統(tǒng)計psyed2018class

1、PSYED 2018: Statistical Methods ILecture 11:Chi-Square TestsReviewIn the linear equation Y = 3X + 1, when X increases by 4 points, Y willincrease by.a. 4 pointsb. 7 pointsc. 12 pointsd. 13 pointsThe regression equation is determined by minimizing.a. the total error between the X and Y valuesb. the t

2、otal error between the predicted Y values and the actual Y valuesc. the total squared error between the X and the Y valuesd. the total squared error between the predicted Y values and the actual Yvalues2If there is a negative correlation between X and Y then the regressionequation, Y = bX + a will h

3、ave.a. b > 0b. b < 0c. a > 0d. a < 0Assuthat sample size and SSY is constant, which of the followingcorrelations would have the smallest standard error of estimate?a. r= -0.10b. r=+0.40c.r= -0.70d. Cannot determine3Chi-square Tests4The null hypothesis: there is no preference among the th

4、ree brands (or Preference in the population are equally divided among the three soft drinks). A researcher obtained a random sample of 48 s todetermine whether there were any significant preferences among three leading brands of soft drinks. Do the data indicate any preferences among the three brand

5、s?5If there is no preferences among the three brands, we would expect that in the population, the proportion for each brand will be the same.H0: there is no preference among the three brands (or Preference in the population are equally divided among the three soft drinks).Expected frequency: Hypothe

6、tical, ideal frequencies that arepredicted from the null hypothesis and the sample size.Chi-square TestsH0: there is no preference among the three brands (or Preference in the population are equally divided among the three soft drinks).Expected frequency: Hypothetical, ideal frequencies that are pre

7、dicted from the null hypothesis and the sample size (fe=pe*n)Observed frequency: The actual frequencies that are found in the sample data.Chi-square statistic (c2) = S(fo fe)2/fec 2 = 116+ 9 + 41616= 1416= .8756Whatdoesthis chi - square statistictell us?Chi-square TestsChi-square statistic is a test

8、 statistic that evaluates the discrepancy between a set of observed frequencies and a set of expected frequencies (under the null hypothesis).c2 = S(fo fe)2/feChi-square distribution is the theoretical distribution of the chi- square statistics that would be obtained through repeated random samples

9、if the null hypothesis was true.7Chi-square distribution is the theoretical distribution of the chi- square statistics that would be obtained through repeated random samples if the null hypothesis was true.Ø Chi-square distribution is positively skewed.Ø If H0 is true, we expect the data (

10、fo) to be to the hypothesis (fe). Thusthe c2 statistic will besmall when H0 is true.Ø Chi-square statistic isnon-negative.8If H0 is true, the c2 statistic should be small and large values of c2 statistic are very unlikely. Thus, unusually large chi-square values form the critical region for the

11、 hypothesis test.9Chi-square distribution is a family of distribution. The shape of thedistribution depends on the df. The shape of the chi-square distribution for different values of df. As the number of categories increases (thus df increases), the peak (mode) of the distribution has a larger chi-

12、square value. Unimodal Positively skewed As df grows infinitely large, the distribution approaches the normal distribution10Chi-square TestsIn the soft drink example:H0: no preference in the population.df = number of categories 1 = C 1 (C is the number of categories).Obtained c2 = S(fo fe)2/fe = 1/1

13、6+9/16+4/16 =.875,is this chi-square statistic large or small?11In the example, c2 = .875, df = number of categories 1=2, is this chi-square statistic large or small? Critical c2(df=2, a=.05)=5.99. Chi-square distribution (Table B.8, p.737) 12Chi-square Tests The soft drink example is an example of

14、Chi-square test for goodness of fit. Chi-square test for goodness of fit: A test that uses the sample data to test a hypothesis about the shape or proportions in teral population. The test determine how well the obtained sample data fit the population proportions specified by the null hypothesis. Th

