2021 AP Exam Administration Sample Student Responses - AP Statistics: Free-Response Question 4

2021

Statistics

Sample Student Responses

and Scoring Commentary

Inside:

Free Response Question 4

•• Scoring Guideline

•• Student Samples

•• Scoring Commentary

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AP® Statistics 2021 Scoring Guidelines

Question 4: Focus on Inference 4 points

General Scoring Notes

• This question is scored in four sections. Each section is initially scored by determining if it meets the

criteria for essentially correct (E), partially correct (P), or incorrect (I). The first section includes

statements of the null and alternative hypotheses and identification of the appropriate hypothesis test in

part (a). The second section includes verifying the conditions for the test identified in part (a) and

calculating the value of the test statistic and the corresponding p-value. The third section includes the

conclusion for the test identified in part (a). The fourth section includes the response to part (b). The

response is then categorized based on the scores assigned to each section and awarded an integer score

between 0 and 4 (see the table at the end of the question).

• The model solution represents an ideal response to each section of the question, and the scoring criteria

identify the specific components of the model solution that are used to determine the score.

Model Solution Scoring

(a)

Section

Let p represent the proportion of all customers

of the pet supply company who would place an

order within 30 days after receiving an e-mail

with a coupon for $10 off the next purchase.

The null hypothesis is

H :

0.40,p =

and the

alternative hypothesis is

H :

0.40.p >

An appropriate test is a one-sample z-test for a

population proportion.

Essentially correct (E)

if the response satisfies

the following three components:

1. States the correct equality for the null

hypothesis for a proportion (e.g.,

0.40p =

)

AND the correct direction of the one-sided

alternative hypothesis for a proportion (e.g.,

0.40p >

)

2. Provides sufficient context for the parameter

by including reference to the population

proportion AND the sampling units

(customers) AND the response variable

(placing an order after receiving a coupon)

3. Identifies a one-sample z-test for a

population proportion by name (e.g., “one-

proportion z-test” but not merely “one-

sample z-test”) or by formula

Partially correct (P) if the response does not

meet the criteria for E but satisfies either

component 1 and/or component 3.

Incorrect (I) if the response does not meet the

criteria for E or P.

Additional Notes:

• The elements of component 2 do not have to be satisfied with the statement of the hypotheses. They may

be satisfied by work presented anywhere in the response, most likely by the statement of the conclusion.

•

If the statement of the hypotheses refers to population proportion and the conclusion refers to sample

proportion (or vice versa), then the population aspect of component 2 is not satisfied.

AP® Statistics 2021 Scoring Guidelines

•

A response that states the null hypothesis as

0.40p ≤

may satisfy component 1.

• To satisfy component 1, the hypotheses must be stated in terms of a proportion. If a symbol other than p

is used to denote the proportion, it must be clearly defined as a proportion (but does not need to

reflect the context of customers who would place an order within 30 days after receiving a coupon) in

order for the response to satisfy component 1. It is acceptable to use “

” to denote the proportion.

• A response that states the hypotheses in words (e.g., “the null hypothesis is that the proportion is 0.40, and

the alternative hypothesis is that the proportion is greater than 0.40”) may satisfy component 1. Neither

context nor the concept of the population is required to satisfy component 1.

• A response that states the hypotheses in words (e.g., “the null hypothesis is that the proportion of all

customers who would place an order within 30 days after receiving a coupon is equal to 0.40, and the

alternative hypothesis is that the proportion is greater than 0.40”) may satisfy component 1 and

component 2.

• If the response clearly refers to the sample proportion instead of the population proportion using words or

a symbol (e.g.,

), then component 2 is not satisfied unless the symbol used is defined as the population

proportion.

• A response may satisfy the population aspect of component 2 by doing the following:

o referring to population in the statement of the conclusion of the inferential procedure

o using notation such as p,

, or

when defining the hypothesis statements.

• A response may satisfy the sampling units aspect of component 2 by referring to “people who place an

order” or similar statement.

• If the response identifies the correct test by name, but also states an incorrect formula, then component 3

is not satisfied.

• If the response identifies the test by formula using a t-percentile instead of a z-percentile, then

component 3 is not satisfied.

