AP Statistics 2.2: Two-Way Table Statistics

Calculate joint, marginal, and conditional relative frequencies from a two-way table, keeping each denominator tied to the question.

[Maximum number: 4]

Baseball cards are trading cards that feature data on a player's performance in baseball games. 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.

	Fewer Than 6 Months	6-10 Months	11-15 Months	16-20 Months	21 or More Months	Total
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

Majority Type of Baseball Cards and Months of Collecting Baseball Cards

(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.

Model Solution
$P(\text{11+ months and majority regular})=\frac{71+76+112}{500}=\frac{259}{500}=0.518.$

Scoring components
1. Provides the correct probability.
2. Shows how the numerator 259 is computed.

(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.

$\begin{aligned} P(\text { majority regular cards } \mid \text { fewer than } 6 \text { months }) =\frac{P(\text { majority regular cards } \cap \text { fewer than } 6 \text { months })}{P(\text { fewer than } 6 \text { months })} =\frac{\frac{80}{500}}{\frac{91}{500}} =\frac{80}{91} =0.879 \end{aligned}$

Essentially correct (E) if the response satisfies the following two components:

Provides the correct probability

Shows work for the correct probability

Partially correct (P) if the response satisfies only one of the two components required for E.

Incorrect (I) if the response does not meet the criteria for E or P.

Additional Notes:

- An arithmetic or transcription error in a response can be ignored if correct work is shown.

- A response of $\frac{80}{91}$ satisfies both components 1 and 2.

- A response that satisfies components 1 and 2 for the probability that given a randomly selected collector has been collecting baseball cards for fewer than six months, they have a majority of rare baseball cards may be scored P.

- A specific probability statement is not required, but if correctly given should be considered a positive in holistic scoring.

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2.2.A—Calculate summary statistics from two-way tables question 1

Question (a)

Question (b)