AP Statistics Student Samples from the 2023 Exam Administration

2023

Statistics

Sample Student Responses

and Scoring Commentary

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Inside:

Free-Response Question 6

 Scoring Guidelines

 Student Samples

 Scoring Commentary

AP® Statistics 2023 Scoring Guidelines

Question 6: Investigative Task 4 points

General Scoring Notes

• Each part of the question (indicated by a letter) is initially scored by determining if it meets the criteria for

essentially correct (E), partially correct (P), or incorrect (I). The response is then categorized based on the

scores assigned to each letter part 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 part 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)

Let X represent the amount of gold applied to a

necklace randomly selected from necklaces

produced with this machine. The random

variable X has an approximately normal

distribution with mean 300 mg and standard

deviation 5 mg.

Then,

( ) ( )

(296 304)

( 304) ( 296)

304 300 296 300

( 0.8) ( 0.8)

0.7881 0.2119 0.5763.

PX PX

PZ PZ

=<−≤

−−

< −≤

= < − ≤−

≈−≈

Essentially correct (E) if the response satisfies the

following three components:

1. Indicates the use of a normal distribution with

mean 300 and standard deviation 5

2. Specifies the correct event (boundary value and

direction) or an event consistent with values

reported in component 1

3. Provides the correct probability of 0.5763 or a

probability consistent with components 1 and 2

Partially correct (P) if the response satisfies only

two of the three components.

Incorrect (I) if the response does not satisfy the

criteria for E or P.

Additional Notes:

Component 1

• A response may satisfy component 1 by any of the following or a combination of the following:

o G

raphical: Displaying a graph of a normal density function with the appropriate scale on the horizontal

axis showing the mean and the standard deviation for the distribution of the amount of gold applied to

the necklaces.

o Calculator function syntax: Labeling correct values of the mean and standard deviation in a

“

normalcdf

” statement, such as:

( )

normalcdf lower 296, upper 304, mean 300, standard deviation 5 .= = = =

Correct specification of the lower and upper bounds is not required to satisfy component 1.

o W

ords: Using a statement such as “normal distribution with mean 300 and standard deviation 5.”

o Standard Notation: Using standard notation such as

(300,5)N

( )

(300, 5 ).N

o Z-score: Displaying the correct mean and standard deviation in a z-score calculation that includes “z,”

such as

304 300

−

AP® Statistics 2023 Scoring Guidelines

Component 2

• A response may satisfy component 2 by any of the following or a combination of the following:

o Graphical: Displaying a graph of a normal density function with the region of interest

(

296 amount of gold 304<<

0.8 0.8Z− <<

) clearly identified. The shaded area does not need to

be proportional, but the boundaries should be on the proper side of the mean and the shading should be

in the proper direction.

o Calculator function syntax: Labeling the lower and upper bounds of the region of interest in a

“normalcdf” statement, such as:



( )

normalcdf lower 296, upper 304, mean 300, standard deviation 5 .= = = =



( )

normalcdf lower 0.8, upper 0.8, 0, 1 .

µσ

=−===

o Words: Specifying the correct event in words with correct numerical value for the boundary value and

correct direction, such as “the probability that the amount of gold applied to a randomly selected

necklace is between 296 and 304 mg” or

(296 amount of gold 304).P <<

o Standard Notation: Using standard notation such as

( )

296 304PX<<

( )

296 300 304 300

−−

( 0.8 0.8).PZ− <<

General

• Minor errors in statistical notation may be ignored. However, this is considered poor communication if

holistic scoring is required.

• If the only error in the response to part (a) is the reversal of the numerator for the z-score,

( )

300 296−

( )

300 304 ,−

the response is scored P.

• An arithmetic or transcription error in a response can be ignored if correct work is shown.

• A response that satisfies components 1 and 2 by including more than calculator syntax (e.g., standard

random variable notation and probability notation) is considered good communication and may be

considered if holistic scoring is required.

AP® Statistics 2023 Scoring Guidelines

Model Solution

Scoring

(b)

(i) If the machine is working properly, and the sample

mean amount of gold,

, has a sampling

distribution that follows a normal distribution with

mean 300 mg and standard deviation

3.5355 mg,

≈

then

( 303)

303 300

( 0.8485)

0.198.





