Question

A company that sells baseball cards claims that 40% of the cards are rookies, 50% are veterans, and 10% are all stars. Suppose a random sample of 100 cards has 55 rookies, 40 veterans, and 5 all-stars.

Card Type	Observed Frequency O	Expected Frequency E	O-E	(O-E)2	(O-E)2 E
Rookie
Veteran
All-Star

Test the company's claim using a 0.01 level of significance. You may use the empty columns of the table above to assist you.

a) State the null and alternative hypothesis

b) Find the value of the test statistic and the critical value. You may use the empty columns of the table above to assist you.

c) State whether you should reject or fail to reject the null hypothesis. Justify your answer.

d) State you conclusion in non-technical terms

Please fill in the table with corresponding values and show all work.

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 =0.4, p2 =0.5, p3​=0.1 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=3-1=2 (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=0.01, the number of degrees of freedom is df=n-1=3-1=2, so the critical value becomes 9.2103. Then the rejection region for this test is R={χ2:χ2>9.2103}. (5)P-value The P-value is the probability that a chi-square statistic having 2 degrees of freedom is more extreme than 9.2103. The p-value is p=Pr(χ2>10.125)=0.0063 (6) The decision about the null hypothesis Since it is observed that χ2=10.125>χ2_crit​=9.2103, it is then concluded that the null hypothesis is rejected. (7) 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 α=0.01 significance level.

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