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AP Statistics 1.4 Graphical Representations for One Categorical Variable- Exam Style Questions - FRQs - New Syllabus

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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.
Segmented bar graphs for bottles sold by elementary and middle school students
(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.
(c) A mosaic plot for the distribution of the number of bottles sold by each of the high schools is shown here.
Mosaic plot for bottles sold by two high schools
(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 (AP Statistics):

• TOPIC 2.1: Tabular and Graphical Representations for the Distributions of Two Categorical Variables
• TOPIC 2.2: Summary Statistics for Two Categorical Variables
▶️ Answer/Explanation
Detailed solution
(a)
The completed segmented bar graphs are shown below.
Completed segmented bar graphs for bottles sold by elementary and middle school students
For the elementary school, the relative frequencies are \(0.5\), \(0.3\), and \(0.2\). Therefore, the segmented bar should be divided at \(0.5\), then at \(0.5+0.3=0.8\), and finally at \(1.0\). The segments represent small, medium, and large bottles, respectively.
For the middle school, the proportions are equal for all three sizes. Therefore, each size represents \(\frac{1}{3}\) of the total number of bottles sold. The segmented bar should be divided at \(\frac{1}{3}\approx 0.33\), then at \(\frac{2}{3}\approx 0.67\), and finally at \(1.0\).
(b)
No, the elementary school administrator’s conclusion is incorrect.
Although the elementary school had a greater proportion of small bottles sold, the middle school sold three times as many bottles in total. 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 the elementary school:
\(0.5N\)
Number of small bottles sold by the middle school:
\(\frac{1}{3}(3N)=N\)
Since \(N>0.5N\), the middle school sold more small bottles than the elementary school. The administrator confused a greater proportion with a greater actual number.
(c)(i)
High School A sold a greater proportion of large bottles.
In a mosaic plot, the proportion within a school is represented by the vertical height of the corresponding segment. For High School A, the large-bottle segment extends from about \(0.7\) to \(1.0\), giving a height of \(0.3\). For High School B, the large-bottle segment extends from about \(0.6\) to \(0.8\), giving a height of \(0.2\). Since \(0.3>0.2\), High School A sold a greater proportion of large bottles.
(c)(ii)
High School B sold a greater number of large bottles.
In a mosaic plot, the number of items in a category is represented by the area of the rectangle, not just the height. The area is based on both the width of the school group and the height of the large-bottle segment. High School B has a much wider rectangle than High School A, so even though its large-bottle proportion is smaller, the area of its large-bottle rectangle is greater. Therefore, High School B sold a greater number of large bottles.
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