AP Statistics Unit 3: Inference for Proportions

Review AP Statistics Unit 3 through proportion estimates, confidence intervals, hypothesis tests, error risks, two-proportion inference, and chi-square methods.

Covered syllabus points

3.1 Estimators 3.2 Sampling Distributions for Sample Proportions 3.3 Constructing a Confidence Interval for a Population Proportion 3.4 Justifying a Claim Based on a Confidence Interval for a Population Proportion 3.5 Setting Up a Test for a Population Proportion 3.6 p-Values 3.7 Carrying Out a Test for a Population Proportion 3.8 Potential Errors When Performing Tests 3.9 Sampling Distributions for the Difference Between Two Population Proportions 3.10 Constructing a Confidence Interval for the Difference Between Two Population Proportions 3.11 Justifying a Claim Based on a Confidence Interval for the Difference Between Two Population Proportions 3.12 Setting Up a Test for the Difference Between Two Population Proportions 3.13 Carrying Out a Test for the Difference Between Two Population Proportions 3.14 Setting Up a Chi-Square Test for Homogeneity or Independence 3.15 Carrying Out a Chi-Square Test for Homogeneity or Independence

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 Inference for Categorical Data: Proportions 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

[Maximum number: 4]

A large company produces an equal number of brand-name lightbulbs and generic lightbulbs. The director of

quality control sets guidelines that production will be stopped if there is evidence that the proportion of all

lightbulbs that are defective is greater than 0.10. The director also believes that the proportion of brand-name

lightbulbs that are defective is not equal to the proportion of generic lightbulbs that are defective. Therefore, the

director wants to estimate the average of the two proportions.

To estimate the proportion of brand-name lightbulbs that are defective, a simple random sample of

400 brand-name lightbulbs is taken and 44 are found to be defective. Let X represent the number of

brand-name lightbulbs that are defective in a sample of 400, and let $p_{X}$ represent the proportion of all

brand-name lightbulbs that are defective. It is reasonable to assume that X is a binomial random variable.

(a)

One condition for obtaining an interval estimate for $p_{X}$ is that the distribution of $\hat{p}_{X}$ is approximately

normal. Is it reasonable to assume that the condition is met? Justify your answer.

(b)

The standard error of $\hat{p}_{X}$ is approximately 0.0156 . Show how the value of the standard error is calculated.

Part (b):

Note that $\hat{p}_{X}=\frac{44}{400}=0.11$ . So the standard error of $\hat{p}_{X}$ is $\sqrt{\frac{\hat{p}_{X}\left(1-\hat{p}_{X}\right)}{n}}=\sqrt{\frac{(0.11)(0.89)}{400}}=0.0156$ .

(c)

How many standard errors is the observed value of $\hat{p}_{X}$ from 0.10 ?

To estimate the proportion of generic lightbulbs that are defective, a simple random sample of 400 generic

lightbulbs is taken and 104 are found to be defective. Let Y represent the number of generic lightbulbs that are

defective in a sample of 400. It is reasonable to assume that Y is a binomial random variable and the distribution

of $\hat{p}_{Y}$ is approximately normal, with an approximate standard error of 0.0219. It is also reasonable to assume

that X and Y are independent.

The parameter of interest for the manager of quality control is D, the average proportion of defective lightbulbs

for the brand-name and the generic lightbulbs. D is defined as $D=\frac{p_{X}+p_{Y}}{2}$ .

Part (c):

Since $\hat{p}_{X}=0.11,0.11-0.10=0.01$ , so that $\hat{p}_{X}$ is $\frac{0.01}{0.0156}=0.64$ standard error away from 0.10.

(d)

Consider $\hat{D}$ , the point estimate of D.

(i)

Calculate $\hat{D}$ using data from the sample of brand-name lightbulbs and the sample of generic

lightbulbs.