−





≈>

≈

(ii) Observing a sample mean amount of 303 mg

would not provide convincing evidence that the

population mean amount of gold being applied by

the machine is something other than 300 mg

because the probability of observing a sample

mean that differs from 300 mg by 3 mg or more is

large, around

0.198(2) 0.396.=

Essentially correct (E) if the response satisfies

components 1, 4, and 5 AND at least one of

components 2 or 3:

1. In part (b-i), indicates the use of a normal (or

an approximately normal) distribution and

identifies the correct parameter values (mean

300 mg and standard deviation

3.5355 mg

≈

) with work shown or a

correct formula

2. In part (b-i), specifies the correct event

(boundary value and direction)

3. In part (b-i), provides the correct probability

of 0.198 or the probability consistent with

components 1 and 2

4. In part (b-ii), indicates whether observing a

sample mean of 303 mg would provide

convincing evidence that the population mean

is not 300 mg, consistent with the response to

part (b-i)

5. In part (b-ii), provides justification based on

the probability obtained in part (b-i) or a

correctly computed probability in part (b-ii)

Partially correct (P) if the response does not

meet the requirements for E and satisfies at least

three of the five components

the response satisfies only components 4 and 5,

consistent with the response to part (b-i)

the response satisfies only component 1 and one

of the other four components.

Incorrect (I) if the response does not meet the

criteria for E or P.

AP® Statistics 2023 Scoring Guidelines

Additional Notes:

Component 1

• A response may satisfy component 1 by any of the following or a combination of the following:

o Graphical: Displaying a graph of a normal density function with the appropriate scale on the horizontal

axis showing mean and standard deviation for the distribution of mean amount of gold.

o Calculator function syntax: Labeling correct values of the mean and standard deviation in a “normalcdf”

statement, such as “normalcdf” or “ncdf”, e.g.,



( )

normalcdf lower 303, upper , mean 300, standard deviation .

= =∞= =

Correct specification of the upper and lower bounds is not required to satisfy component 1.

o Words: Using a statement such as “approximately normal distribution with mean 300 and standard

deviation

”

o Standard Notation: Using standard notation such as

( )

300,

( )

300, .







o Z-score: Displaying the correct mean and standard deviation in a z-score calculation, which includes “z,”

such as

303 300

−

o To satisfy the supporting work criterion of component 1 the response must clearly indicate that the

standard deviation for

is computed by dividing 5 by the square root of 2. This may be indicated with

words, or standard notation such as

Component 2

• A response may satisfy component 2 by any of the following or a combination of the following:

o Graphical: Displaying a graph of a normal density function with the region of interest (

303X >

0.8485Z >

) clearly identified. The shaded area does not need to be proportional, but the boundary

should be on the proper side of the mean and the shading should be in the proper direction.

o Calculator function syntax: Labeling the lower and upper bounds of the region of interest in a

“normalcdf” statement, such as:



( )

normalcdf lower 303, upper , mean 300, standard deviation .

= =∞= =



( )

normalcdf lower 0.8485, upper , 0, 1 .

µσ

= =∞= =

o Words: Specifying the correct event in words with correct numerical value for the boundary and correct

direction, such as the “probability that the mean amount of gold applied to two randomly selected

necklaces is greater than 303 mg” or

(mean amount of gold 303).P >

o Standard Notation: Using standard notation such as or

( )

303PX>

( )

303 300

3.5355

−

( )

.0.8485PZ>

Component 3

• A response may satisfy component 3 if the reported probability is consistent with components 1 and 2. If a

normal distribution with a mean of 300 mg and standard deviation 5 mg is used, the probability reported

should be 0.274.

AP® Statistics 2023 Scoring Guidelines

General

• Minor errors in statistical notation may be ignored. However, this is considered poor communication and

may be considered if holistic scoring is required.

• If the only error in part (b) is the reversal of the numerator for the z-score

( )

300 303 ,−

the response is

scored P.

• An arithmetic or transcription error in a response can be ignored if correct work is shown.

Alternate Solution:

• A response that calculates the probability that the total amount of gold on the two randomly selected

necklaces is greater than 606 mg

( )

2 303mg×

can be scored essentially correct if the response

demonstrates:

o The use of a normal (or approximately normal) distribution and the correct mean and standard deviation

for the distribution of the total amount of gold applied to two randomly selected necklaces, (e.g.,

( )

600,7.071 ,N

or mean

2(300) 600 mg= =

and standard deviation

( )

2 5 7.071 mg,= ≈

606 600

0.8485

7.071

−

= ≈

), satisfying component 1.

o Specifies the correct event, including the correct boundary value of

2(303) 606

total

x = =

0.8485,z ≈

and the correct direction, satisfying component 2.

o The probability is correctly computed using the mean and the standard deviation of the total amount of

gold applied to two randomly selected necklaces. An arithmetic error can be ignored if correct work is

shown and the result is between 0 and 0.5.