Confidence Interval Approach:

•

If a one-sample z-interval for a population proportion is identified correctly by name (e.g., “one-

proportion z-interval” but not merely “one-sample z-interval”) or by formula, then component 3 is

satisfied.

• If a response uses a one-sample z-interval for a population proportion, then component 2 is satisfied if the

esponse indicates that it is a confidence interval for the proportion of all customers who would place an

order within 30 days after receiving a coupon, even if the hypotheses are not stated.

AP® Statistics 2021 Scoring Guidelines

Model Solution Scoring

(a)

Section

The independent observations condition for

performing the one-sample z-test for a

population proportion is satisfied because the

data were obtained from a random sample of

90 customers who placed an order in the past

year and, because sampling of customers is done

without replacement, it is assumed that this

large online company has more than

10(90) = 900

customers.

The sample size is large enough to support an

assumption that the sampling distribution of

is approximately normal because

(90)(0.4) 36=

and

(90)(1 0.4) 54−=

are both

at least 10.

The value of the sample proportion is

0.422

p = ≈

and the value of the test

statistic is

0.40

0.430.

(0.40)(0.60)

−

= ≈

The

corresponding p-value is

( 0.430) 0.333.Pz>≈

Essentially correct (E) if the response satisfies

the following four components:

1. Checks the independence condition by

ferring to the random selection of

90 customers AND indicating that the

company is assumed to have at least

900 customers (i.e.,

90 0.10N≤

)

2. Checks that the sample size is large enough

to support the assumption that the sampling

distribution of

is approximately normal

by verifying that

(90)(0.4)

and

(90)(1 0.4)−

are both at least 10 (or 5)

3. Correctly reports the value of the z-statistic

4. Correctly reports the p-value, consistent

with the reported test statistic and stated

alternative hypothesis

Partially correct (P) if the response satisfies

only two or three of the four components.

Incorrect (I) if the response does not meet the

crit

eria for E or P.

Additional Notes:

• In order to satisfy the reference to the random selection of 90 customers in component 1 it is minimally

acceptable to state “random sample – check” or “SRS – check.” However, component 1 is not satisfied if

the response implies that random assignment was used or only states “random - check.”

• In order to satisfy component 2, the response must include actual values of the observed successes and

failures, or values for the expected successes and failures, or formulas for the expected number of

successes and failures with values inserted AND the response must make a comparison of the two values

with some standard criterion, such as 5 or 10. If expressions such as

(90)(0.4)

and

(90)(1 0.4)−

are used,

simplification is not required.

o Examples of acceptable quantities (comparisons must still be made):

• 38 and 52 (observed counts)

• 36 and 54 (expected counts under the null hypothesis)

•

(90)(0.4)

and

(90)(1 0.4)−

•

( )

90 (0.4222)

and

(90)(1 0.422)−

o Unless values of all parameters are explicitly defined in the response, the following quantities are

unacceptable:

•

90 p

90(1 )p−

(1 )np−

•

90 p

90(1 )p−

(1 )np−

•

(0.4)n

(0.6)n

(0.4222)n

(1 0.4222)n −

AP® Statistics 2021 Scoring Guidelines

•

When computing the test statistic, using a

of 0.42 in the numerator results in a test statistic equal to

0.39 with a p-value of 0.36. These values satisfy components 3 and 4.

• If the response uses

in the null standard error formula to calculate the z-statistic, component 3 may be

satisfied.

• A response that reports the correct value for the z-statistic but contains errors in supporting work may still

satisfy component 3.

• If the response satisfies component 4, any supporting work for the p-value may be treated as extraneous.

• If the response compares the value of the test statistic to a critical value instead of reporting a p-value,

then the critical value (1.645), or a critical value consistent with the stated alternative hypothesis, satisfies

component 4.

• If a two-tailed alternative hypothesis is stated or the direction of the stated one-tailed alternative

hypothesis is incorrect, then the p-value must be consistent with the stated alternative hypothesis to satisfy

component 4.

• If the response omits identifying the hypotheses, the correct one-sided alternative hypothesis is assumed

when scoring component 4.

• If an incorrect alternative hypothesis is stated, then the p-value must be consistent with the stated

alternative hypothesis to satisfy component 4.

