Question

goodness of fit A buyer for a t-shirt shop wants to compare the proportion of t-shirts of each size that are sold to the proportion that were ordered. The buyer counts the number of t-shirts of each size that are sold in a week. size Sold Percentage ordered Small 25 10% Medium 41 20% Large 91 40% Extra Large 68 30% Does the distribution of the sold items follow the distribution of the ordered items?

State the null and alternative hypothesis

Give the p-value

Give a conclusion for the hypothesis test.

Answer 1

Chi-Square Goodness of Fit test - Run at 5%

(1) Null and Alternative Hypotheses The following null and alternative hypotheses need to be tested: H0: p1 =0.1, p2 =0.2, p3​=0.4, p4​=0.3 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. Degrees of Freedom The number of degrees of freedom is df=n-1=4-1=3 Test Statistics The Chi-Squared statistic is computed as follows: P-value The P-value is the probability that a chi-square statistic having 3 degrees of freedom is more extreme than 7.8147. The p-value is p=Pr(χ2>0.6481)=0.8853 The decision about the null hypothesis Since it is observed that p-value = 0.8853 which is very high, it is then concluded that the null hypothesis is not rejected. 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=0.05,0.01,0.10,0.50 significance level.)