Homework ReviewHomogeneous PopulationsIndependent VariablesMendel's ExperimentsAssignmentTests ofHomogeneityandIndependenceRobb T.KoetherHomeworkReviewHomogeneousPopulationsIndependentVariablesMendel’sExperimentsAssignmentTests of Homogeneity and IndependenceLecture 50Sections 14.4 - 14.5Robb T. KoetherHampden-Sydney CollegeWed, Apr 22, 2009Tests ofHomogeneityandIndependenceRobb T.KoetherHomeworkReviewHomogeneousPopulationsIndependentVariablesMendel’sExperimentsAssignmentOutline1Homework Review2Homogeneous Populations3Independent Variables4Mendel’s Experiments5AssignmentTests ofHomogeneityandIndependenceRobb T.KoetherHomeworkReviewHomogeneousPopulationsIndependentVariablesMendel’sExperimentsAssignmentExercise 14.7, page 935Exercise 14.7, page 935An experiment is conducted to try to determine themost effective over-the-counter pain relief medication.The five best-selling brands of pain relief are tested onn = 200 randomly selected subjects.For each subject, the “best” pain relief medicine isrecorded, based on various criteria.The results are presented below.You are asked to test the claim that the five brands(K = 5) of pain relief medicine are equally effective.Tests ofHomogeneityandIndependenceRobb T.KoetherHomeworkReviewHomogeneousPopulationsIndependentVariablesMendel’sExperimentsAssignmentExercise 14.7, page 935Exercise 14.7, page 935Brand of Pain ReliefA B C D EObserved Counts 38 44 56 37 25Tests ofHomogeneityandIndependenceRobb T.KoetherHomeworkReviewHomogeneousPopulationsIndependentVariablesMendel’sExperimentsAssignmentExercise 14.7, page 935Exercise 14.7, page 935(a) State the null hypothesis.(b) Compute the expected counts.(c) Compute the observed test statistic.(d) Find the p-value.(e) State your decision and conclusion using a 5%significance level.Tests ofHomogeneityandIndependenceRobb T.KoetherHomeworkReviewHomogeneousPopulationsIndependentVariablesMendel’sExperimentsAssignmentExercise 14.7, page 935Solution(a) If the brands are equally effective, then we would expectthem to be selected equally often as “best” by thesubjects. Therefore,H0: p1= p2= p3= p4= p5= 0.20.(b) The sample size is n = 200, so the expected counts areeach 40.Tests ofHomogeneityandIndependenceRobb T.KoetherHomeworkReviewHomogeneousPopulationsIndependentVariablesMendel’sExperimentsAssignmentExercise 14.7, page 935Solution(c) The test statistic is χ2=Pall cells(O−E)2E. Its value isχ2=(38 − 40)240+(44 − 40)240+(56 − 40)240+(37 − 40)240+(25 − 40)240=440+1640+25640+940+22540=51040= 12.75.Tests ofHomogeneityandIndependenceRobb T.KoetherHomeworkReviewHomogeneousPopulationsIndependentVariablesMendel’sExperimentsAssignmentExercise 14.7, page 935Solution(d) The p-value is χ2cdf(12.75,E99,4) = 0.0126.(e) Reject H0and conclude that the brands of pain relieverare not equally effective.Tests ofHomogeneityandIndependenceRobb T.KoetherHomeworkReviewHomogeneousPopulationsIndependentVariablesMendel’sExperimentsAssignmentHomogeneous PopulationsNow we wish to compare two or more populations byobserving the values of a qualitative variable for each ofthem.For example, is the distribution of types of crime inRichmond the same in 2009 as it was in 2008 (Jan -Mar)?Category 2008 2009Homicide 11 9Rape 10 8Robbery 135 150Assault 132 143Arson 21 10Burglary 334 284Larceny 930 1032Auto Theft 234 208Tests ofHomogeneityandIndependenceRobb T.KoetherHomeworkReviewHomogeneousPopulationsIndependentVariablesMendel’sExperimentsAssignmentHomogeneous PopulationsNow we wish to compare two or more populations byobserving the values of a qualitative variable for each ofthem.For example, is the distribution of types of crime inRichmond the same in 2009 as it was in 2008 (Jan -Mar)?Category 2008 2009Homicide 11 9Rape 10 8Robbery 135 150Assault 132 143Arson 21 10Burglary 334 284Larceny 930 1032Auto Theft 234 208Tests ofHomogeneityandIndependenceRobb T.KoetherHomeworkReviewHomogeneousPopulationsIndependentVariablesMendel’sExperimentsAssignmentHomogeneous PopulationsNow we wish to compare two or more populations byobserving the values of a qualitative variable for each ofthem.For example, is the distribution of types of crime inRichmond the same in 2009 as it was in 2008 (Jan -Mar)?Category 2008 2009Homicide 11 9Rape 10 8Robbery 135 150Assault 132 143Arson 21 10Burglary 334 284Larceny 930 1032Auto Theft 234 208Tests ofHomogeneityandIndependenceRobb T.KoetherHomeworkReviewHomogeneousPopulationsIndependentVariablesMendel’sExperimentsAssignmentHomogeneous PopulationsDefinition (Homogeneous)Two populations are called homogeneous if they exhibit thesame distributions over the same categories.If there was no significant change in the distribution oftypes of crime in Richmond, then those two populations(2008, 2009) are homogeneous.Tests ofHomogeneityandIndependenceRobb T.KoetherHomeworkReviewHomogeneousPopulationsIndependentVariablesMendel’sExperimentsAssignmentThe Test of HomogeneityThe null hypothesis is that the populations arehomogeneous.The alternative hypothesis is that the populations arenot homogeneous.H0: The populations are homogeneous.H1: The populations are not homogeneous.Tests ofHomogeneityandIndependenceRobb T.KoetherHomeworkReviewHomogeneousPopulationsIndependentVariablesMendel’sExperimentsAssignmentIndependent VariablesIn single-vehicle fatal accidents in 2007, consider thefollowing two variables.Whether the driver was alcohol-impaired.The time of day or night.Alcohol- Not Alcohol-Time Impaired Impaired12am - 3am 2496 13373am - 6am 1221 11006am - 9am 343 15079am - 12pm 182 142112pm - 3pm 332 19063pm - 6pm 635 22036pm - 9pm 1189 23379pm - 12am 1616 1924Tests ofHomogeneityandIndependenceRobb T.KoetherHomeworkReviewHomogeneousPopulationsIndependentVariablesMendel’sExperimentsAssignmentIndependent VariablesIn single-vehicle fatal accidents in 2007, consider thefollowing two variables.Whether the driver was alcohol-impaired.The time of day or night.Alcohol- Not Alcohol-Time Impaired Impaired12am - 3am 2496 13373am - 6am 1221 11006am - 9am 343 15079am - 12pm 182 142112pm - 3pm 332 19063pm - 6pm 635 22036pm - 9pm 1189 23379pm - 12am 1616 1924Tests ofHomogeneityandIndependenceRobb T.KoetherHomeworkReviewHomogeneousPopulationsIndependentVariablesMendel’sExperimentsAssignmentIndependent VariablesIn single-vehicle fatal accidents in 2007, consider thefollowing two variables.Whether the driver was alcohol-impaired.The time of day or night.Alcohol- Not Alcohol-Time Impaired Impaired12am - 3am
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