AP Statistics 1.11: Random Sampling

C The farmer should write the region numbers

from row A, 1 through 5, onto same-size slips of

paper, then put the numbers into a hat, mix well,

and select one of the numbers. The farmer should

repeat this process for the region numbers of

each of the other rows (i.e., row B, 6 through 10;

row C, 11 through 15; row D, 16 through 20; row

E, 21 through 25) and select one number from

each row. This process will result in the

selection of one region from each row. The

farmer will examine every cabbage plant in each

of the selected regions for aphid damage to

determine the proportion of cabbage plants in the

selected regions that are damaged by aphids.

Alternative Solution:

The farmer should use a random number

generator to generate one two-digit integer from

01 to 05, one two-digit integer from 06 to 10, one

two-digit integer from 11 to 15, one two-digit

integer from 16 to 20, and one two-digit integer

from 21 to 25. For each integer selected, the

farmer should select the corresponding

numbered region and examine every cabbage

plant in each of the selected regions for aphid

damage to determine the proportion of cabbage

plants in the selected regions that are damaged

by aphids.

Scoring

Essentially correct (E) if the response satisfies the

following four components:

1. Describes a random selection process that

indicates the groupings used for the strata (by

rows)

2. Describes how to correctly implement a random

selection process for which the selection of

regions within each stratum is equally likely

3. Describes a random selection process for which

the selections across strata are independent

4. Describes a random selection process that results

in the selection of one region from each stratum

Partially correct (P) if the response satisfies only

two or three of the four components required for E.

Incorrect (I) if the response does not meet the

criteria for E or P.

Scoring Notes:

- A response may satisfy component 1 by referring to the five regions in the rows instead of using row letters.

- For responses that use cards or slips of paper:

○ If the number of slips of paper (number of cards) does not equal 5 for each random selection, then

component 1 is not satisfied. Slips of paper (cards) do not need to be specifically identified as

equally sized.

○ If the response does not describe a thorough mixing (shuffling) of the slips of paper (cards), then

component 2 is not satisfied.

- For responses that use a random number generator or table of random digits:

○ If it is not clear that a random selection process allows the selection of regions within each stratum to

be equally likely, then component 2 is not satisfied.

○ If the response does not clearly indicate that a random number is generated from the region numbers

within each of the five rows (e.g., only describes the generation of five random numbers from the

two-digit integers from 01 to 25), then component 4 is not satisfied.

- For responses that use a fair die:

○ If a five-sided fair die is rolled for each of the five rows and the response clearly indicates the region

numbers assigned to the values on the die for each roll, then component 4 is satisfied.

○ If a six-sided die is rolled for each of the five rows and the response clearly indicates which number

is excluded (e.g., "if a 6 is rolled, roll again until a non-6 number is achieved") AND the response

clearly indicates the region numbers (or columns) assigned to the values on the die for each roll, then

component 4 is satisfied.

○ If a 25-sided fair die is rolled for each of the five selections, then component 4 is not satisfied

without sufficient further justification because the selection of one region from each stratum is not

guaranteed.

- If a response describes two separate random selection processes in detail (e.g., describes how to use a

random number generator and slips of paper in a hat), score both descriptions according to the four

components and use the lower score.

- If a response indicates that separate samples were taken from each of the strata, then component 3 is

satisfied.

- A response that selects columns instead of regions within strata cannot satisfy component 3.