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H-SC MATH 121 - Lecture 49 - Goodness-of-Fit Test

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The Goodness-of-Fit TestGoodness-of-Fit Test on the TI-83The Effect of Sample SizeGoodness-of-Fit ExampleAssignmentTheGoodness-of-FitTestRobb T.KoetherTheGoodness-of-FitTestGoodness-of-Fit Test on theTI-83The Effect ofSample SizeGoodness-of-FitExampleAssignmentThe Goodness-of-Fit TestLecture 49Section 14.3Robb T. KoetherHampden-Sydney CollegeTue, Dec 1, 2009TheGoodness-of-FitTestRobb T.KoetherTheGoodness-of-FitTestGoodness-of-Fit Test on theTI-83The Effect ofSample SizeGoodness-of-FitExampleAssignmentOutline1The Goodness-of-Fit Test2Goodness-of-Fit Test on the TI-833The Effect of Sample Size4Goodness-of-Fit Example5AssignmentTheGoodness-of-FitTestRobb T.KoetherTheGoodness-of-FitTestGoodness-of-Fit Test on theTI-83The Effect ofSample SizeGoodness-of-FitExampleAssignmentOutline1The Goodness-of-Fit Test2Goodness-of-Fit Test on the TI-833The Effect of Sample Size4Goodness-of-Fit Example5AssignmentTheGoodness-of-FitTestRobb T.KoetherTheGoodness-of-FitTestGoodness-of-Fit Test on theTI-83The Effect ofSample SizeGoodness-of-FitExampleAssignmentThe HypothesesExample (Steps 1 - 3)(1) H0: p1= 1/6, p2= 1/6, p3= 1/6, p4= 1/6, p5= 1/6,p2= 1/6H1: H0is false.(2) α = 0.05.(3) χ2=Xall cells(O − E)2E.TheGoodness-of-FitTestRobb T.KoetherTheGoodness-of-FitTestGoodness-of-Fit Test on theTI-83The Effect ofSample SizeGoodness-of-FitExampleAssignmentThe HypothesesExample (Step 4)(4)Number 1 2 3 4 5 6Observed 20 20 14 8 14 14(Expected) (15) (15) (15) (15) (15) (15)χ2= 6.8.TheGoodness-of-FitTestRobb T.KoetherTheGoodness-of-FitTestGoodness-of-Fit Test on theTI-83The Effect ofSample SizeGoodness-of-FitExampleAssignmentTI-83 - Chi-Square ProbabilitiesTI-83 Chi-square ProbabilitiesPress 2nd DISTR.Select χ2cdf.Enter the lower endpoint, the upper endpoint, and thedegrees of freedom.Press ENTER. The probability appears in the display.TheGoodness-of-FitTestRobb T.KoetherTheGoodness-of-FitTestGoodness-of-Fit Test on theTI-83The Effect ofSample SizeGoodness-of-FitExampleAssignmentTI-83 - Chi-Square ProbabilitiesPracticeFind P (χ23> 6) with df = 3.Find P (20 < χ225< 30) with df = 25.Find P (χ26< 10) with df = 6.Find the probability that χ2512is within one standarddeviation of its mean.TheGoodness-of-FitTestRobb T.KoetherTheGoodness-of-FitTestGoodness-of-Fit Test on theTI-83The Effect ofSample SizeGoodness-of-FitExampleAssignmentThe Goodness-of-Fit TestIn our example, we found χ2= 6.8.There are 6 categories (1-6), so there are 5 degrees offreedom.TheGoodness-of-FitTestRobb T.KoetherTheGoodness-of-FitTestGoodness-of-Fit Test on theTI-83The Effect ofSample SizeGoodness-of-FitExampleAssignmentThe Goodness-of-Fit TestExample (Steps 5, 6, and 7)(5) p-value = χ2cdf(6.8,E99,5) = 0.2359.(6) Accept H0.(7) The die is fair.TheGoodness-of-FitTestRobb T.KoetherTheGoodness-of-FitTestGoodness-of-Fit Test on theTI-83The Effect ofSample SizeGoodness-of-FitExampleAssignmentOutline1The Goodness-of-Fit Test2Goodness-of-Fit Test on the TI-833The Effect of Sample Size4Goodness-of-Fit Example5AssignmentTheGoodness-of-FitTestRobb T.KoetherTheGoodness-of-FitTestGoodness-of-Fit Test on theTI-83The Effect ofSample SizeGoodness-of-FitExampleAssignmentGoodness-of-Fit Test on the TI-83Be careful when using the TI-83!There is a function