Question

Essential Wear is a company that sends 5 items of apparel every 3 months. They specialize in women's clothes and the boxes consist of undergarments, accessories, and loungewear. The subscriber pays a flat fee but they do not know what they will be getting in each box. The company claims that 20% of the goods sent are underwear, and 10% are bras, 18% are socks, 36% are loungewear, and 16% are accessories. Suppose a suspicious customer tally's the totals for the last 3 years and finds that the company sent 18 pairs of underwear, 4 bras, 16 pairs of socks, 12 loungewear pieces and 10 accessories. Test the claim that the actual proportions of content differ from Essential Wear's claim. Use a 0.05 level of significance. a. Write the null and alternative hypotheses. Label the claim. b. Draw the graph and shade the critical area(s). Place the critical value(s) on the graph. c. Fill in the following: Critical Value(s) Test Statistic P Value d. Circle one: REJECT H or FAIL TO REJECT H. e. Statement of conclusion:

Answer 1

Categories Observed Expected (fo-fe)</fe Category 1 18 60*0.20=12 (18-12)2/12 = 3 Category 2 4 60*0.10-6 (4-6)2/6 = 0.667 Category 3 16 60*0.18=10.8 (16-10.8)2/10.8 = 2.504 Category 4 12 60*0.36=21.6 (12-21.6)2/21.6 = 4.267 Category 5 10 60*0.16=9.6 (10-9.6)/9.6 = 0.017 Sum = 60 60 10.454 (1) Null and Alternative Hypotheses The following null and alternative hypotheses need to be tested: Ho: p1 = 0.20, p2 = 0.10, p3 = 0.18, p4 = 0.36, p5 = 0.16 H: 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) Rejection Region Based on the information provided, the significance level is a = 0.05, the number of degrees of freedom is df = 5-1 = 4, so then the rejection region for this test is R= {x? : x? > 9.488}. : (3) Test Statistics The Chi-Squared statistic is computed as follows: n (O; - - E;) E; = 3 +0.667 + 2.504 + 4.267 + 0.017 = 10.454 i=1 (4) Decision about the null hypothesis = 10.454 > x2 = 9.488, it is then concluded that the Since it is observed that x? null hypothesis is rejected. (5) Conclusion It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that some of the population proportions differ from those stated in the null hypothesis, at the a = 0.05 significance level.

Chi-square = 10.454, df = 4 Right-tail p-value is 0.03344 R command: pchisq(10.454, 4, lower.tail=FALSE pchisq(10.454, 4) 0 3 6 99.488 12 15

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