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- 1. Chi-Square Test What is chi-square testing? o Identifiessignificantdifferencesamongthe observedfrequenciesandthe expected frequenciesof aparticulargroup o Attemptstoidentifywhetherany differencesbetweenthe expectedandobserved frequenciesare due tochance,or some otherfactor that isaffectingit. o There are actuallymanytypesof Chi-square tests,butthe mostcommonone isthe Pearson Chi-squareTest. Terms and Definitions o Categorical Data- 2 types a. Numerical data- informof numbers.(ex.1,2,3,4) b. Categorical data- comesinformof divisions.(ex.Yesorno) o ExpectedFrequencies -valuesforparametersthatare hypothesizedtooccur -can be determinedthrough: 1) Hypothesizingthatthe frequencies are equal foreach category. 2) Hypothesizingthe valuesonthe basisof some prior knowledge. 3) A mathematical method (seePage 3) Two applications ofPearson Chi-Square Test 1) Chi-square testforIndependence -Thistestswhetherthe “category”fromwhichthe data comesfromaffectsthe data. -May alsobe thoughtof as testingwhetherthe categoriesinthe experiment“prefer”certain kindsof data. Example:Isthere a difference inthe carchoicesof male and females? 2) Chi-square testforgoodness-of-fit -Thistestswhetherthe observed data“fit”the expected data. Example:Dothe car salesthisyearmatch the car saleslastyear?(ie.Didwe still sell around50 blue cars? 25 redcars?) Requirementsofthe Chi-squaredTest 1. The valuesof the parameterstobe comparedare quantitative andnominal. 2. There shouldbe one or more categoriesinthe setup. 3. The observationsshouldbe independentof eachother. 4. An adequate sample size.(Atleast10) 5. Most of the time,itis the frequencyof the observationsthatare used.
- 2. Example A studentwantstosee whetherthe foodpreferencesof malesandfemalesdiffered.He triedtosee whethermalesorfemaleshadageneral difference inthe preference forcookedandraw foods. A survey was conductedwiththe followingresults: Twelve malespreferredCookedfoods. Eightmalespreferred Rawfoods. Five femalespreferred Cookedfoods. Five femalespreferred Cawfoods. Step 1: State the null hypothesisand the alternative hypothesis. Ho: There isno significantdifference betweenthe food preferencesof malesandfemales. Or Foodpreference isindependentof gender. Ha: There is a significantdifferencebetweenthe foodpreferencesof malesandfemales. Or Foodpreference isaffectedbygender. Step 2: State the level ofsignificance. (FishThingy) α = 0.05 0.05 is the level of significance for most scientific experiments. Step 3: Set up a contingencytable: The contingencytable summarizesthe data. The categoriesonthe columnsare the “preferences”thatyouare checking. The categoriesonthe rows are the “populations”whosepreferencesare beingchecked.A row total and columntotal isalwaysincludedaswell. Preference Male Female Total (Row) Cooked 12 5 17 Raw 8 5 13 Total (Column) 20 10 30
- 3. Step 4: Compute for the expectedfrequencies. The chi-square testforindependenceusuallyusesthe thirdmethodof gettingexpectedfrequencies. ExpectedFrequency=(RowTotal)(ColumnTotal) Grand total ThisexpectedfrequencyiscomputedforEACH cell. Preference Male Female Total (Row) Cooked (20)(17)/30 = 11.33 (10)(17)/30 = 5.67 17 Raw (13)(20)/30 = 8.67 (13)(10)/30 = 4.33 13 Total (Column) 20 10 30 The fundamental formulaforthe Chi-squaredtestis: Where O isthe observedfrequencies E is the expectedfrequencies Andx2 isthe chi-square value Step 5: Rearrange the table to show the observedand expectedfrequenciesonthe columns,and the subcategorieson the rows. Preference Observed Expected Chi-square CookedMales 12 11.33 0.0396 CookedFemales 5 5.67 0.0792 Raw Males 8 8.67 0.0518 Raw Females 5 4.33 0.1037 Total 0.2743
- 4. Step 6: Determine the degreesoffreedom The degreesof freedomis: df = (Rows – 1)(Columns– 1) df = (2 – 1)(2 – 1) = 1 Step 7: Check the tabular Chi-squaredvalue with your df and level ofsignificance. Checkingthe table,we see thatthe tabular chi-squaredvaluefordf = 1, and α = 0.05 is3.841. Since our calculatedchi-squareislessthanthis,the conclusionisto acceptthe null hypothesis. Hence, foodpreference isindependentof gender. If it were greater, we would rejectthe null hypothesis.

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