AP Statistics 1.9: Comparing Distributions

Compare quantitative distributions by combining graph features, summary statistics, z-scores, and contextual evidence about relative position.

Covered syllabus points

1.9.A—Compare multiple quantitative one-variable graphical representations 1.9.B—Compare multiple quantitative one-variable graphical representations of summary statistics 1.9.C—Justify a claim using multiple quantitative one-variable graphical representations 1.9.D—Calculate z-scores with population parameters 1.9.E—Compare z-scores as measures of relative position for distributions

The manager of an automotive company is interested in comparing the gas mileages for cars

manufactured in Country A and cars manufactured in Country B. The manager selected a

random sample of 100 cars manufactured in Country A and a random sample of 100 cars

manufactured in Country B. The gas mileages for each sample, in miles per gallon (mpg), are

summarized in the boxplots.

Boxplots of Gas Mileage for Each Country

Compare the distributions of gas mileage for the sample of cars manufactured in Country A

and the sample of cars manufactured in Country B.

	Model Solution	Scoring
A	The distribution of gas mileage for the sample of cars manufactured in Country A has a lower center than the distribution of gas mileage for the sample of cars manufactured in Country B. The median gas mileage for the sample of cars manufactured in Country A (18 mpg) is less than the median gas mileage for the sample of cars manufactured in Country B (32 mpg).<br>The range of the gas mileages for the sample of cars manufactured in Country A (24 mpg) is slightly greater than the range of the gas mileages for the sample of cars manufactured in Country B (22 mpg). However, the IQR of the gas mileages for the sample of cars manufactured in Country A (8 mpg) is less than the IQR of the gas mileages for the sample of the cars manufactured in Country B (12 mpg).<br>The car manufactured in Country A with 38 mpg (the maximum of the sample of cars manufactured in Country A) is an outlier, while the distribution of gas mileage for the sample of cars manufactured in Country B has no outliers.	Essentially correct (E) if the response satisfies at least three of the following four components: 1. Directly compares the center for the two distributions 2. Directly compares the spread (either IQR or range) for the two distributions 3. Indicates that the gas mileage of one of the cars manufactured in Country A is an outlier 4. Provides sufficient context, which includes the manufacturing countries ("Country A" and "Country B") AND the dependent variable ("gas mileage" or "mpg") Partially correct (P) if the response satisfies only two of the four components required for E. Incorrect (I) if the response does not meet the criteria for E or P.

Scoring Notes:

- The response need not include specific numerical values to satisfy any given component or to score E on

part A.

- A response that only uses the "means" in the comparison of center would not satisfy component 1.

- Any acceptable mention of shape in the response should be ignored because complete shape information

cannot be determined from a boxplot. Acceptable mentions of shape include:

○ The shape of the distribution of the gas mileage for cars manufactured in Country A can be

described as skewed, positively skewed, or right skewed.

○ The shape of the distribution of the gas mileage for cars manufactured in Country B can be

described as skewed, negatively skewed, left skewed, or approximately symmetric.

- If the response describes the shape of either distribution as just "symmetric," "normal," "unimodal," or an

incorrect shape (e.g., "the distribution of gas mileages for Country A is left skewed" or "the distribution of

gas mileages for Country B is right skewed"), then part A cannot be scored E.

- A response that only lists values for center and/or spread and does not directly compare them does not

satisfy components 1 and/or 2.

- A response that just refers to "A" or "B" AND the dependent variable ("gas mileage" or "mpg") may

satisfy component 4.

[Maximum number: 1]

Consider the following back-to-back stemplot: \begin{tabular}{r|l|ll}
& 0 & 348 & Key: 2|5| 6 represents \\
1 & 01256 & a value of 52 on the left \\
843 & 2 & 29 & and 56 on the right \\
65210 & 3 & 2557 & \\
92 & 4 & & \\
7552 & 5 & 6 & \\
6 & 6 & 1458 & \\
6 & 7 & 09 & \\
8541 & 8 & & \\
90 & 9 & &
\end{tabular} Which of the following is a correct statement?

The distributions have the same mean.

The distributions have the same median.

The interquartile range of the distribution to the left is 20 greater than the interquartile range of the distribution to the right.

The distributions have the same variance.

[Maximum number: 4]

The chemicals in clay used to make pottery can differ depending on the geographical region where the clay originated. Sometimes, archaeologists use a chemical analysis of clay to help identify where a piece of pottery originated. Such an analysis measures the amount of a chemical in the clay as a percent of the total weight of the piece of pottery. The boxplots below summarize analyses done for three chemicals-X, Y, and Z-on pieces of pottery that originated at one of three sites: I, II, or III.

Figure for Question 1.9 Comparisons of the Distributions for One Quantitative Variable question 3 — AP Statistics

(a)

For chemical Z, describe how the percents found in the pieces of pottery are similar and how they differ among the three sites.

(b)

Consider a piece of pottery known to have originated at one of the three sites, but the actual site is not known.

(i)

Suppose an analysis of the clay reveals that the sum of the percents of the three chemicals X, Y, and Z is 20.5\%. Based on the boxplots, which site-I, II, or III-is the most likely site where the piece of pottery originated? Justify your choice.

(ii)

Suppose only one chemical could be analyzed in the piece of pottery. Which chemical-X, Y, or Z- would be the most useful in identifying the site where the piece of pottery originated? Justify your choice.

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Covered syllabus points

1.9 Comparisons of the Distributions for One Quantitative Variable question 1

1.9 Comparisons of the Distributions for One Quantitative Variable question 2

1.9 Comparisons of the Distributions for One Quantitative Variable question 3

Question (a)

Question (b)

Question (i)

Question (ii)