AP Statistics 1.9 Comparisons of the Distributions for One Quantitative Variable Questions

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.

D

Scoring

This question is scored in three sections. Section 1 consists of part (a), section 2 consists of part (b-i), and section 3 consists of part (b-ii). Each section is scored as essentially correct (E), partially correct (P), or incorrect (I).

Section 1 is scored as follows:
Essentially correct (E) if the response includes the following three components:
1. Recognition that the medians or centers are almost the same for the three sites
2. Recognition that the variability (ranges, IQRs, spread) is different across the three sites
3. Context is included

Partially correct (P) if the response includes only two of the three components.
Incorrect (I) if the response includes at most one of the three components.

Notes:
- In all sections, comments about shape should be ignored because complete shape information is not obtainable from boxplots.
- Responses are not required to give numerical values. If responses provide numerical values, any reasonable approximation from the boxplots is acceptable.
- Because the boxplots are all symmetric, it is acceptable if the response discusses means instead of medians.
- Any discussion of chemical X and chemical Y is considered extraneous.
- Context is satisfied by any of the following references: site, chemical, weight, total weight, X, Y, or Z.

Section 2 is scored as follows:
Essentially correct (E) if the response includes the following three components:
1. Site III is chosen.
2. Sums of the minimum and maximum are computed for the three chemicals at each site.
3. A reasonable numerical justification is given involving sums of a statistical measure across the three chemicals to choose Site III.

Partially correct (P) if the response includes only two of the three components.
Incorrect (I) if the response includes at most one of the three components.
Notes:
- If the response computes only the sum of the minimums for Site I and the sum of the maximums for Site II and recognizes that this is sufficient, the response is scored E.
- If an alternative measure is used that involves sums of the three chemicals, such as the sum of the medians or the sums of the first quartiles and sums of the third quartiles, instead of the minimum and maximum sums, the second component is not satisfied, but the third component might be satisfied.
○ If the response explicitly or implicitly compares the alternate sum to the other two sites (for example, by indicating that the sum is the closest to 20.5 percent or by listing the sums for all three sites) the response is scored P.
○ If the response does not have an implicit or explicit comparison, the response is scored I.
- If either Site I or Site II is identified as the correct choice, no matter how that choice is justified, the response is scored I.
- The approximate sums of the medians are 27.5 for Site I, 16 for Site II, and 20 for Site III.

Section 3 is scored as follows:
Essentially correct (E) if the response chooses chemical Y AND gives a reasonable justification based on the fact that the distributions of chemical Y are distinctive across sites.

Partially correct (P) if the response chooses chemical Y AND provides justification based on the boxplots, but does not clearly explain that the distributions of chemical Y are distinctive across sites; OR
if the response correctly discusses that the distributions of chemical Y are distinctive across sites, but never explicitly chooses chemical Y as the best choice, for instance, by stating only that there is substantial overlap across sites for chemicals X and Z but no overlap for chemical Y.

Incorrect (I) if the response does not meet the criteria for E or P.
Notes:
- To justify that the distributions of chemical Y are distinctive across sites, the justification must address both location and variability of the boxplots; for example, by stating that the boxplots do not overlap for chemical Y.
- If the response chooses chemical X or Z OR chooses chemical Y with no reasonable justification, the response is scored I.
- The justification that the distributions of chemical Y are distinctive across sites:
○ The following are acceptable because both location and variability are addressed. Such responses are scored E.
- The boxplots for chemical Y do not overlap, or the boxplots for chemicals X and Z overlap.
- All values of Site I are high, all values of Site II are low, and all values of Site III are in the middle.
- The ranges never intersect.
- The boxplots share no data.
- Has completely different percentages at each site.
○ The following are incomplete justifications and are scored P.
- The boxplots vary.
- Chemical Y varies the most.
- Chemical Y has the greatest variation.
- The variation between/among sites is the largest.
- The boxplots are different.
- The medians/means differ.
- The medians/means are most variable.
- There is a difference in the percentages of chemical Y for each site.
- The distribution of percents differs the most among the sites.
4 Complete Response
Three sections essentially correct
3 Substantial Response
Two sections essentially correct and one section partially correct
2 Developing Response
Two sections essentially correct and no sections partially correct
OR
One section essentially correct and one or two sections partially correct
OR
Three sections partially correct
1 Minimal Response
One section essentially correct
OR
No sections essentially correct and two sections partially correct

Part (b):
The z-score for a player who can lift a weight of 370 pounds is $z=\frac{370-310}{25}=2.4$. The z-score indicates that the amount of weight the player can lift is 2.4 standard deviations above the mean for all previous players in this position.