15、e df for this: the number of categories 1.13Chi-square Tests14Another example of Chi-square test for Goodness of Fit: A social psychologist tend to be much older ths that people who serve on juriestizens in teral population. Toverify this speculation, he obtains voter registration records and finds

16、that 20% of registered voters are 18-29 years old, 45% are 30-49 years old, and 35% are age 50 or older. The psychologist also monitors jury composition over several weeks and observes the following distribution of ages for actual juries.Are these data sufficient to conclude that the age distributio

17、n for jurors is significantly different from the distribution for the population of registered votes? (Jurors are selected from the population of registered votes.)Chi-square Tests15H0: the age distribution for jurors matches the distribution for the population of registered votes.df=2, critical c2(

18、df=2, a=.05)=5.99, the observed c2 is bigger than the critical c2 therefore we conclude that the age distribution for jurors issignificantly different from the distribution for the population of registered votes (c2(df=2, n=80)=6.385, p<.05).Chi-square statistic (c2) = S(fo fe)2/fe = 64/16+64/28

19、=6.385Chi-square TestsChi-square test for independence: A test that uses the frequencies found in sample data to test a hypothesis about the relationship between two variables in the population.The test determine whether the distribution in one variabledepends on the distribution of the other variab

20、le in the population.1617observationalityColor preference1IntrovertBlue2IntrovertRed3ExtrovertYellow4IntrovertGreen5ExtrovertRed6IntrovertBlue7IntrovertYellow8ExtrovertRed1Contingency tables: Frequency tables of two variables presented simultaneously are called contingency tables. Contingency tables

21、 are constructed by listing all the levels of one variable as rows in a table and the levels of the other variables as columns, then finding the joint or cell frequency for each cell.marginal (column)frequency (fc)Totalfrequency8marginal row frequency (fr)Cell frequency19H0: there is no relationship

22、 betweenality and colorpreference in general population; (The distribution of colorpreference does not depend onality in general population); (The distribution of color preference is the same for introvert as that for extrovert in the population). Is there a relationship between Do color preferences

23、 depend on ality and color preference?ality?Chi-square TestsH0: The distribution of color preference is the same for introvertas that for extrovert in the population.Observed frequency (fo)20Total frequency (n)marginal (column) frequency (fc)Expected frequency (fe)marginal rowfrequency (fr)Chi-squar

24、e TestsH0: The distribution of color preference is the same for introvertas that for extrovert in the population.Expected proportionof color preference in each cell (pe) = fc/n .RedYellowGreenBlue(marginal frequency)Introvert50%10%20%20%50Extrovert50%10%20%20%150(marginal frequency)100204040200Expec

25、ted frequency fe = (fc/n) *fr when the null is true.21Chi-square TestsObserved frequency (fo)Expected frequency fe = (fc/n) *fr when the null is true.Chi-square statistic (c2) = S(fe fo)2/fe = 35.6df=(R-1)(C-1)=3Critical c2 (df=3, a=.05)=7.81. We conclude that22Chi-square Testsdf=(R-1)*(C-1). Degree

26、s ofdom and expected frequencies.(Once three values have been selected, all the remaining expectedfrequencies are determined by the row totals and the column totals. This example has only threechoices, so df = 3.)23Ø Independence of observationsØ Size of expected cell frequency: the expect

27、ed cell frequency is greater than 5 with df 2 and greater than 10 for df=1. When not satisfied, Fishers exact test may be used for 2x2 contingency table. For larger contingency tables, collapse levels of a variable when collapsing makes sense.Assumptions related to Chi-square tests2425Parametric vs.

28、 non-parametric tests:Non-parametric test generally is not as sensitive as parametric test. If possible, you should always choose the parametric test.Non-parametric test: Make few (if any) assumptions about population distribution or population parameters; the data usually consist of frequencies (va

29、riables measured in ordinal or nominal scale). For example, Chi-square test of goodness of fit and of independence.Parametric test: Test hypotheses about specific population parameters; make assumptions about the shape of population distribution and about other population parameters; require numerical sc