Statistical Inference for Frequency Data Chapter 16

Chapter Topics Applications of Pearson’s Chi-Square  Goodness of Fit Computations Practical Significance  Independence Computations Practical Significance  Testing equality of 2 or more proportions Computations Test Extensions  JMP Procedures

Pearson’s Chi-Square Statistic Statistic developed by Karl Pearson to test hypotheses about frequency data  Do not confuse with the chi-square distribution, which was derived by Helmert Used for three purposes  Testing goodness of fit Did our sample really come from that distribution?  Testing independence Are gender and alcohol consumption really independent?  Testing equality of proportions Is the proportion of residents who are college grads the same in every state?

Testing Goodness of Fit Did our sample really come from that population? Null Hypothesis  States that the sample did come from that population Alternative Hypothesis  States that the sample did not come from that population The test statistic is given by: Degrees of freedom: k-1 Involves a value for each observation as well as a value for what the observation should have been if the sample really did come from that population

Testing Goodness of Fit The Chi-Square Distribution We reject the null hypothesis if the test statistic computed on the previous page is greater than the critical value. (continued) χ2χ2 df=1 df=2 df=5 df=9

Testing Goodness of Fit Did the following data come from a binomial population with 25% graduate students? If the null hypothesis were true, then we would expect to have 75% undergraduates (150) and 25% graduates (50). The critical value is (α=.05, ν =1); therefore, we reject the null hypothesis. There is evidence the sample did not come from that population.  Note: the degrees of freedom are k-1 CategoryNumber Undergraduate168 Graduate32 Total200

Testing Goodness of Fit Cohen developed the following measure of effect size  is the observed proportion  is the expected proportion Guidelines are interpreted as follows:  0.1 – small effect  0.3 – medium effect  0.5 – large effect

Testing Goodness of Fit Assumptions  Every sample observation must fall in one and only one category  The observations must be independent  The sample n must be large Rules of thumb: When we have one degree of freedom, the expected value of all cells should be at least ten When we have two or more degrees of freedom, the expected value of all cells should be at least five Yates’ Correction for Continuity  Apply this correction when we have one degree of freedom and the expected value of one or some cells is not appreciably greater than ten

Testing Independence Are gender and classification independent? Null Hypothesis  States that the variables are independent Alternative Hypothesis  States that the variables are not independent The test statistic is given by: Degrees of freedom: (r-1)(c-1) Involves a value for each observation as well as a value for what the observation should have been if the variables were independent

Testing Independence Determining the Expected Value  If the variables were independent, the probability of begin a male undergraduate would be the probability of being a male times the probability of being an undergraduate This product gives a probability. To determine the number, simply multiply that probability by the sample size Computational Example CategoryNumber Undergraduate Men77 Undergraduate Women91 Graduate Men19 Graduate Women13 Total200

Testing Independence Put into a contingency table & compute expected values UndergraduateGraduate Male Female Expected Values: The Test Statistic:

Testing Independence Cramér developed the following measure of association  When the variables are independent, the numerator is 0  When there is a perfect association, the numerator is (s-1) Cohen’s measure of effect size is the same, and the values are interpreted the same:  0.1 – small effect  0.3 – medium effect  0.5 – large effect

Testing Equality of c≥2 Proportions Null Hypothesis  States that all proportions are equal Alternative Hypothesis  States that some proportions are not equal The test statistic is given by: Degrees of freedom: c-1 Involves a value for each observation as well as a value for what the observation should have been if the proportions are equal

JMP Procedures Independence  Three columns: Two nominal and one continuous  First column: Gender  Second column: Classification  Third column: Count  Analyze | Fit Y by X | Y – Gender | X – Classification | Freq – Count | OK |

Chapter Review Applications of Pearson’s Chi-Square  Goodness of Fit Computations Practical Significance  Independence Computations Practical Significance  Testing equality of 2 or more proportions Computations Test Extensions  JMP Procedures