First compute $\hat{p}_{Y}=\frac{104}{400}=0.26$ . So $\hat{D}=\frac{0.11+0.26}{2}=0.185$ .

(ii)

Calculate $s_{\hat{D}}$ , the standard error of $\hat{D}$ .

The standard error of $\hat{p}_{X}=0.0156$ is obtained from part (b). The standard error of $\hat{p}_{Y}$ is

$\sqrt{\frac{\hat{p}_{Y}\left(1-\hat{p}_{Y}\right)}{n}}=\sqrt{\frac{(0.26)(0.74)}{400}}=0.0219$ . So the standard error of $\hat{D}$ is

$S_{\hat{D}}=\sqrt{\frac{1}{4}\left(0.0156^{2}+0.0219^{2}\right)}=0.0134$ .

Part (e)

W=\frac{0.185-0.10}{0.0134}=6.34 .

AP ${ }^{\text {® }}$ STATISTICS

2018 SCORING GUIDELINES

Question 6 (continued)

Part (f)

Suppose the true mean D is 0.10. Then the observed value of $\hat{D}=0.185$ is 6.34 standard errors from

the mean D. Using Chebyshev's inequality, the probability of observing a value of $\hat{D}$ within 6.34 standard

errors of the mean of 0.10 is at least $1-\frac{1}{6.34^{2}}=0.975$ . So the probability of observing a value as far from

0.10 as the one observed, or farther, is at most 0.025 if the true mean really is 0.10. Therefore, the p-value

for this test is at most 0.025, which is less than 0.05, so the null hypothesis can be rejected. There is

sufficient statistical evidence at the 0.05 level to conclude that the average proportion for all products that

are defective is greater than 0.10.

[Maximum number: 4]

A polling agency showed the following two statements to a random sample of 1,048 adults in the United States.

Environment statement: Protection of the environment should be given priority over economic growth.
Economy statement: Economic growth should be given priority over protection of the environment.

The order in which the statements were shown was randomly selected for each person in the sample. After reading the statements, each person was asked to choose the statement that was most consistent with his or her opinion. The results are shown in the table.

Table for Question 3 Inference for Categorical Data: Proportions question 3 — AP Statistics

(a)

Assume the conditions for inference have been met. Construct and interpret a 95 percent confidence interval for the proportion of all adults in the United States who would have chosen the economy statement.

Part (a):

The appropriate procedure is a one-sample z-interval for a population proportion. The problem stated the conditions for inference have been met, so they do not need to be checked. A 95 percent confidence interval for the population proportion is given as $\hat{p} \pm z^{*} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$ , which is

0.37 \pm 1.96 \sqrt{\frac{(0.37)(0.63)}{1,048}} \approx 0.37 \pm 0.03=(0.34,0.40) .

We are 95 percent confident that the population proportion of all adults in the U.S. who would have chosen the economy statement is between 0.34 and 0.40.

(b)

One of the conditions for inference that was met is that the number who chose the economy statement and the number who did not choose the economy statement are both greater than 10. Explain why it is necessary to satisfy that condition.

Part (b):

The condition is necessary because the formula for the confidence interval relies on the fact that the binomial distribution can be approximated by a normal distribution which then results in the sampling distribution of $\hat{p}$ being approximately normal. The approximation does not work well unless both $n \hat{p}$ and $n(1-\hat{p})$ are at least 10.

(c)

A suggestion was made to use a two-sample z-interval for a difference between proportions to investigate whether the difference in proportions between adults in the United States who would have chosen the environment statement and adults in the United States who would have chosen the economy statement is statistically significant. Is the two-sample z-interval for a difference between proportions an appropriate procedure to investigate the difference? Justify your answer.

Part (c):

The suggested procedure is not appropriate because one of the requirements for using a two-sample z-interval for a difference between proportions is that the two proportions are based on two independent samples. In the situation described the two proportions come from a single sample and thus are not independent.

