AP Statistics 1.11 Random Sampling Questions

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.

Scoring

This question is scored in three sections. Section 1 consists of parts (a), (b), and (c), section 2 consists of part (d), and section 3 consists of parts (e) and (f). Sections 1, 2, and 3 are scored as essentially correct (E), partially correct (P), or incorrect (I).

Section 1 is scored as follows:
Essentially correct (E) if the response includes the following three components:
1. In part (a) the response says no A N D either argues that the sample will only be selected from one production line and not the entire population (that is, the sample will only represent one production line) O R argues that the tortillas from the two productions lines are different.
2. In part (b) the response chooses Method 1 AND refers to a relevant characteristic of the histogram (shape, center, or variability) that matches what would be expected when using Method 1 (or that does not match what would be expected when using Method 2).
3. In part (c) the response chooses Method 2 AND either justifies by stating that the sample comes from only one production line (does not come from both production lines) OR justifies by comparing the possible range of diameters for the two methods.

Partially correct (P) if the response includes only two of the three components.
Incorrect (I) if the response includes at most one of the three components.
Note: If a response includes more than one justification in an individual part, score the weaker of the two justifications. For example, a response for part (a) that says "No, because only one line was selected and because the sample size was too small" does not satisfy the first component because the sample size argument is incorrect.

Section 2 is scored as follows:
Essentially correct (E) if the response includes the following three components:
1. The shape is approximately normal.
2. The mean is 6 inches.
3. The standard deviation is $\frac{0.11}{\sqrt{200}}$ inch.

Partially correct (P) if the response includes only two of the three components.
Incorrect (I) if the response includes at most one of the three components.
Note: It is not necessary to include units (inches) for the mean or standard deviation.
Section 3 is scored as follows:
Essentially correct (E) if the response includes the following two components:
1. In part (e) the response chooses Method 1 AND describes the sampling distribution of the sample mean for Method 2 as having some sample means close to the mean of production line A (5.9 inches) and the other sample means close to the mean of production line B (6.1 inches).
2. In part (f) the response chooses Method 1 AND
- refers to a correct answer in part (e);
OR
- describes the sampling distribution of the sample mean for Method 2 as having some sample means close to the mean of production line A (5.9, less than 6) and the other sample means close to the mean of production line B (6.1, greater than 6);
OR
- argues that on a single day, it would be preferable to get a sample with a mean around 6 rather than getting a sample that would have a mean around 5.9 (less than 6) or a mean around 6.1 (greater than 6).

Partially correct (P) if the response includes one of the two components.
Incorrect (I) if the response does not meet the criteria for E or P.
Notes:
- In part (e), the response must be clear that there is more than one mean being described (365 sample means).
- Correct: The sample means will be around 5.9 or 6.1.
○ Not correct: The sample mean will be around 5.9 or 6.1.
- Parts (e) and (f) are not satisfied if the response does not imply the sample means vary from the population means in both methods (or does not imply that the sample mean varies from the population mean when making the single day argument in part (f)). However, if the mistake is made in part (e), do not penalize the response in part (f) for the same mistake.
- Correct: The sample means will be around 5.9 or 6.1 instead of close to 6 .
- Not correct: The sample means will be 5.9 or 6.1 instead of close to 6.
- Not correct: The sample means will be 6 instead of around 5.9 or 6.1.
4 Complete Response
All three sections essentially correct
3 Substantial Response
Two sections essentially correct and one section partially correct
2 Developing Response
Two sections essentially correct and one section incorrect
OR
One section essentially correct and either one or two sections partially correct
OR
All three sections partially correct
1 Minimal Response
One section essentially correct and two sections incorrect
OR
Two sections partially correct and one section incorrect