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Social Statistics: Chi-square This week What is chi-square CHIDIST Non-parameteric statistics 2 Parametric statistics A main branch of statistics 3 Assuming data with a type of probability distribution (e.g. normal distribution) Making inferences about the parameters of the distribution (e.g. sample size, factors in the test) Assumption: the sample is large enough to represent the population (e.g. sample size around 30). They are not distribution-free (they require a probability distribution) Nonparametric statistics Nonparametric statistics (distribution-free statistics) 4 Do not rely on assumptions that the data are drawn from a given probability distribution (data model is not specified). It was widely used for studying populations that take on a ranked order (e.g. movie reviews from one to four stars, opinions about hotel ranking). Fits for ordinal data. It makes less assumption.Therefore it can be applied in situations where less is known about the application. It might require to draw conclusion on a larger sample size with the same degree of confidence comparing with parametric statistics. Nonparametric statistics Nonparametric statistics (distribution-free statistics) Data with frequencies or percentage Number of kids in difference grades The percentage of people receiving social security 5 One-sample/Two-sample chi-square One-sample chi-square includes only one dimension Two-sample chi-square includes two dimensions 6 Whether the number of respondents is equally distributed across all levels of education. Whether the voting for the school voucher has a pattern of preference. Whether preference for the school voucher is independent of political party affiliation and gender Compute chi-square One-sample chi-square test 2 (O E ) 2 E O: the observed frequency E: the expected frequency 7 Example Question: Whether the number of respondents is equally distributed across all opinions? One-sample chi-square for 23 8 Preference for School Voucher maybe against 17 50 total 90 Chi-square steps Step1: a statement of null and research hypothesis There is no difference in the frequency or proportion in each category H 0 : P1 P2 P3 There is difference in the frequency or proportion in each category H 1 : P1 P2 P3 9 Chi-square steps Step2: setting the level of risk (or the level of significance or Type I error) associated with the null hypothesis 10 0.05 Chi-square steps Step3: selection of proper test statistic 11 Frequencynonparametric procedureschisquare Chi-square steps Step4. Computation of the test statistic value (called the obtained value) category for maybe against Total 12 observed frequency (O) 23 17 50 90 expected frequency (E) D(difference) 30 30 30 90 7 13 20 (O-E)2 49 169 400 (O-E)2/E 1.63 5.63 13.33 20.60 Chi-square steps Step5: determination of the value needed for rejection of the null hypothesis using the appropriate table of critical values for the particular statistic 13 Distribution of Chi-Square df = r-1 (r= number of categories) If the obtained value > the critical value reject the null hypothesis If the obtained value < the critical value accept the null hypothesis Chi-square steps 14 Chi-square steps Step6: a comparison of the obtained value and the critical value is made 15 20.6 and 5.991 Chi-square steps Step 7 and 8: decision time 16 What is your conclusion, why and how to interpret? Another example 17 We’ll settle the age-old debate of whether people can actually detect their favorite cola based solely on taste. For 30 coke-lovers, I blindfold them, and have them sample 3 colas…is there a true difference, or are these preference differences explainable by chance? Hypothesis Null: There are no preferences: The population is divided evenly among the brands Alternate: There are preferences: The population is not divided evenly among the brands 18 Chance Model df = C -1 = 3 -1 = 2, set α = .05 For df = 2, X2-crit = 5.99 19 Calculate Chi-Square category Coke Pepsi RC Cola Total 20 observed frequency (O) 13 9 8 30 expected frequency (E) D(difference) 10 10 10 30 (O-E)2 3 1 2 9 1 4 (O-E)2/E 0.9 0.1 0.4 1.4 Decision and Conclusion 21 2 crit 5 . 99 2 obt 1 . 40 2 obt 2 crit Conclude that the preferences are evenly divided among the colas when the logos are removed. Excel functions CHIDIST (x,degrees_freedom) CHIDIST(20.6,2) CHIDIST(1.40,2) 22 0.000036<0.05 0.496585304>0.05