Part (c):
Because the two variables - time to run 40 yards and amount of weight lifted - are recorded on different scales, it is important not only to compare the players' values but also to take into account the standard deviations of the distributions of the variables. One reasonable way to do this is with z-scores.

The z-scores for the 40-yard running times are as follows:

The z-scores for the amount of weight lifted are as follows:

The z-scores indicate that both players are faster than average in the 40-yard running time and both are well above average in the amount of weight lifted. Player A is better in running time, and Player B is better in weight lifting. But the z-scores also indicate that the difference in their weight lifting (a difference of 0.2 standard deviation) is quite small compared with the difference in their running times (a difference of 1.0 standard deviation). Therefore, Player A is the better choice, because Player A is much faster than Player B and only slightly less strong.

Scoring

Parts (a), (b) and (c) are scored as essentially correct (E), partially correct (P) or incorrect (I).
Part (a) is scored as follows:
Essentially correct (E) if the answer is "no" AND the response provides a reasonable explanation, based on the relationship between the mean, standard deviation, and minimum value of a data set whose distribution can be approximated by a normal distribution.

Partially correct (P) if the answer is "no" but the explanation is weak.
Incorrect (I) if the answer is "no" without an explanation or with an unreasonable explanation, OR if the response concludes that it is reasonable to believe that the distribution is approximately normal.

Notes

- A reasonable explanation should describe a characteristic of a normal distribution that is substantially contradictory to the information given for the running time data so that the running time distribution cannot be reasonably approximated by a normal distribution.

- Plausible comments about the distribution of running times are considered extraneous.

- Incorrect comments about the distribution of running times can lower the score one level (that is, from E to P or from P to I), depending on the severity of the comment.

Part (b) is scored as follows:
Essentially correct (E) if the response calculates the z-score correctly AND provides a correct interpretation that includes direction.

Partially correct (P) if the response has only one of the two components (calculation and interpretation) correct.

Incorrect (I) if the response fails to meet the criteria for E or P.
Notes

- Calculating a probability from a normal distribution for the weights is considered extraneous and is not a sufficient interpretation of a z-score.

- Percentiles are extraneous and cannot be used to indicate direction from the mean, because the distribution cannot be determined from the information provided.

- Context is provided in the stem of problem and is not required for the response to be considered correct.

- Either the formula with correct symbols or with correct numerical values is needed in addition to the value 2.4 in the calculation of the z-score.

- A diagram can show direction from the mean, if the quantities are appropriately labeled.

Part (c) is scored as follows:
Essentially correct (E) if the response addresses the following three components:

Correct selection of Player A.

Numerical adjustments of the scales so that the players' values can be compared for BOTH variables: time to run 40 yards and amount of weight lifted.

Justification of the selection in component 1 by using the players' values on both variables with respect to the adjusted scales.

Partially correct (P) if the response has exactly two of the three components listed above.
Incorrect (I) if the response fails to meet the criteria for E or P.
Notes

- It is not necessary to calculate z-scores. For example, the following response is scored as essentially correct (E): "Players A and B are close in weight lifting, because the difference of 5 pounds is much less than 1 standard deviation (25 pounds), but much less close in running time because the difference is 0.15 seconds, which is exactly one standard deviation. Therefore, player A should be selected since he is considerably faster and almost as strong as player B."

- Component 3 is not satisfied by the statement, "Player A should be selected since the weights lifted are close and running times are less close," because the adjusted scales are not mentioned. Such a statement could apply to the original data, where the values are on different scales.

- The justification in component 3 must reference the adjusted scale for at least one variable AND at least be implied for the other variable.

- Normal probability calculations can be used in establishing the numerical scale adjustments for component 2 and for justifying the selection of the players in component 3. However, this results in a lowering of scores (that is, from E to P or from P to I) unless the student has concluded in part (a) that it was reasonable to believe that the distribution of running times was approximately normal.

- Conceptual miscalculation of z-scores or probabilities (for example, using the wrong mean, reversing the order of subtraction, or multiplying probabilities) results in the loss of credit for component 2, whereas minor arithmetic mistakes are overlooked.

Complete Response
All three parts essentially correct

Substantial Response
Two parts essentially correct and one part partially correct

Question 1

Developing Response
Two parts essentially correct and one part incorrect

OR
One part essentially correct and one or two parts partially correct

OR
Three parts partially correct

Minimal Response
One part essentially correct and two parts incorrect

OR
Two parts partially correct and one part incorrect