Erica Chauvet is a Statistics Professor at Waynesburg University and a former high school AP Statistics teacher. She is on the textbook team for *The Practice of Statistics* and *Statistics and Probability with Applications*. She has been an AP Statistics Exam reader for the past 16 years where she enjoys hosting the 1.96-mile 95% Confidence (Prediction) Fun Run among other social events. Erica loves to play tennis and enjoyed playing doubles with Luke Wilcox in Tampa at this year's reading.

**The Question - **__2024 #2__

__2024 #2__

*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.*

**Part (a)**

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

**WOULD THIS GET CREDIT?**

To earn an (E – Essentially correct), both graphs must be partitioned and shaded correctly. If only one of the two graphs were partitioned and shaded correctly, then the score was a (P – Partially correct). Otherwise, incorrect.

### Teaching Tips:

Teach your students to read carefully.

Ask students to be neat when making graphs. The solid region should not look like stripes. They should use vertical lines to mark the boundaries to make it clear they partitioned the regions correctly.

Students need to have a basic understanding of fractions (how to divide a region into 3 equal pieces).

**Part (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.*

**WOULD THIS GET CREDIT?**

To earn an (E – Essentially correct), the student must satisfy three components: (**1**) answer “No”, (**2**) provide a correct calculation to support their answer, and (**3**) provide context, specifically “elementary,” “middle,” and “bottles.”

If the graphs in part (a) were identical, the student’s response in part (b) could earn an E if answered in a manner consistent with their work in part (a).

All three components are needed for an (E).

Two of the three components are needed for a (P).

### Teaching Tips:

Remind your students to use the words exactly as they are given in the problem. They should not call bottles, “cups.” They should not refer to elementary school students as “lower school students.”

Remind your students to answer the question directly. This was a “Yes/No” question and a direct Yes/No or “The administrator is incorrect” was needed for full credit.

When explaining their response, they should show their work. Although (1/3)(3) is a simple calculation, there are many incorrect ways to get a value of 1, so they need to show they carried out a correct calculation.

Always include context in solutions for free response questions.

**Part (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.*

**WOULD THIS GET CREDIT?**

These two parts were scored together and there were four components: (**1**) answer “A” for the first question and (**2**) support it with either 0.7 > 0.6 or that the “relative frequency” is greater for A than B. For the second question, students needed to (**3**) answer “B” and (**4**) support the answer by showing or describing that the large bottle area in the mosaic plot is greater for High School B than for High School A.

All four components were needed for an (E). Two or three of the four components were needed for a (P).

Let’s focus on some of the ways students could get component 4:

### Teaching Tips:

Be sure students understand the difference between frequencies and relative frequencies (see this lesson) and the difference between segmented bar graphs and mosaic plots (see this lesson).

Remind students to be specific. There are many different “areas” in a mosaic plot. Students could have even drawn arrows to the large bottle areas if they struggled to describe them in words.

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