AP Statistics 1.9: Comparing Relative 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