AP Statistics 2.1: Categorical Data Displays

Compare two categorical variables with two-way tables, segmented bars, and mosaic plots, using conditional proportions to discuss association.

[Maximum number: 4]

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.

No, the segment for small bottles for the elementary school is wider than the segment for small bottles for the middle school; however, the middle school students sold three times as many bottles as the elementary school students. So, if the elementary school sold x number of bottles, the middle school sold 3 x number of bottles.

For example, if the elementary students sold 100 bottles total, then they sold 0.5(100) or 50 small bottles. However, because the middle school students sold three times the total number of bottles as the elementary students, they would have sold 300 bottles total and $(0. \overline{3})(300)=100$ small bottles. Because 100 > 50, the middle school sold more small bottles, and the elementary school's administrator is not correct.

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

States that the elementary school's administrator is incorrect

Provides correct mathematical support verifying the middle school sold more small bottles than the elementary school, consistent with the response to component 1

Includes context

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

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

Additional Notes:

- A response that shows mathematical support by showing the inequalities $0.5 x<0. \overline{3}(3 x), 0.5 x<x$ , $0.5(1)<0. \overline{3}(3)$ , or $(0.5)\left(\frac{1}{4}\right)<(0. \overline{3})\left(\frac{3}{4}\right)$ may satisfy component 2.

- In order to satisfy component 3, a response must include "elementary," "middle," and "bottles."

- Part (b) should be scored consistent with the response to part (a).

(c)

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

A mosaic plot for the distribution of the number of bottles sold by each of the high schools is shown here.

(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.

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Covered syllabus points

2.1 Tabular and Graphical Representations for the Distributions of Two Categorical Variables question 1

Question (a)

Question (b)

Question (c)

Question (i)

Question (ii)