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AP Statistics 8.6 Carrying Out a Chi-Square Test for Homogeneity or Independence- FRQs - Exam Style Questions

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Question

Michelle is at a national baseball card collector’s convention with approximately \(20,000\) attendees. She notices that some collectors have both regular cards, which are easily obtained, and rare cards, which are harder to obtain. Michelle believes that there is a relationship between the number of months a collector has been collecting baseball cards and whether the majority of the cards (cards appearing more often) in their collection are regular or rare. She obtains information from a random sample of \(500\) baseball card collectors at the convention and records how many full months they have been collecting baseball cards and whether the majority of the cards in their card collection are regular or rare. Her results are displayed in a two-way table.
Majority Type of Baseball Cards and Months of Collecting Baseball Cards
 Fewer Than 6 Months6-10 Months11-15 Months16-20 Months21 or More MonthsTotal
Has a Majority of Regular Baseball Cards\(80\)\(84\)\(71\)\(76\)\(112\)\(423\)
Has a Majority of Rare Baseball Cards\(11\)\(16\)\(9\)\(6\)\(35\)\(77\)
Total\(91\)\(100\)\(80\)\(82\)\(147\)\(500\)
(a) If one collector from the sample is selected at random, what is the probability that the collector has been collecting baseball cards for \(11\) or more months and has a majority of regular baseball cards? Show your work.
(b) Given that a randomly selected collector from the sample has been collecting baseball cards for fewer than \(6\) months, what is the probability the collector has a majority of regular baseball cards? Show your work.
(c) Michelle believes there is a relationship between the number of months spent collecting baseball cards and which type of card is the majority in the collection (regular or rare).
(i) Name the hypothesis test Michelle should use to investigate her belief. Do not perform the hypothesis test.
(ii) State the appropriate null and alternative hypotheses for the hypothesis test you identified in (c-i). Do not perform the hypothesis test.
(d) After completing the hypothesis test described in part (c), Michelle obtains a p-value of \(0.0075\). Assuming the conditions for inference are met, what conclusion should Michelle make about her belief? Justify your response.

Most-appropriate topic codes (CED):

• TOPIC 4.3: Introduction to Probability
• TOPIC 4.5: Conditional Probability
• TOPIC 8.5: Setting Up a Chi-Square Test for Homogeneity or Independence
• TOPIC 8.6: Carrying Out a Chi-Square Test for Homogeneity or Independence
▶️ Answer/Explanation
Detailed solution

(a)
The number of collectors who have been collecting for \(11\) or more months and have a majority of regular cards is the sum of the counts in the corresponding cells: \(71 + 76 + 112 = 259\).
The total number of collectors in the sample is \(500\).
The probability is \(\frac{259}{500} = 0.518\).

(b)
This is a conditional probability. The sample is restricted to the \(91\) collectors who have been collecting for fewer than \(6\) months. Of these, \(80\) have a majority of regular cards.
The probability is \(\frac{80}{91} \approx 0.879\).

(c)
(i) The appropriate test is a chi-square test for independence (or association).
(ii) The hypotheses are:
– \(H_0\): There is no association between the number of months spent collecting baseball cards and the majority type of card in the collection for all baseball card collectors at the convention.
– \(H_a\): There is an association between the number of months spent collecting baseball cards and the majority type of card in the collection for all baseball card collectors at the convention.

(d)
Since the p-value of \(0.0075\) is less than any common significance level (e.g., \(\alpha=0.05\)), Michelle should reject the null hypothesis.

There is convincing statistical evidence of an association between the number of months spent collecting baseball cards and the majority type of card in the collection for all baseball card collectors at the convention.

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