Confidence Interval Approach:

• If either a one-sided 95 percent confidence interval is correctly calculated as

(0.337, 1)

, or a two-sided

90 percent confidence interval is correctly calculated as

(0.337, 0.508)

, then component 3 is satisfied.

• If only the lower end of a confidence interval is used to reach a conclusion, then component 4 is satisfied.

• Application of a confidence interval approach must be consistent with the stated alternative to satisfy

component 4. A two-sided 95 percent confidence interval is

(0.320, 0.524)

, and a lower one-sided

95 percent confidence interval is

(0, 0.508)

AP® Statistics 2021 Scoring Guidelines

Model Solution Scoring

(a)

Section

Because the p-value is greater than

0.05,

the null hypothesis should not be rejected. The

results from this study do not provide

convincing statistical evidence that the

manager’s belief is correct. That is, there is not

convincing statistical evidence that more than

40 percent of all customers of the pet supply

company would place an order within 30 days

after receiving an e-mail with a coupon for $10

off the next purchase.

Essentially correct (E) if the response satisfies

the following two components:

1. Provides a correct justification of the

conclusion based on whether the p-value is

less than

0.05

(or a comparison of the

value of the test statistic to an appropriate

critical value, e.g.,

1.645z <−

)

2. States a correct conclusion consistent with

the stated alternative hypothesis OR states a

conclusion that answers the inference

question (e.g., states the conclusion in terms

of the manager’s belief)

Partially correct (P) if the response satisfies

only one of the two components.

Incorrect (I) if the response does not meet the

criteria for E or P.

Additional Notes:

• Although including proper context (the concept of population proportion and referencing the response

variable) is important in stating the conclusion, context displayed in stating the conclusion is considered

in scoring component 2 of Section 1.

• The response need not make an explicit decision about the null hypothesis (reject

or fail to reject

)

in order to satisfy component 1. However, if an incorrect decision is stated, then component 1 is not

satisfied.

• If the conclusion and justification are consistent with an incorrect p-value (or an incorrect value of the test

statistic, or an incorrect confidence interval), the response may satisfy component 1 and component 2.

• If the response omits hypotheses, assume the correct alternative hypothesis,

H :

0.40,p >

was provided

when scoring component 1 and component 2.

• If the conclusion includes a definitive statement (e.g., “this proves that we do not have enough evidence to

claim...” or “there is no evidence…”), then component 2 is not satisfied.

• If the response includes a statement that is equivalent to accepting the null hypothesis (e.g., “we conclude

that the proportion of customers who will place an order is 0.40”), then component 2 is not satisfied.

• If the response includes an incorrect interpretation of the p-value, then the score for Section 3 is lowered

one level (that is, from E to P or from P to I).

• The clarity and quality of the statement of the conclusion and the statement of the justification may be

used in a holistic approach to decide whether to score up or down (e.g., raising a score of 2.5 to 3 or

reducing a score of 2.5 to 2).

Confidence Interval Approach:

• If the alternative hypothesis is specified correctly as

H :

0.40,p >

then component 1 is satisfied if the

justification is based on whether 0.40 is below the lower end of the confidence interval. If the alternative

hypothesis is stated in the wrong direction, then component 1 is satisfied if the justification is based on

whether 0.40 is above the upper end of the confidence interval.

AP® Statistics 2021 Scoring Guidelines

•

If no alternative hypothesis is specified in the response, then assume the correct alternative hypothesis is

provided when scoring component 2.

• If an incorrect two-sided alternative hypothesis is specified, then component 2 is satisfied if the

justification is based on whether 0.40 is included in the confidence interval.

• If the response includes an incorrect interpretation of the confidence interval, then the score for Section 3

is lowered one level (that is, from E to P or from P to I).

AP® Statistics 2021 Scoring Guidelines

Model Solution Scoring

(b)

Section

Because the null hypothesis was not rejected in

part (a), a Type II error could have been made.

A Type II error occurs when the null hypothesis

is false and is not rejected. In this case, a

Type II error is made by failing to reject the null

hypothesis that 40 percent (or less) of all

customers of the pet supply company would

place an order within 30 days after receiving an

e-mail with a coupon for $10 off the next

purchase, when in fact, more than 40 percent

would do so.

