AP Statistics 1.4 Representing a Categorical Variable with Graphs- FRQs - Exam Style Questions

Question

A local elementary school decided to sell bottles printed with the school district’s logo as a fund-raiser. The students in the elementary school were asked to sell bottles in three different sizes (small, medium, and large). The relative frequencies of the number of bottles sold for each size by the elementary school were \(0.5\) for small bottles, \(0.3\) for medium bottles, and \(0.2\) for large bottles.

A local middle school also decided to sell bottles as a fund-raiser, using the same three sizes (small, medium, and large). The middle school students sold three times the number of bottles that the elementary school students sold. For the middle school students, the proportion of bottles sold was equal for all three sizes.

(a) Complete the segmented bar graphs representing the relative frequencies of the number of bottles sold for each size by students at each school.

(b) An administrator at the elementary school concluded that the elementary school students sold more small bottles than the middle school students did. Is the elementary school administrator’s conclusion correct? Explain your response.

Two high schools are also selling the bottles and are competing to see which one sold more large bottles.

(i) Which of the two high schools sold a greater proportion of large bottles? Justify your answer.

(ii) Which of the two high schools sold a greater number of large bottles? Justify your answer.

Most-appropriate topic codes (CED):

• TOPIC 1.4: Representing a Categorical Variable with Graphs
• TOPIC 2.2: Representing Two Categorical Variables
• TOPIC 2.3: Statistics for Two Categorical Variables

▶️ Answer/Explanation

Detailed solution

(a)
The completed segmented bar graphs are shown below.

– For the Elementary School, the segments are partitioned at \(0.5\) (for Small), \(0.8\) (for Medium, \(0.5+0.3\)), and \(1.0\) (for Large, \(0.8+0.2\)).
– For the Middle School, the proportions are equal for all three sizes, so each size represents \(\frac{1}{3}\) of the total. The segments are partitioned at \(\frac{1}{3} \approx 0.33\) and \(\frac{2}{3} \approx 0.67\).

(b)
No, the elementary school administrator’s conclusion is incorrect.

Explanation:
Although the proportion of small bottles sold by the elementary school (\(0.5\)) is greater than the proportion sold by the middle school (\(\approx 0.33\)), the middle school sold three times as many total bottles. Let \(N\) be the total number of bottles sold by the elementary school. Then the middle school sold \(3N\) bottles.
– Number of small bottles sold by Elementary: \(0.5 \times N\)
– Number of small bottles sold by Middle: \(\frac{1}{3} \times (3N) = N\)
Since \(N > 0.5N\), the middle school sold more small bottles.

(c)
(i) High School A sold a greater proportion of large bottles.
Justification: The proportion is represented by the height of the segment. The segment for “Large bottles” is taller for High School A (from \(0.7\) to \(1.0\), a height of \(0.3\)) than for High School B (from \(0.6\) to \(0.8\), a height of \(0.2\)).

(ii) High School B sold a greater number of large bottles.
Justification: The number of bottles is represented by the area of the rectangle (height \(\times\) width). The rectangle for large bottles at High School B is visibly wider than the rectangle for High School A. Although it is shorter, its greater width gives it a larger overall area, representing a greater number of large bottles sold.

AP Statistics 1.4 Representing a Categorical Variable with Graphs- FRQs - Exam Style Questions

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