AP
®
STATISTICS
2012 SCORING COMMENTARY
Question 6
Overview
The primary goals of this question were to assess students’ ability to (1) implement simple random
sampling; (2) calculate an estimated standard deviation for a sample mean; (3) use properties of variances
to determine the estim
ated standard deviation for an estimator; (4) explain why stratification reduces a
standard error in a particular study.
Sample: 6A
Score: 4
In part (a) the 2,000 students in the high school are assigned numbers from 0000 to 1,999. A random
selection of 100 unique numbers from
a random integers table is clearly described by ignoring repeated
four-digit numbers and by ignoring the four-digit numbers greater than 1,999. The 100 unique random
numbers are used to select the 100 students with the corresponding numbers. Thus, part (a) was scored as
essentially correct. In part (b) the formula for the standard deviation of the sample mean is correctly
identified by the appropriate statistics from the sample, and these statistics are used to calculate the
correct standard error of Peter’s point estimator. Thus, part (b) was scored as essentially correct. The
variances, weights, and sample sizes for female responses and male responses in Rania’s samples are
correctly combined in part (c) to produce the correct estimated standard deviation of the stratified sample
mean. Thus, part (c) was scored as essentially correct. In part (d) the standard deviations for each gender
are identified as being smaller in Rania’s sample data than the standard deviation in Peter’s sample data.
The two smaller variances for each gender are explicitly linked to the smaller sample standard deviation for
Rania’s point estimator with the phrase “the rule for variances needed to calculate Rania’s.” Hence,
part (d) was scored as essentially correct. With all four parts scored as essentially correct, the response
earned a score of 4.
Sample: 6B
Score: 3
In part (a) the response implicitly identifies each student with a number from 1 to 2,000. A random number
generator is used to produce 100 unique numbers, and the students corresponding to the 100 unique
numbers are selected for the simple random sample. Thus, part (a) was scored as essentially correct.
In
part (b) the formula for the standard deviation of the sample mean is correctly identified by use of the
appropriate sample statistics, and the correct standard error of Peter’s point estimator is calculated. Thus,
part (b) was scored as essentially correct. In part (c) the standard errors for the sample means for the
female responses and male responses are correctly calculated, but the attempt to combine the standard
deviations rather than the variances produces an incorrect estimate for the standard deviation of Rania’s
point estimator. Thus, part (c) was scored as partially correct. The response in part (d) identifies the
difference in sample means for the female and male responses, but the linkage of the reduced variability in
the stratified sample to a smaller estimated standard deviation of the estimator for the mean is insufficient.
Thus, part (d) was scored as partially correct. With two parts scored as essentially correct and two parts
scored as partially correct, the response earned a score of 3.
© 2012 The College Board.
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