AP® STATISTICS

STATISTICS

2010 SCORING GUIDELINES (Form B)

Question 2

Intent of Question

The primary goals of this question were to assess students’ ability to (1) describe a simple random

sam

pling procedure; (2) identify an effective stratification variable; (3) provide a statistical advantage of a

stratified random sample over a simple random sample in context.

Solution

Part (a):

The administrators could number an alphabetical list of students from 1 to 2,500. They could then use a

random

number generator from a calculator or computer to generate 200 unique random integers from

1 to 2,500. The students corresponding to those 200 numbers would be asked to participate in the

survey.

Part (b):

One possible stratification variable might be the school level of the student (elementary, middle, high

school). The students’ perceptions of the importance of good nutrition in food served may differ

depending on the students’ ages and therefore on school levels. For exam

ple, there may be a

difference between what elementary students value in food served as opposed to middle school and

high school students.

Part (c):

One statistical advantage of using stratified random sampling as opposed to simple random sampling

is, for example, if the elementary, middle and high sch

ool strata create groups that differ with respect

to what they value ⎯ and are therefore more homogeneous with respect to opinion on this issue ⎯

then for the same overall sample size a more accurate estimate of the overall proportion of students

who are satisfied with the food under this contract may result. Another advantage is that stratified

random sampling guarantees that each of the school-level strata will have some representation,

because it is possible that a simple random sample would miss one or more of the strata completely.

Scoring

Parts (a), (b) and (c) are each scored as essentially correct (E), partially correct (P) or incorrect (I).

Part (a) is scored as follows:

Essentially correct (E) if the response describes a sampling procedure for generating a simple random

sample and the description of the sampling procedur

e includes sufficient detail for implementation.

Partially correct (P) if random selection is used correctly for generating a simple random sample, but

the description of the sampling procedure does not

provide sufficient detail for implementation.

Incorrect (I) if random selection is not used in a correct way for a simple random sample.

Note: A response in which objects are placed into a hat or a box and then drawn out can only earn an “E”

if the response explicitly states that the objects are mi

xed or that they are drawn out at random.

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STATISTICS

2010 SCORING GUIDELINES (Form B)

Question 2 (continued)

Part (b) is scored as follows:

Essentially correct (E) if the response identifies a reasonable stratification variable and provides a

reasonable justification in context (such as stating, “the groups (strata) might differ with respect to

food preferences or nutritional awareness”).

Partially correct (P) if the response identifies a r

easonable stratification variable but provides a weak

justification (such as stating only, “the groups (strata) differ”).

Incorrect (I) if the response identifies an unreasonable stratification variable, or provides an

unreasonable justification or no justification.

Part (c) is scored as follows:

Essentially correct (E) if the response provides a reasonable statistical advantage of stratified random

sampling that is not also true of random sampling, and that is clearly communicated and in context.

Partially correct (P) if the response provides a reasonable statistical advantage that is either not well

comm

unicated or that is not in context.

Incorrect (I) if the response includes only a vague potential statistical advantage, such as “data more

accurate” or “stratified random sampling is better.”

Note: Responses to part (c) such as “stratified random sampling allows for inferences to be drawn for

the three grade levels separately about the feelings of students in those grade levels” should be

considered incorrect unless also accompanied by a statistical advantage specific to stratified random

sampling.

4 Complete Response

All three parts essentially correct

3 Substantial Response

Two parts essentially correct and one part partially correct

Developing Response

Two parts essentially correct and one part incorrect

One part essentially correct and one or two parts partially correct

Three parts partially correct

1 Minimal Response

One part essentially correct and two parts incorrect

Two parts partially correct and one part incorrect

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STATISTICS

2010 SCORING COMMENTARY (Form B)

Question 2

Sample: 2A

Score: 4

In part (a) the student assigns a number to each of the 2,500 students in the district, then proposes to use a

comp

uter to generate 200 random numbers to identify the students to be selected in the simple random

sample. This proposed procedure can be implemented given access to the random number generator on a

computer. The student identifies school level as an effective variable on which to stratify in part (b) and then

explains that the students at different levels may vary in food satisfaction owing to maturity and level of

awareness. In part (c) the student states that the “lurking variable” of school level would be eliminated by

stratifying, when in fact differences in food satisfaction according to school level are modeled, not eliminated.

The student recognizes that grouping students into strata according to knowledge regarding health can help

to provide accurate conclusions. Parts (a), (b) and (c) were all scored as essentially correct. The entire

answer, based on all three parts, was judged a complete response and earned a score of 4.

Sample: 2B

Score: 3

In part (a) the student proposes to use a random number table to identify which of the 2,500 students in the

school distric

t will be selected for the simple random sample. The student carefully numbers each member of

the population, using four digits to ensure that they are equally likely to be selected. It is true that school level

is an example of an effective variable on which to stratify, as requested in part (b), but the response fails to

link the variable to students’ opinions about satisfaction with the food under this contract. The response in

part (c) describes the possibility of missing all of the students from one of the three school buildings. Parts (a)

and (c) were scored as essentially correct, but part (b) was scored as only partially correct. The entire answer,

based on all three parts, was judged a substantial response and earned a score of 3.

Sample: 2C

Score: 2

In the solution to part (a) the student proposes to number the 2,500 students in the school district and then to

use a

random number generator to select 200 students at random. This procedure will yield a simple random

sample of the students in the district. In part (b) the student identifies an effective variable on which to

stratify and notes that students in the different school levels may like to eat different kinds of food. In part (c)

the student fails to notice that a simple random sample is also likely to provide information about how

students in the different strata differ in their satisfaction with the food. Parts (a) and (b) were scored as

essentially correct, but part (c) was scored as incorrect. The entire answer, based on all three parts, was

judged a developing response and earned a score of 2.

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