identify the parameter of interest and describe it in the context of the problem situation.2. Determine the null value and state the null hypothesis.3. State the alternative hypothesis.Hypothesis-Testing Analysis4. Give the formula for the computed value of the test statistic.5. State the rejection region for the selected significance level6. Compute any necessary sample quantities, substitute into the formula for the test statistic value, and compute that value.Hypothesis-Testing Analysis7. Decide whether H0 should be rejected and state this conclusion in the problem context.The formulation of hypotheses (steps 2 and 3) should be done before examining the data.Type II Probability for a Level Testa( )b m�Alt. Hypothesisa 0:H m m>a 0:H m m<a 0:H m m�Type II Probability ( )b m�0/znam ms�-� �F +� �� �01/znam ms�-� �- F - +� �� �0 0/ 2 / 2/ /z zn na am m m ms s� �- -� � � �F + - F - +� � � �� � � �Sample SizeThe sample size n for which a level test also has at the alternative value is( )b m b�=m�a202/ 20( )( )z znz za ba bsm msm m�+� ��� ��-�� �=�+� ��� ���-� ��one-tailed testtwo-tailed testCase II: Large-Sample TestsWhen the sample size is large, the z tests for case I are modified to yield valid test procedures without requiring either a normal population distribution or a known .sLarge Sample Tests (n > 40)For large n, s is close to .sTest Statistic:0/XZS nm-=The use of rejection regions for case I results in a test procedure for which the significance level is approximately.aCase III: A Normal Population DistributionIf X1,…,Xn is a random sample from a normal distribution, the standardized variablehas a t distribution with n – 1 degrees of freedom./XTS nm-=The One-Sample t TestNull hypothesis:0 0:H m m=Test statistic value:0/xts nm-=a 0:H m m>Alternative HypothesisRejection Region for Level Testa 0:H m m<a 0:H m m�a, 1nt ta -�, 1nt ta -�-/ 2, 1nt ta -�orThe One-Sample t Test/ 2, 1nt ta -�-A Typical Curve for the t Testbcurve for n – 1 dfbValue of d corresponding to specified alternative to m�0 when bm m�=8.3Tests Concerning a Population ProportionA Population ProportionLet p denote the proportion of individuals or objects in a population who possess a specified property.Large-Sample TestsLarge-sample tests concerning p are a special case of the more general large-sample procedures for a parameter.qLarge-Samples Concerning pNull hypothesis:0 0:H p p=Test statistic value:( )00 0ˆ1 /p pzp p n-=-a 0:H p p>Alternative HypothesisRejection Regiona 0:H p p<a 0:H p p�z za�z za�-/ 2z za�/ 2z za�-orLarge-Samples Concerning pValid provided 0 010 and (1 ) 10.np n p� - �( )pb�Alt. Hypothesisa 0:H p p>a 0:H p p<( )pb�0 0 0(1 ) /(1 ) /p p z p p np p na� ��- + -F� �� �� �-� �General Expressions for 0 0 0(1 ) /1(1 ) /p p z p p np p na� ��- - -- F� �� �� �-� �( )pb�Alt. Hypothesisa 0:H p p�( )pb�General Expressions for 0 0 0(1 ) /(1 ) /p p z p p np p na� ��- + -F� �� �� �-� �0 0 0(1 ) /(1 ) /p p z p p np p na� ��- - -- F� �� �� �-� �Sample SizeThe sample size n for which a level test also has( )p pb�=a20 002/ 2 0 00(1 ) (1 )(1 ) (1 )z p p z p pp pnz p p z p pp pa ba b�� �� �- + -�� ���-� ��� �=�� ��� �- + -� ���-� ��� ��two-tailed testone-tailed testSmall-Sample TestsTest procedures when the sample size n is small are based directly on the binomial distribution rather than the normal


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PSU STAT 401 - Tests of Hypotheses Based on a Single Sample

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