Chi-Square Test for Goodness of Fit, Exams of Statistics

Download Chi-Square Test for Goodness of Fit and more Exams Statistics in PDF only on Docsity! Chi-Square Predition Text 2023  Chi-Square Test for Goodness of Fit - Chi-Square Goodness of Fit Test:  Used when:  Test method. Use the chi-square goodness of fit test to determine whether observed sample frequencies differ significantly from expected frequencies specified in the null hypothesis.  -1 categorical variable  -1- population  -Appropriate when these conditions are met:  -The sampling method is simple random sampling.  -The population is at least 10 times as large as the sample.  -The variable under study is categorical.  -The expected value of the number of sample observations in each level of the variable is at least 5.  (Q): Hypothesis Test  (S): Sample Test  (T): 1 categorical variable  (N): 1 sample from 1 population  Hypothesis:  That is, if one is true, the other must be false; and vice versa.  H0: The data are consistent with a specified distribution.  Ha: The data are not consistent with a specified distribution.  Analyze Sample Data  Using sample data, find the degrees of freedom, expected frequency counts, test statistic, and the P-value associated with the test statistic.  Degrees of freedom: k-1  is equal to the number of levels (k) of the categorical variable minus  Expected frequency counts. The expected frequency counts at each level of the categorical variable are equal to the sample size times the hypothesized proportion from the null hypothesis  Ei = npi  where Ei is the expected frequency count for the ith level of the categorical variable, n is the total sample size, and pi is the hypothesized proportion of observations in level i.  Test statistic. The test statistic is a chi-square random variable (Χ2) defined by the following equation.  Χ2 = Σ [ (Oi - Ei)2 / Ei ]  where Oi is the observed frequency count for the ith level of the categorical variable, and Ei is the expected frequency count for the ith level of the categorical variable.  Formulate an Analysis Plan  The analysis plan describes how to use sample data to accept or reject the null hypothesis. The plan should specify the following elements.  Significance level. Often, researchers choose significance levels equal to 0.01, 0.05, or 0.10; but any value between 0 and 1 can be used.  Test method. Use the chi-square test for independence to determine whether there is a significant relationship between two categorical variables.  Analyze Sample Data  Using sample data, find the degrees of freedom, expected frequencies, test statistic, and the P-value associated with the test statistic. The approach described in this section is illustrated in the sample problem at the end of this lesson.  Degrees of freedom. The degrees of freedom (DF) is equal to:  DF = (r - 1) * (c - 1)  where r is the number of levels for one catagorical variable, and c is the number of levels for the other categorical variable.  Expected frequencies. The expected frequency counts are computed separately for each level of one categorical variable at each level of the other categorical variable. Compute r * c expected frequencies, according to the following formula.  Er,c = (nr * nc) / n  where Er,c is the expected frequency count for level r of Variable A and level c of Variable B, nr is the total number of sample observations at level r of Variable A, nc is the total number of sample observations at level c of Variable B, and n is the total sample size.  Test statistic. The test statistic is a chi-square random variable (Χ2) defined by the following equation.  Χ2 = Σ [ (Or,c - Er,c)2 / Er,c ]  where Or,c is the observed frequency count at level r of Variable A and level c of Variable B, and Er,c is the expected frequency count at level r of Variable A and level c of Variable B.  P-value. The P-value is the probability of observing a sample statistic as extreme as the test statistic. Since the test statistic is a chi-square, use the Chi-Square Distribution Calculator to assess the probability associated with the test statistic. Use the degrees of freedom computed above.  Interpret Results  If the sample findings are unlikely, given the null hypothesis, the researcher rejects the null hypothesis. Typically, this involves comparing the P-value to the significance level, and rejecting the null hypothesis when the P-value is less than the significance level.  Test Your Understanding of This Lesson  Problem  A public opinion poll surveyed a simple random sample of 1000 voters. Respondents were classified by gender (male or female) and by voting preference (Republican, Democrat, or Independent). Results are shown in the contingency table below.  Voting Preferences Row total  Republican Democrat Independent  Male 200 150 50 400  Female 250 300 50 600  Column total 450 450 100 1000  Is there a gender gap? Do the men's voting preferences differ significantly from the women's preferences? Use a 0.05 level of significance.  