AP
®
STATISTICS
2011 SCORING COMMENTARY (Form B)
Question 5
Sample: 5A
Score: 4
Part (a) correctly identifies the procedure to be a “[o]ne sample z confidence interval for [a] proportion.” The
response notes that random sampling is given in the statement of the problem
and includes appropriate
checks of the sample-size conditions based on the observed counts of successes and failures. Step 1 of the
solution to part (a) was scored as essentially correct. In step 2 of part (a) the mechanics are shown and are
correct for the calculation of the interval identified in step 1, so step 2 of the solution to part (a) was scored as
essentially correct. After stating that the interval is a 99 percent confidence interval, the response notes that
“0.45 is not contained in the interval,” and so it is not reasonable to believe that 45 percent of the vaccine-
eligible people had received flu vaccine. The solution to step 3 of part (a) was scored as essentially correct.
The solution to part (b) incorporates 2.576, the correct critical value for a 99 percent confidence interval based
on a normal distribution, and also includes 0.5 as the value of the proportion. The conservative choice of 0.5
guarantees that the margin of error based on the resulting sample size calculation will be no greater than
0.02. Part (b) was scored as essentially correct. Because the three steps in part (a) were scored as essentially
correct and part (b) was also scored as essentially correct, the response earned a score of 4.
Sample: 5B
Score: 3
This response notes that “[t]he sample is random” but also that “2350 is more than 10% of the population.”
This sample-size restriction is both incomplete (because it does not restrict the numbers of successes and
failures) and incorrect (because it should read “less than” rather than “more than”). The procedure is
identified as a “one proportion z interval test.” Because of the incorrect sample-size condition, step 1 of
part (a) was scored as partially correct. Because the correct “one proportion z interval test” procedure is
named, and the correct interval is provided, step 2 of part (a) was scored as essentially correct. The response
then compares 0.45 to the resulting confidence interval by indicating that the interval of people that receive
this flu vaccine is less than 45 percent. The response includes context and a direct comparison of 0.45 to the
interval, so step 3 of part (a) was scored as essentially correct. The response states that “the belief is wrong,”
which is not necessarily correct, although the evidence indicates that 0.45 is not a plausible value for the true
proportion. In part (b) the response substitutes the observed proportion 0.41617 into the formula for the
margin of error. The observed proportion is not as conservative as using the value 0.5, so the resulting sample
size does not guarantee that the margin of error will be no greater than 0.02. Part (b) was scored partially
correct. Because one step in part (a) was scored as essentially correct, and part (b) was scored as essentially
correct, while two steps in part (a) were scored as partially correct, the response earned a score of 3.
Sample: 5C
Score: 2
The solution to part (a) includes no consideration of conditions for inference, so step 1 of part (a) was scored
as incorrect. The correct procedure (“1 proportion z interval”) is name
d, and the correct interval is calculated.
Step 2 of part (a) was thus scored as essentially correct. The response notes that 45 percent is not included in
the interval and that “we can’t say that 45% of vaccine-eligible people received flu vaccine.” This response
was scored essentially correct for step 3 of part (a). The solution in part (b) does not provide a formula,
includes 0.45 in the numerator of the standard error instead of the more statistically conservative value 0.5,
and does not include the square root in the standard error. Therefore, part (b) was scored as incorrect.
Because two steps in part (a) were scored as essentially correct, one step in part (a) was scored as incorrect,
and part (b) was scored as incorrect, the response earned a score of 2.
© 2011 The College Board.
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