Question

problem on chi-square test for homogeneity

2. Students often claim that the distribution of grades varies depending on the subject. From past records a university registrar chose random samples of grades from selected first year courses in English, Mathematics and Biology. Care was taken to exclude any student who may have taken two or more of these courses. The distribution of grades for each sample is given below. English Mathematics Biology 13 17 10 GRADEc42 15 24 (a) Test at .05 if the distribution of grades differs among the three subjects. (b) If Ho is rejected in (a), determine which cells contribute most to the decision, and give an interpretation of each (c) Estimate the proportion of students in this mathematics course who obtained a grade of B or better using a 95% confidence interval.

Answer 1

Solution Conduct the test. Null hypothesis: H0: The distribution of grades is same in all the three subjects Alternative hypothesis: Ha: The distribution of grades is not same in all the three subjects. MINITAB Procedure: Step 1: Choose Stat > Tables > Chi-Square Test (Two-Way Table in Worksheet). Step 2: In Columns containing the table, enter the columns of English, Mathematics and Biology Step 3: Click OK. MINITAB output:

Chl-Square Test: English, Mathematics, Biology Expected count aro printod below obserred count Chi-Square contribations are printed below expected counts Engliah Mathematics Biology Total 26 11.26 3.477 13 9.00 1.773 0.890 30 27.71 0-190 80 34-63 0.077 17.66 0.025 o. 42 32.90 2.517 24 16.78 26.32 2.739 .20S 10 76 15 12.99 О. 312 30 6.62 0.286 10.39 1.106 19 8.23 1.265 4.19 3.452 6.58 0.051 Total 100 52 80 231 i cells with expected counts less than 5. From MINITAB output, the p-value is, p-value 0.019 Since, p-value < 0.05, thus, the null hypothesis is rejected. Therefore, it can be concluded that the distribution of grades is not same in all the three subjects. Conclusion

Grade A of English contribute mostly to the Chi-square test statistic, as the difference between the observed frequency and expected value is quite wide, which gives rise to the high value of the chi-square value (3.477). Also, Grade F of Mathematics contributes greatly to the Chi-square test statistic, as the difference between the observed frequency and expected value is quite wide, which gives rise to the high value of the chi-square value (3.452). C. Obtain the proportion of students. MINITAB procedure: Step1: Choose Stat> Basic Statistics >1 Proportion. Step2: Choose Summarized data Step3: In Number of events, enter 25. In Number of trials, enter 51 Step5: Click Options. Under Alternative, and choose not equal. Stepó: Click OK in each dialog box. MINITAB output: Test and CI for One Proportion Sample xN Sample p 95 CI 25 51 0.490196 (0.352998, 0.627395) Using the normal approximation From the MINITAB output, the 95% confidence interval is,

CI-(0.3530.0.6274 Thus, with 95% confidence it can be concluded that the proportion of students who obtained grade B or better in mathematics lie between 35.30% and 62.74%