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Unformatted text preview: ChiSquare Χ
Chapter 15 2 ChiSquare Χ 2 Used to test hypotheses about the dependent tendencies between two variables Variables need not be IV Dependent variable is not an interval or ratio scale of measurement (usually frequency counts) Unlike z, t, or F: not comparing means and making no assumptions about normal distributions of scores or equal variances, etc. Dependent Variables Two variables are dependent (or related) if knowledge of one variable leads you to make different predictions about the various levels of the other variable Example: – Are sex and hair length dependent (or related)? – Yes, if we know a person’s sex, we would make different predictions about their hair length ChiSquare Statistical Hypotheses
H0: The row variable and the column variable in a contingency table are independent (or unrelated) H1: The row variable and the column variable in a contingency table are dependent (or related) Hair Length Less than 4 More than inches 4 inches Men Sex Women 25 15 15 45 Hair Length Less than 4 More than inches 4 inches Men Sex Women Marginals Marginals 40 60 100 25 15 40 15 45 60 ChiSquare test statistic Provides a test of the statistical null hypothesis of no dependency (or relationship) between the column and row variables of a contingency table Formula
Χ2 = ∑ ∑ (Orc Erc )2
r=1 c=1 Erc r c Orc = observed frequency in the row and column cell Erc = expected frequency in the row and column cell r = number of categories of the row variable c = number of categories of the column variable 5step Hypothesis testing with Chi Square
1. Statistical Hypotheses H0: sex and hair length are independent H1: sex and hair length are dependent α = .05 1. 3. Set stage 5step Hypothesis testing with Chi Square
a. Use ChiSquare because we have frequency counts of the number of men and women with short and long hair b. Χ2 = has (r1)(c1) df c. Χ2 crit(1)= 3.84. P. 498 Table A7 where r=# of categories of sex (2) c=# of categories of hair length (2) df = (21)(21) = 1 3. Decision rule: reject H0 if Χ2obs≥ 3.84 4. Compute Χ2 a. Collect data and get observed frequencies Less than More than 4 inches 4 inches Men Women 25 15 15 45 Computing Χ (continued)
2 4.b. Compute expected frequencies work backward from row and column marginals to determine what cell frequencies would be expected if H0 is true!
Expected = (row marginal for cell)(column marginal for cell) frequency total number of responses of a cell Less than 4 inches Men Women (40)(40) = 16 100 (60)(40) = 24 100 40 More than 4 inches (40)(60) = 24 100 (60)(60) = 36 100 60 40 60 100 Computing Χ (continued)
2 c. Subtract E from O d. Square each OE difference e. Divide each squared OE difference by E for that cell f. Sum the resulting (OE)2/E values over all cells in contingency table to get Χ2obs Cells r c 1 1 2 2 1 2 1 2 Obsrc 25 15 15 45 Exprc (OE) (OE)2 (OE)2 E 16 24 24 36 9 9 9 9 81 81 81 81 5.06 3.38 3.38 2.25 Χ2obs = ∑∑(OE)2 = 14.07 E 5. Decision Reject H0 and accept H1 because Χ2obs = 14.07 is > Χ2crit(1) = 3.84 We have discovered that sex and hair length are dependent in the population from which the sample was drawn. Can be used to test hypothesis of dependency for any number of levels of two variables 1. H0: The opinions a person holds on “freeing Hollywood” are independent on the community that person lives in H1: The opinions a person holds on “freeing Hollywood” are dependent on the community that person lives in 2. α = .01 1. a. Use Χ2: looking at hypotheses between two categorical variables. b. Χ2crit(31) (31) = 4 df c. Χ2crit(4) = 13.3 d. Decision rule: reject H0: if Χ2obs ≥ 13.3 4. Collect Data, compute Χ
Favorable Neutral Unfavorable Hollywood Los Feliz Pasadena 2 Free Hollywood? 15 70 10 95 60 55 75 190 25 75 15 115 100 200 100 400 Compute expected frequencies Favorable Neutral Unfavorable Holly (100)(190)= 47.5 (100)(95)= 23.8 400 wood 400 Los Feliz
(200)(190)= 95 400 (200)(95)= 47.5 400 (100)(115)=28.8 400 (200)(115)=57.5 400 (100)(115)=28.8 400 Free Hollywood? 100 200 100 400 Pasa (100)(190)= 47.5 (100)(95)= 23.8 400 dena 400 190 95 115 r 1 1 1 2 2 2 3 3 3 c 1 2 3 1 2 3 1 2 3 Obsrc Exprc (OE) (OE)2 (OE)2 E 60 15 25 55 70 75 75 10 15 47.5 23.8 28.8 95 47.5 57.5 47.5 23.8 28.8 12.5 8.8 3.8 40 22.5 17.5 27.5 13.8 13.8 156.25 77.44 14.44 1600 306.25 190.44 190.44 3.29 3.25 .50 16.84 5.33 8.00 6.61 506.25 10.66 756.25 15.92 Χ2 =70.40 5. Decision Reject H0 and accept H1 because Χ2obs = 70.40 is > Χ2crit(4) = 13.3 We have discovered that the opinions a person holds on “freeing Hollywood” are dependent on the community that person lives in Χ Goodness of fit
2 Used to test hypothesis about specific predictions for a single variable Examples: – Proportions of freshman, sophomores, juniors, and seniors enrolled in a class A. equal proportions (25% each) B. different proportions (10% f, 15% sophs, 65% j, 10 seniors) 5step procedure
H0: The proportion of F, Soph, J, and S is represented as 10%, 15%, 65%, and 10% H1: The proportion of F, Soph, J, and S is NOT represented as 10%, 15%, 65%, and 10% 2. α = .01
1. 3. a. Use Χ2 Goodness of fit b. Χ2crit(r1) or Χ2(41) = 11.3 c. Decision rule: reject H0: if Χ2obs ≥ 11.3 1. Collect data, compute Χ2 Fresh Soph Junior Senior Obs: 10 50 30 30 = 120 Exp: 10%(120) 15%(120) 65%(120) 10%(120) 12 18 78 12 Expected frequencies come from H0 statement, NOT from marginals Cells 1 2 3 4 Obsrc 10 50 30 30 Exprc 12 18 78 12 (OE) (OE)2 (OE)2 E 2 32 48 18 4 .33 1024 56.89 2304 29.54 324 27.00 Χ2 = 113.76 5. Decision Reject H0: The proportion of F, Soph, J, and S is not 10%, 15%, 65%, and 10% ...
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This note was uploaded on 01/16/2011 for the course PSYC 274 at USC.
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