called χ2-Test, but it does notperform the goodness-of-fit test.Some TI-84s have a GOF-Test function.The GOF-Test function does perform thegoodness-of-fit test.TheGoodness-of-FitTestRobb T.KoetherTheGoodness-of-FitTestGoodness-of-Fit Test on theTI-83The Effect ofSample SizeGoodness-of-FitExampleAssignmentGoodness-of-Fit Test on the TI-83TI-83 Goodness-of-fit testPut the observed counts in list L1.Put the hypothetical proportions in list L2.Multiply L2by the sample size and store as L2. Theseare the expected counts.Calculate (L1-L2)2/L2(either all at once or step bystep).Go to LIST > MATH and select sum (item #5).Enter Ans and press ENTER. The value of χ2appears.Then use χ2cdf to find the p-value.TheGoodness-of-FitTestRobb T.KoetherTheGoodness-of-FitTestGoodness-of-Fit Test on theTI-83The Effect ofSample SizeGoodness-of-FitExampleAssignmentOutline1The Goodness-of-Fit Test2Goodness-of-Fit Test on the TI-833The Effect of Sample Size4Goodness-of-Fit Example5AssignmentTheGoodness-of-FitTestRobb T.KoetherTheGoodness-of-FitTestGoodness-of-Fit Test on theTI-83The Effect ofSample SizeGoodness-of-FitExampleAssignmentThe Effect of Sample SizeIn the previous example, 1’s and 2’s wereoverrepresented and 4’s were underrepresented.Yet that was not significant.What if we observed the same proportions, but for alarger sample?That is, what if we had observedNumber 1 2 3 4 5 6Observed 200 200 140 80 140 140(Expected) (150) (150) (150) (150) (150) (150)Would it be signficant?TheGoodness-of-FitTestRobb T.KoetherTheGoodness-of-FitTestGoodness-of-Fit Test on theTI-83The Effect ofSample SizeGoodness-of-FitExampleAssignmentThe Effect of Sample SizeIn the previous example, 1’s and 2’s wereoverrepresented and 4’s were underrepresented.Yet that was not significant.What if we observed the same proportions, but for alarger sample?That is, what if we had observedNumber 1 2 3 4 5 6Observed 200 200 140 80 140 140(Expected) (150) (150) (150) (150) (150) (150)Would it be signficant?TheGoodness-of-FitTestRobb T.KoetherTheGoodness-of-FitTestGoodness-of-Fit Test on theTI-83The Effect ofSample SizeGoodness-of-FitExampleAssignmentThe Effect of Sample SizeIn the previous example, 1’s and 2’s wereoverrepresented and 4’s were underrepresented.Yet that was not significant.What if we observed the same proportions, but for alarger sample?That is, what if we had observedNumber 1 2 3 4 5 6Observed 200 200 140 80 140 140(Expected) (150) (150) (150) (150) (150) (150)Would it be signficant?TheGoodness-of-FitTestRobb T.KoetherTheGoodness-of-FitTestGoodness-of-Fit Test on theTI-83The Effect ofSample SizeGoodness-of-FitExampleAssignmentThe Effect of Sample SizeIn the previous example, 1’s and 2’s wereoverrepresented and 4’s were underrepresented.Yet that was not significant.What if we observed the same proportions, but for alarger sample?That is, what if we had observedNumber 1 2 3 4 5 6Observed 200 200 140 80 140 140(Expected) (150) (150) (150) (150) (150) (150)Would it be signficant?TheGoodness-of-FitTestRobb T.KoetherTheGoodness-of-FitTestGoodness-of-Fit Test on theTI-83The Effect ofSample SizeGoodness-of-FitExampleAssignmentThe Effect of Sample SizeIn the previous example, 1’s and 2’s wereoverrepresented and 4’s were


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H-SC MATH 121 - Lecture 49 - Goodness-of-Fit Test

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