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Ch15_Nonparametric

# To chi square test statistic chi square j o j e j ej

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Unformatted text preview: tic Chi-square ∑ j (O j − E j ) Ej 2 where sum is over each category. Each squared difference is weighted by the Each inverse of expected frequency, Ej. Distribution under the null hypothesis of no difference hypothesis If Os and Es are all “big” (>5) the Chi-square If and are statistic has a χ 2 distribution with J-1 df (J=5 in the above example). in Test of smoking patterns Test Calculate χ2obs Calculate Determine df (J-1 where J is number of Determine categories) categories) Choose significance level (e.g., α=.05) Choose =.05) Look up χ 2crit Table C.7, p. 504. Look crit Table If χ2obs > χ 2crit then reject the hypothesis If that recent smoking pattern is the same as previous. previous. If χ2obs < χ 2crit then don’t reject the If Calculations Calculations A0 A1 A2 A3 A4 Total Observed 406 164 189 78 26 863 Expected 371.09 146.71 207.12 86.3 51.78 863 (O-E)2/E 3.28 2.04 1.59 0.80 12.84 20.54 •χ 2obs = 20.54, df = 5-1 = 4 •χ 2crit(4) =9.48 •Since χ 2obs > χ 2crit(4) reject H0. Therefore we conclude that smoking patterns have changed. Comparison involving two variables Comparison Adult female TV characters by hair color and career level Hair Color Blonde Dark Career Professional Career Level Level Nonprofessional 36 24 48 72 Raw data Raw B1 B2 total B1 B2 total A1 36 24 60 A1 A2 48 72 120 A2 total 84 96 180 total 84 96 180 84*60/180 84*120/180 96*60/180 96*120/180 60 120 Expected values under independence independence B1 B2 total A1 84*60/180 96*60/180 60 A2 84*120/180 96*120/180 120 total 84 96 180 B1 B2 total A1 28 32 60 A2 56 64 120 total 84 96 180 Chi-square table Chi-square Also called a rows-by-columns Also contingency table contingency Each data point fits into one cell Row and column totals are called Row marginals marginals We use marginals to calculate expected We marginals expected cell totals, and compare to observed cell observed totals totals Expected Values Expected Hair Color E = row marg.*col marg. total responses Blonde Dark Marginals Career Professional O = 36 Career Level Level E = 28 NonO = 24 professional E = 32 Marginals O = 48 E = 56 O = 72 E = 64 120 84 96 N = 180 60 Chi-square test Chi-square χ = ∑∑ 2 i =1 j =1 c r (Oij − Eij ) Eij 2 2 (36 − 28) (24 − 32) χ= + + 28 32 2 2 (48 − 56) (72 − 64)...
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