Solution  The solution to this problem takes four steps: (1) state the hypotheses, (2) formulate an analysis plan, (3) analyze sample data, and (4) interpret results. We work through those steps below:  State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.  H0: Gender and voting preferences are independent.  Ha: Gender and voting preferences are not independent.  Formulate an analysis plan. For this analysis, the significance level is 0.05. Using sample data, we will conduct a chi-square test for independence.  Analyze sample data. Applying the chi-square test for independence to sample data, we compute the degrees of freedom, the expected frequency counts, and the chi-square test statistic. Based on the chi-square statistic and the degrees of freedom, we determine the P-value.  DF = (r - 1) * (c - 1) = (2 - 1) * (3 - 1) = 2  Er,c = (nr * nc) / n  E1,1 = (400 * 450) / 1000 = 180000/1000 = 180  If sample data are displayed in a contingency table (Populations x -----Category levels), the expected frequency count for each cell of the table is at least 5.  This approach consists of four steps:  state the hypotheses,  formulate an analysis plan,  analyze sample data, and  interpret results.  State the Hypotheses  Every hypothesis test requires the analyst to state a null hypothesis and an alternative hypothesis. The hypotheses are stated in such a way that they are mutually exclusive. That is, if one is true, the other must be false; and vice versa.  Suppose that data were sampled from r populations, and assume that the categorical variable had c levels. At any specified level of the categorical variable, the null hypothesis states that each population has the same proportion of observations. Thus,  H0: Plevel 1 of population 1 = Plevel 1 of population 2 = . . . = Plevel 1 of population r  H0: Plevel 2 of population 1 = Plevel 2 of population 2 = . . . = Plevel 2 of population r  . . .  H0: Plevel c of population 1 = Plevel c of population 2 = . . . = Plevel c of population r  The alternative hypothesis (Ha) is that at least one of the null hypothesis statements is false.  Formulate an Analysis Plan  The analysis plan describes how to use sample data to accept or reject the null hypothesis. The plan should specify the following elements.  Significance level. Often, researchers choose significance levels equal to 0.01, 0.05, or 0.10; but any value between 0 and 1 can be used.  Test method. Use the chi-square test for homogeneity to determine whether observed sample frequencies differ significantly from expected frequencies specified in the null hypothesis. The chi-square test for homogeneity is described in the next section.  Analyze Sample Data  Using sample data from the contingency tables, find the degrees of freedom, expected frequency counts, test statistic, and the P-value associated with the test statistic. The analysis described in this section is illustrated in the sample problem at the end of this lesson.  Degrees of freedom. The degrees of freedom (DF) is equal to:  DF = (r - 1) * (c - 1)  where r is the number of populations, and c is the number of levels for the categorical variable.  Expected frequency counts. The expected frequency counts are computed separately for each population at each level of the categorical variable, according to the following formula.  Er,c = (nr * nc) / n  where Er,c is the expected frequency count for population r at level c of the categorical variable, nr is the total number of observations from population r, nc is the total number of observations at treatment level c, and n is the total sample size.  Test statistic. The test statistic is a chi-square random variable (Χ2) defined by the following equation.  Χ2 = Σ [ (Or,c - Er,c)2 / Er,c ]  where Or,c is the observed frequency count in population r for level c of the categorical variable, and Er,c is the expected frequency count in population r for level c of the categorical variable.  P-value. The P-value is the probability of observing a sample statistic as extreme as the test statistic. Since the test statistic is a chi-square, use the Chi-Square Distribution Calculator to assess the probability associated with the test statistic. Use the degrees of freedom computed above.  Interpret Results  If the sample findings are unlikely, given the null hypothesis, the researcher rejects the null hypothesis. Typically, this involves comparing the P-value to the significance level, and rejecting the null hypothesis when the P-value is less than the significance level.  Test Your Understanding of This Lesson  Problem  In a study of the television viewing habits of children, a developmental psychologist selects a random sample of 300 first graders - 100 boys and 200 girls. Each child is asked which of the following TV programs they like best: The Lone Ranger, Sesame Street, or The Simpsons. Results are shown in the contingency table below.  Viewing Preferences Row total