AP Statistics 2.2: Two-Way Table Summaries

Calculate joint, marginal, and conditional relative frequencies from two-way tables, then compare categorical distributions in context.

[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.

[Maximum number: 4]

To compare success rates for treating allergies at two clinics that specialize in treating allergy sufferers, researchers selected random samples of patient records from the two clinics. The following table summarizes the data.

Table for Question 2.2 Summary Statistics for Two Categorical Variables question 2 — AP Statistics

(a)

(i) Complete the following table by recording the relative frequencies of successful and unsuccessful treatments at each clinic.

The relative frequencies of successful and unsuccessful treatments in each clinic are presented in the following table:

Clinic A

Clinic B

Unsuccessful
Treatment

$\frac{51}{139} \approx 0.3669$ $\frac{33}{68} \approx 0.4853$

Successful
Treatment

$\frac{88}{139} \approx 0.6331$ $\frac{35}{68} \approx 0.5147$

(b)

Based on the relative frequency table in part (a-i), which clinic is more successful in treating allergy sufferers? Justify your answer.

(c)

A physician who worked at both clinics believed that it was important to separate the patients in the study by severity of the patient's allergy (severe or mild). The physician constructed the following mosaic plot. The values in the mosaic plot represent the number of patients who were either successfully treated or unsuccessfully treated in each allergy severity group within each clinic. For example, the value 78 represents the number of patients successfully treated in the mild group within Clinic A.
CLINIC A
Based on the mosaic plot, the physician concluded the following:
For mild allergy sufferers, Clinic B was more successful in treating allergies.
For severe allergy sufferers, Clinic B was more successful in treating allergies.

Official question visual

(i)

(i) For each clinic, which allergy severity is treated more successfully? Justify your answer.

- Clinic A:

- Clinic B:

Model Solution
Clinic A treats mild allergies more successfully: $78/104=75.0\%$ for mild cases versus $10/35\approx28.6\%$ for severe cases.

Clinic B also treats mild allergies more successfully: $11/12\approx91.7\%$ for mild cases versus $24/56\approx42.9\%$ for severe cases.

Scoring components for part (c): correctly identifies and justifies the more successfully treated severity for both clinics using within-severity success rates.

(ii)

For each clinic, which allergy severity is more likely to be treated? Justify your answer.

- Clinic A:

- Clinic B:

Model Solution
Clinic A is more likely to treat mild cases: $104/139\approx74.8\%$ mild versus $35/139\approx25.2\%$ severe.

Clinic B is more likely to treat severe cases: $56/68\approx82.4\%$ severe versus $12/68\approx17.6\%$ mild.

Scoring components for part (c): correctly identifies and justifies the more common severity at each clinic using within-clinic proportions.

．A car company is investigating whether offering an optional navigation system in their cars at a lower price would

increase sales of the optional navigation system．The navigation system would give the driver directions to a

destination．

A researcher at the company surveyed a random sample of car owners who owned exactly one of their cars．Some

of the cars already had the optional navigation system，and some did not．The researcher asked the car owners

whether they would purchase the optional navigation system if offered at the lower price when they purchase their

next car．

The researcher found that 60 percent of owners had the optional navigation system．The researcher also found

that 18 percent of owners said，＂No，I would not purchase＂the optional navigation system at the lower price．Six

percent of the owners had a car with the optional navigation system and said，＂No，I would not purchase＂the

optional navigation system at the lower price．The following table summarizes this information．

Table for Question 2.2 Summary Statistics for Two Categorical Variables question 3 — AP Statistics

(a)

（a）Determine the probabilities for the empty cells in the table．Write your answers in the table．

(b)

（d）Consider your previous results．Should the car company offer the optional navigation system at the lower

price？Justify your answer．

Question 4

Begin your response to QUESTION 4 on this page．

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Covered syllabus points

2.2 Summary Statistics for Two Categorical Variables question 1

Question (a)

Question (b)

2.2 Summary Statistics for Two Categorical Variables question 2

Question (a)

Question (b)

Question (c)

Question (i)

Question (ii)

2.2 Summary Statistics for Two Categorical Variables question 3

Question (a)

Question (b)