Consequently, the manager may decide not to

use the coupon promotion when it actually

would result in more than 40 percent of their

customers making a purchase within 30 days.

Essentially correct (E) if the response satisfies

the following two components:

1. States that a Type II error could have been

made

2. Provides a reasonable interpretation of the

consequence of the stated error AND uses

sufficient context, by including at minimum

“those who would place an order” or

“coupon” or “sales” (e.g., indicating the

manager may not use the coupon promotion

when it would actually lead to more than

40% of customers placing an order)

Partially correct (P) if the response satisfies

only one of the two components.

Incorrect (I) if the response does not meet the

criteria for E or P.

Additional Notes:

• If the response to part (a) rejects the null hypothesis, then

o Component 1 is satisfied if the response states that a Type I error could have been made.

o Component 2 is satisfied if the response provides a reasonable interpretation of the consequence of a

Type I error AND uses sufficient context by including at minimum “those who would place an order”

or “coupon” or “sales.”

• If the response states that a Type II error could have been made, followed by an incorrect description of a

Type II error (e.g., “did not find convincing evidence that more than 40% of customers will place an order

when there actually was evidence of more than 40%”), component 1 is not satisfied.

• The clarity and quality of the statement of the consequence may be used in a holistic approach to decide

whether to score up or down (e.g., raising a score of 2.5 to 3 or reducing a score of 2.5 to 2).

AP® Statistics 2021 Scoring Guidelines

Scoring for Question 4

Each essentially correct (E) part counts as 1 point, and each partially correct (P) part counts as ½ point.

Score

Complete Response 4

Substantial Response 3

Developing Response 2

Minimal Response 1

If a response is between two scores (for example, 2 ½ points), use a holistic approach to decide whether to

score up or down, depending on the strength of the response and quality of the communication.

Sample 4A, pg 1 of 2

Sample 4A, pg 2 of 2

Sample 4B, pg 1 of 2

Sample 4B, pg 2 of 2

Sample 4C, pg 1 of 2

Sample 4C, pg 2 of 2

Statistics 2021 Scoring Commentary

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Question 4

Note: Student samples are quoted verbatim and may contain spelling and grammatical errors.

Overview

The primary goals of this question were to assess a student’s ability to (1) identify an appropriate inference

procedure to test a claim about a population proportion; (2) identify the appropriate null hypothesis and the

appropriate alternative hypothesis; (3) check conditions required for accurate application of the identified

inference procedure; (4) compute the value of a test statistic and the corresponding p-value; (5) state and justify a

conclusion about the claim; and (6) determine whether a Type I or Type II error could have been made and

describe a consequence of the identified type of error.

This question primarily assesses skills associated with inference, including skills in skill category 1: Selecting

Statistical Methods; skill category 3: Using Probability and Simulation; and skill category 4: Statistical

Argumentation. Skills required for responding to this question include (1.B) Identify key and relevant information

to answer a question or solve a problem, (1.E) Identify an appropriate inference method for significance tests,

(1.F) Identify null and alternative hypotheses, (3.E) Calculate a test statistic and find a p-value, provided

conditions for inference are met, (4.A) Make an appropriate claim or draw an appropriate conclusion, (4.C) Verify

that inference procedures apply in a given situation, and (4.E) Justify a claim using a decision based on

significance tests.

This question covers content from Unit 6: Inference for Categorical Data: Proportions of the course framework in

the AP Statistics Course and Exam Description. Refer to topics 6.4, 6.5, 6.6, and 6.7, and learning objectives

DAT-3.B, UNC-5.A, VAR-6.D, VAR-6.E, VAR-6.F, and VAR-6.G.

Sample: 4A

Score: 4

The response earned the following: Section 1 – E; Section 2 – E; Section 3 – E; Section 4 – E.

In section 1, the response satisfies all three components. The response satisfies component 1 by correctly

identifying the hypothesis statements. The response satisfies component 2 by stating, “the true proportion of past

customers who placed an order after being offered a $10 coupon.” The words “true proportion” specify the

population proportion, but the response also does this by referring to p in the hypothesis statements. The response

satisfies component 3 because the response identifies the correct hypothesis test by name, “1 prop z test.”

