Question

A department store wanted to see if the distribution of t-shirt sizes sold followed the proportions: 14% Small, 22% Medium, 38% Large, and 26% Extra Large. A record of the sales of t-shirts is shown in the given table. At the 0.025 level of significance, can it be concluded that the percentages are equal to those stated? Show your work to receive credit.

Small Medium Large Extra Large

28 32 53 47

Answer 1

Chi-Square Goodness of Fit test

(1) Null and Alternative Hypotheses The following null and alternative hypotheses need to be tested: H0: p1 =0.14, p2 =0.22, p3​=0.38, p4​=0.26 Ha​: Some of the population proportions differ from the values stated in the null hypothesis This corresponds to a Chi-Square test for Goodness of Fit. (2) Degrees of Freedom The number of degrees of freedom is df=n-1=4-1=3 (3) Test Statistics The Chi-Squared statistic is computed as follows: (4)Critical Value and Rejection Region Based on the information provided, the significance level is α=0.025, the number of degrees of freedom is df=n-1=4-1=3, so the critical value becomes 9.3484. Then the rejection region for this test is R={χ2:χ2>9.3484}. (5)P-value The P-value is the probability that a chi-square statistic having 3 degrees of freedom is more extreme than 9.3484. The p-value is p=Pr(χ2>3.3925)=0.335 (6) The decision about the null hypothesis Since it is observed that χ2=3.3925≤χc2​=9.3484, it is then concluded that the null hypothesis is not rejected. (7) Conclusion It is concluded that the null hypothesis Ho is not rejected. Therefore, there is NOT enough evidence to claim that some of the population proportions differ from those stated in the null hypothesis, at the α=0.025 significance level.