AP
®
STATISTICS
2010 SCORING COMMENTARY
Question 2
Overview
The primary goals of this question were to assess students’ ability to (1) describe a sampling distribution of
a sample mean; (2) set up and perform a normal probability calculation based on the sampling distribution.
Sample: 2A
Score: 4
This is a very efficient, well-written response. The student describes all three components of the sampling
distribution — shape, center and spread — correctly in part (a). In addition, the student provides an accurate
justification (with good notation) for the values of the mean and standard deviation of this distribution. This
part was scored as essentially correct. In part (b) the student begins by computing the mean and standard
deviation of total airtim
e for random samples of 40 songs by using rules for linear transformation of a random
variable. The student then sets up the z-score correctly and obtains the desired probability. This part was
scored as essentially correct. Because parts (a) and (b) were both scored as essentially correct, the response
earned a score of 4.
Sample: 2B
Score: 3
The response includes an accurate description with strong justification for the shape, center and spread of
the sampling distribution in part (a). Although the student uses the notation “N(3.9, .174)” in the last line of
the answer, the narrative comments clarify that the distribution “is approximately normal.” This part was
scored as essentially correct. The student correctly calculates the expected total airtime for 40 songs in
part (b). However, the student’s computation of the standard deviation is not correct. Using this incorrect
standard deviation results in a plausible but incorrect z-score of 3.636. Because the student obtains the
correct upper-tail probability that corresponds to this z-score, this part was scored as partially correct. With
part (a) scored as essentially correct and part (b) scored as partially correct, the response earned a score of 3.
Sample: 2C
Score: 2
This response shows developing but incomplete understanding of sampling distributions. The first sentence
in part (a) gives a correct value for the center (“3.9 minutes”) but seems to be referring to the distribution of
song lengths in a single random sample. The “standard deviation of the sample” remark in the second
sentence confirms that the student is not describing the sampling distribution of the sample mean. As a
result, this part was scored as incorrect. In part (b) the student gives the correct expected time required to
play 40 randomly selected songs and computes the standard deviation of the total airtime correctly, by
multiplying the values obtained in part (a) by 40. The student then sets up and performs the normal
probability calculation correctly, so this part was scored as essentially correct. Because part (a) was scored as
incorrect and part (b) was scored as essentially correct, the response earned a score of 2.
© 2010 The College Board.
Visit the College Board on the Web: www.collegeboard.com.