Section 1 was scored essentially correct (E).

In section 2 the response satisfies all four components. For component 1 the response correctly states “SRS” and

“Assume that the company has at least 900 customers.” For component 2 the response correctly compares

“

( )

90 .4

” and “

( )

90 .6

” to 10. For components 3 and 4 the response reports the correct values of the z-statistic and

p-value, “

.43z =

” and “

.333p =

,” respectively. Section 2 was scored essentially correct (E).

In section 3 the response satisfies both components. For component 1 the response correctly compares the p-value

by stating “our p-value (.33) is greater than

(.05).” For component 2 the response provides a correct

conclusion by stating, “We do not have convincing evidence that the true proportion of past customers who

placed an order after being offered $10 off their next purchase is greater than .4.” Section 3 was scored essentially

correct (E).

Statistics 2021 Scoring Commentary

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Question 4 (continued)

In section 4 the response satisfies both components. For component 1 the response correctly states a “type II error

could have been made.” For component 2 the response provides a reasonable interpretation of the consequence of

the stated error with sufficient context in two ways. First, the response states, “If this occured, the manager would

decide to not offer the $10 coupon because it is believed that it doesn’t increase sales enough when in reality the

coupon led to greater than 40% of those being offered it placing an order.” Second the response states there was

a “missed opportunity for sales,” which alone also satisfies component 2. Section 4 was scored essentially

correct (E).

Sample: 4B

Score: 3

The response earned the following: Section 1 – P; Section 2 – P; Section 3 – E; Section 4 – E.

In section 1 the response satisfies component 1 by specifying correct hypotheses for a proportion. Recall the third

Additional Note states

H :

0.40p ≤

may satisfy component 1. The response does not satisfy component 2

because there is no reference to the coupon. The response satisfies component 3 because the response correctly

names the test by using a z-statistic formula “

0.422 0.40

0.4(1 .4)

−

.” Section 1 was scored partially correct (P).

In section 2 the response does not satisfy component 1 because the response does not correctly check the

independence condition. Simply stating “Random?” with a checkmark is not sufficient for the independence

condition. The response satisfies component 2 by correctly checking the sample size condition. The response

satisfies component 3 because the response reports the correct value of the z-statistic. The response does not

satisfy component 4 because the reported p-value does not match the z-statistic and is not consistent with the

stated alternative hypothesis. Section 2 was scored partially correct (P).

In section 3 the response satisfies both components by correctly comparing the p-value to

and providing a

correct conclusion. Remember that context is not required in the conclusion for section 3. Context was required in

section 1. Section 3 was scored essentially correct (E).

In section 4 the response satisfies component 1 by correctly indicating a Type II error could have been made. The

response satisfies component 2 by providing a reasonable interpretation of the consequence of the stated error

with sufficient context: “We would have incorrectly assumed that few customers would place orders if they were

rewarded for coming back, which could have costed the business a lot of money in orders & customers.” Section

4 was scored essentially correct (E).

Sample: 4C

Score: 2

The response earned the following: Section 1 – P; Section 2 – E; Section 3 – I; Section 4 – P.

In section 1 the response satisfies component 1 by specifying correct hypotheses for a proportion as a “rate” and

percent. The response does not satisfy component 2 because there is no explicit reference to the population using

words or the notation p for the population proportion. The response does not satisfy component 3 because,

although the response correctly names the test, the z-statistic formula provided is not correct. Section 1 was scored

partially correct (P).

Statistics 2021 Scoring Commentary

Question 4 (continued)

In section 2 the response satisfies components 1 and 2 by correctly checking the conditions. The response reports

the correct values of the z-statistic and p-value, satisfying components 3 and 4. The incorrect formula was taken

into account when scoring component 3 in section 1 and does not affect the scoring of component 3 in section 2.

Section 2 was scored essentially correct (E).

In section 3 the response does not compare the p-value to

, so the response does not satisfy component 1. The

response does not satisfy component 2 because the conclusion is incorrect. Section 3 was scored incorrect (I).

In section 4 the response satisfies component 1 by correctly indicating a Type II error could have been made. The

response does not satisfy component 2 because the response does not provide a correct consequence of the error.

Section 4 was scored partially correct (P).

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