Problem

Acme Toy Company prints baseball cards. The company claims that 30% of the cards are rookies, 60% veterans, and 10% are All-Stars. The cards are sold in packages of 100.

Suppose a randomly-selected package of cards has 50 rookies, 45 veterans, and 5 All-Stars. Is this consistent with Acme's claim? Use a 0.05 level of significance.

Solution

The solution to this problem takes four steps: (1) state the hypotheses, (2) formulate an analysis plan, (3) analyze sample data, and (4) interpret results. We work through those steps below: * State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis. * Null hypothesis: The proportion of rookies, veterans, and All-Stars is 30%, 60% and 10%, respectively. * Alternative hypothesis: At least one of the proportions in the null hypothesis is false. * Formulate an analysis plan. For this analysis, the significance level is 0.05. Using sample data, we will conduct a chi-square goodness of fit test of the null hypothesis. * Analyze sample data. Applying the chi-square goodness of fit test to sample data, we compute the degrees of freedom, the expected frequency counts, and the chi-square test statistic. Based on the chi-square statistic and the degrees of freedom, we determine the P-value.

DF = k - 1 = 3 - 1 = 2

(Ei) = n * pi

(E1) = 100 * 0.30 = 30

(E2) = 100 * 0.60 = 60

(E3) = 100 * 0.10 = 10

Χ2 = Σ [ (Oi - Ei)2 / Ei ]

Χ2 = [ (50 - 30)2 / 30 ] + [ (45 - 60)2 / 60 ] + [ (5 - 10)2 / 10 ]

Χ2 = (400 / 30) + (225 / 60) + (25 / 10) = 13.33 + 3.75 + 2.50 = 19.58 where DF is the degrees of freedom, k is the number of levels of the categorical variable, n is the number of observations in the sample, Ei is the