AP Statistics 3.1.A Justifying Unbiased Estimators

Practise deciding whether an estimator is unbiased by comparing its expected value with the parameter, weighting strata correctly, and separating bias from variance.

An environmental research agency conducted a study of a certain state's roadsides to estimate the mean number

of discarded cans and bottles per mile of public road. The state's public roads were grouped into three types:

Major highways: major paved roads designed for high traffic volume

Minor highways: smaller paved roads designed for low traffic volume

Unpaved roads: gravel and dirt roads

There are about 100,000 miles of public roads in the state. The environmental research agency defined a

sampling unit to be a one-mile segment of public road. Using a database supplied by the state's department of

transportation, the agency randomly selected 30 one-mile road segments for each of the three types of roads.

Researchers from the agency searched the roadsides along each of the selected one-mile road segments and

recorded the number of discarded cans and bottles. Results are shown in the table below.

Table for Question 3.1.A—Justify why an estimator is or is not unbiased question 1 — AP Statistics

Two methods for estimating the mean number of discarded cans and bottles per mile along all public roads

in the state are given below.

Method 1

\begin{aligned} & 0.24 \bar{x}_{\text {major }}+0.58 \bar{x}_{\text {minor }}+0.18 \bar{x}_{\text {unpaved }}= \\ & (0.24)(11.2)+(0.58)(32.6)+(0.18)(21.7)=25.5 \end{aligned}

Method 2

\frac{\bar{x}_{\text {major }}+\bar{x}_{\text {minor }}+\bar{x}_{\text {unpaved }}}{3}=\frac{11.2+32.6+21.7}{3}=21.83

Which of these methods gives a better estimate of this mean? Explain.

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

All question bank results loaded

3.1.A—Justify why an estimator is or is not unbiased question 1