AP Statistics 3.1: Estimators

Part (c):

The Method I estimate is an unbiased estimate of the mean number of discarded cans and bottles

per mile of public road in the state. The Method I estimate weights the sample means according to

the proportion of each type of road in the population. The Method II estimate is biased in

under-estimating the population mean. By giving equal weight to each type of road, the Method II

estimate assigns too little weight to the minor highways and the minor highways appear to have

more discarded cans and bottles than the other two types of roads.

Question 3 (continued)

Scoring

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

Part (a) is scored as follows:

Essentially correct (E) if both components (identifying the variable of interest and population

parameter) are correct.

Partially correct (P) if only one component is correct.

Notes:

1. The definition of the parameter must include wording that would distinguish it from a sample

mean.

2. The response does not need to include "per mile" in the definition of the variable or parameter.

Part (b) is scored as follows:

Essentially correct (E) if the student identifies a stratified random sample and provides a reasonable

explanation that includes identification of road types as strata and indicates that a sample is selected

from each stratum.

Partially correct (P) if the student identifies a stratified random sample but provides an incomplete

explanation (such as stating that the state highways were grouped into three types but failing to

clearly indicate that separate random samples of one mile road segments were taken within each

group).

OR

The response clearly shows that the student knows the difference between cluster and stratified

sampling by discussing both types, but has the names of the two types of sampling reversed.

Incorrect (I) if the student identifies a stratified random sample with no justification or an incorrect

justification

OR

chooses an incorrect sampling method.

Question 3 (continued)

Part (c) is scored as follows:

Essentially correct (E) if student chooses Method I and provides a justification that explicitly indicates

that the strata sizes are different (for example, the state has more miles of minor highways than miles

of major highways and unpaved roads) and therefore weighting by strata size is preferable to the

unweighted average.

Partially correct (P) if a student chooses Method I but provides a weak justification (one that does not

explicitly indicate that the strata sizes are different, such as "it adjusts for road types" or "it uses the

mileage proportions").

Incorrect (I) if the student chooses Method 1 with no justification or an incorrect justification

OR

chooses Method II.

4 Complete Response

All three parts essentially correct

3 Substantial Response

Two parts essentially correct and one part partially correct

2 Developing Response

Two parts essentially correct and one part incorrect

OR

One part essentially correct and one or two parts partially correct

OR

Three parts partially correct

1 Minimal Response

One part essentially correct and two parts incorrect

OR

Two parts partially correct and one part incorrect

The point estimate is based on expecting 150 students to be required to say no and 150 students to truthfully answer the question. Of the 213 answers of no, we expect that 213-150=63 were from students who truthfully answered the question. That means we expect that the remaining 150-63=87 students truthfully answered the question and responded yes. So the point estimate for the proportion of all students at the high school who would respond yes to the question is $\frac{87}{150}=0.58$.

Scoring

Parts (a), (b), and (c) are scored as essentially correct (E), partially correct (P), or incorrect (I).
Part (a) is scored as follows:
Essentially correct (E) if the response satisfies the following five components:

Uses a standard error in the form $\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$ where $\hat{p}$ is between 0 and 1.

Shows evidence that $\hat{p}=0.7$ was correctly used in the standard error.

Shows evidence that 0.116 was correctly used as the margin of error in the calculation.

Shows evidence that $z^{*}=1.96$ was correctly used as the critical value in the calculation.

Includes a single, positive whole-number answer.

Partially correct (P) if the response satisfies only three or four of the five components.
Incorrect (I) if the response satisfies at most two of the five components.

Notes:

- Using an equation in the form $n=\frac{z^{2} \hat{p}(1-\hat{p})}{\operatorname{MOE}^{2}}$ satisfies component 1.

- A value of 0.21 in the numerator of the standard error implies that $\hat{p}=0.7$ was correctly used in the standard error and satisfies component 2.

- An equation such as 0.816=0.7+ MOE implies that 0.116 was correctly used for the margin of error and satisfies component 3.

- Statements that suggest a whole-number answer is approximate (such as, "about 60" or " $\approx 60$ ") satisfy component 5.

- Algebraic work between the set-up and final answer does not need to be shown to satisfy component 5.

- When calculating the values 0.7, 0.116, or 1.96, ignore minor arithmetic errors or transcription errors if they can be identified by the work shown.

Part (b) is scored as follows:
Essentially correct (E) if the response satisfies the following three components:

Explains why the responses to the survey might differ from the truth about student recycling in this context (for example, the survey was not anonymous, the question was asked by an authority figure).

Explains how the responses to the survey might differ from the truth about student recycling (for example, "students might say yes when they actually don't recycle," "students lie and say yes," "students don't recycle but lie to the teacher").

Describes the effect of the bias on the point estimate (or the proportion, percentage, number of yes responses in the sample) and doesn't contradict the bias described.

Partially correct (P) if the response satisfies only two of the three components.
Incorrect (I) if the response satisfies at most one of the three components.
Notes:

- To satisfy component 1 the response must provide a reason that is based on a bias created by the teacher asking students in person. For example, a response that addresses the wording of the question, voluntary response, or sampling variability does not satisfy component 1.

- To satisfy component 2 the response needs to explicitly contrast what the students say with what they do.

- Evidence used to address component 3 cannot also be used to address component 2. For example, a response that says "Students might lie, producing an estimate that is too high" addresses the effect of the bias on the point estimate but should not be combined with the statement about students lying to infer that students do not actually recycle. However, a response that says "Students may lie and say yes, producing an estimate that is too high" satisfies both components 2 and 3.

- If the response is clearly about the population proportion and not about the point estimate, component 3 cannot be satisfied.

- Statements such as "the interval will be too high" do not satisfy component 3 because they don't specifically address the point estimate.

Part (c) is scored as follows:
Essentially correct (E) if the response gives an answer of 150 in (c-i) and gives an answer of 0.58 (or equivalent) in (c-ii).

Partially correct (P) if the response gives an answer of 150 in (c-i) and gives an answer of 0.42 (or equivalent) in (c-ii);

OR
if the response does not give an answer of 150 in (c-i) but gives an answer of 0.58 (or equivalent) with supporting work in (c-ii).

Incorrect (I) if the response does not meet the criteria for E or P.

Notes:

- In part (c-i) the answer must be a single number. Responses such as "at least 150" or "147-153" are incorrect. However, responses such as "about 150" or " $\approx 150$ " are acceptable.

- In part (c-ii) the proportion can be described verbally (e.g., "87 out of 150").

- In part (c-ii) if the response clearly indicates that 0.58 (or 0.42) is the population proportion, lower the overall score in part (c) by one level (that is, from E to P, or from P to I). Using probability notation such as P( yes ) does not clearly indicate a population proportion.

- In part (c-ii) if the response includes a point estimate of 0.58 or 0.42 but uses a confidence interval as the final answer, lower the overall score in part (c) by one level (that is, from E to P, or from P to I).

- If the answer is incorrect in part (c-i) and the answer in part (c-ii) uses numerator = 87 and denominator =300- answer to (c-i), the response should be scored P.

Complete Response
Three parts essentially correct

Substantial Response
Two parts essentially correct and one part partially correct

Developing Response
Two parts essentially correct and no parts partially correct

OR
One part essentially correct and one or two parts partially correct

OR
Three parts partially correct

Minimal Response
One part essentially correct

OR
No parts essentially correct and one or two parts partially correct

Intent of Question

The primary goals of this question were to assess a student's ability to (1) compute a probability based on a weighted mixture of two populations; (2) compute a conditional probability; and (3) recognize a binomial random variable and compute the probability associated with it.

Solution