Scoring

This question is scored in three sections. Section 1 consists of the mechanics of the confidence interval in part (a), section 2 consists of interpreting the confidence interval in part (a), and section 3 consists of parts (b) and (c).

Section 1 is scored as follows:
Essentially correct (E) if the response includes the following two components:

States the correct procedure by name or formula

Calculates the confidence interval

Partially correct (P) if the response includes only one of the two components.
Incorrect (I) if the response does not meet the criteria for E or P.

Notes:

- A formula with correct values is sufficient for component 2.

- Component 2 can never be satisfied if the response contains an incorrect formula.

Section 2 is scored as follows:
Essentially correct (E) if the response includes the following four components for the interval interpretation:

Estimates a proportion

Infers about the population

States 95 percent confidence

Includes context

Partially correct (P) if the response satisfies components 1 and 2 AND satisfies only one of components 3 or 4;

OR
if the response gives a correct interpretation of the confidence level in context without interpreting the specific interval.

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

- Any indication of an inference to the adults sampled rather than the population does not satisfy component 2 and is scored as I.

- Stating values that are unrealistic as proportions (including blanks) in the interpretation lowers the score by one level (from E to P or from P to I).

- When both the interpretation of the interval and the level are given, only the interpretation of the interval is scored.

Section 3 is scored as follows:
Essentially correct (E) if the response includes the following two components:

Part (b) states that the condition implies the sampling distribution of $\hat{p}$ is approximately normal OR the normal approximation to the binomial distribution is appropriate.

Part (c) indicates the procedure is not appropriate because the two proportions come from a single sample (dependent) rather than two (independent) samples.

Partially correct (P) if the response includes only one of the two components.
Incorrect (I) if the response does not meet the criteria for E or P.
Notes:

- Referring to the sampling distribution needing to be normal without explicitly stating $\hat{p}$ satisfies component 1.

- Referring to normal approximation must specify "to binomial" to satisfy component 1.

- Discussing the two-sample z-interval versus the two-proportion z-interval is extraneous information.

- In part (c) component 2 can never be satisfied if a procedure that is not possible for one sample of categorical data is suggested.

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 two parts partially correct

Intent of Question

The primary goals of this question were to assess a student's ability to (1) use a scatterplot to comment on a report about the relationship between two variables and interpret the slope for the least-squares regression line summarizing this relationship; (2) describe the relationship between two variables in a scatterplot when a categorical variable is introduced and compare a characteristic of the distribution of a variable for different categories of individuals in a scatterplot; and (3) describe how the associations between two variables for each category of individuals in a scatterplot differ from the overall association in the same scatterplot.

Solution

[Maximum number: 1]

In a random sample of 500 American adults, 210 expressed the belief that journalists have "low" ethical standards. A 95\% confidence interval for the proportion of all American adults who believe that journalists have "low" ethical standards is given by (0.3767, 0.4633). Which of the following is a correct interpretation of the confidence level of 95\%?

There is a 0.95 probability that the true proportion of all American adults who believe that journalists have "low" ethical standards is between 0.3767 and 0.4633.

We are 95\% confident that the true proportion of all American adults who believe that journalists have "low" ethical standards is between 0.3767 and 0.4633 .

If the proportion of American adults who believe that journalists have "low" ethical standards is calculated for repeated random samples of 500 adults, 95\% of these sample proportions will fall between 0.3767 and 0.4633 .

If a confidence interval for the proportion of American adults who believe that journalists have "low" ethical standards is calculated for repeated random samples of 500 American adults, approximately 95\% of these intervals will contain the true population proportion.

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Covered syllabus points

3 Inference for Categorical Data: Proportions question 1

3 Inference for Categorical Data: Proportions question 2

Question (a)

Question (b)

Question (c)

Question (d)

Question (i)

Question (ii)

3 Inference for Categorical Data: Proportions question 3

Question (a)

Question (b)

Question (c)

3 Inference for Categorical Data: Proportions question 4