• While the probability of a difference in part (b-ii) is twice the probability found in part (b-i), it is not

necessary for a response to double the probability in part (b-i). However, if the probability in part (b-i) is

doubled, that should be considered if holistic scoring is required.

AP® Statistics 2023 Scoring Guidelines

Model Solution Scoring

(c)

(i) The sampling distribution of the sample

range for random samples of size

2n =

from a normal distribution with standard

deviation

is skewed to the right.

Almost all values of the simulated ranges

are between 0 mg and about 25 mg and the

center of the distribution is about 6 mg.

(ii) As the value of the population standard

deviation increases, the variation (spread) in

the distribution of the sample range

increases and the mean of the distribution of

the sample range also increases.

Essentially correct (E) if the response satisfies four

or five of the following five components:

1. In part (c-i), describes the shape as positively

skewed (or skewed to the right)

2. In part (c-i), describes the center of the

distribution as about 6 mg

3. In part (c-i), describes the spread of the

simulated sample ranges as having most values

between 0 mg and 30 mg

4. In part (c-ii), indicates that the mean (or center)

of the distribution of the sample range increases

as the population standard deviation increases

5. In part (c-ii), indicates that the variation (or

spread) of the distribution of the sample range

increases as the population standard deviation

increases

Partially correct (P) if the response satisfies three

of the five components

the response satisfies only components 4 and 5

the response only satisfies components 2 and 4

the response satisfies only components 3 and 5.

Incorrect (I) if the response does not meet the

criteria for E or P.

Additional Notes:

• The centrality component of the response to part (c-i) may be satisfied with any reasonable description of

center, such as a comment on the approximate value of mean or median of the distribution or indication that

the center is somewhere between 4 and 10 mg.

• The spread component of the response to part (c-i) may be satisfied with a reasonable statement about the

inter-quartile range. Any value between 5 and 7, inclusive, is considered reasonable.

• Although the actual range of the sample distribution of the sample range is infinite, a comment that the range

is any value between 20 and 30 may be scored as satisfying the spread component of the response to

part (c-i).

• The response to part (c-ii) need not comment on the skewness of the distributions. Any comment on the

skewness of the distributions should be considered as extraneous in scoring the response.

AP® Statistics 2023 Scoring Guidelines

Model Solution Scoring

(d)

(i) No, a sample range of 10 mg is not unusual if

the machine is working properly with a

standard deviation of 5 mg.

Although observing a sample range around

10 mg or greater is much more likely if the

population standard deviation is 8 mg or

12 mg than when the population standard

deviation is 5 mg, the graph of the sampling

distribution of the sample range for samples

of size 2 from a normal distribution with

5 mg

indicates that 10 mg is not an

unusual value for the range when

5 mg

There is about a 20% chance that a random

sample of two necklaces would yield a range

of 10 mg or more when the machine is

working properly.

(ii) No, Cleo’s sample mean of 303 mg and range

of 10 mg do not indicate that the machine is

not working properly. As noted in part (b-i),

the probability that the sample mean would

be equal to or greater than 303 mg when the

machine is working properly is almost 20%

so having a sample mean of 303 mg is not

unusual. Furthermore, it is less than one

standard deviation,

3.5355 mg,

away

from 300 mg. As indicated in part (d-i), the

probability of a range of 10 mg or greater

when the population standard deviation is 5

mg is also about 20%, so not unusual. There

is not statistically significant evidence to

show the machine is not working properly.

Essentially correct (E) if the response satisfies

four or five of the following five components:

1. In part (d-i), indicates a sample range of 10 mg

is not unusual

2. In part (d-i), justifies the response to

component 1 by indicating that the graph of

the sampling distribution of the sample range

when

mg shows that values of 10 mg

or greater occur often

3. In part (d-ii), indicates there is not convincing

evidence that the machine is not working

properly

4. In part (d-ii), justifies the response by

indicating that a sample mean of 303 mg

would not be unusual when the machine is

working properly with one of the following

• 303 mg is less than one standard deviation

away from 300 mg

• The probability that the mean is greater

than or equal to 303 mg was shown in (b)

to have probability of 0.198, which is not

unusual

• Some other reasonable justification

5. In part (d-ii), justifies the response by

indicating that a sample range of 10 mg is not

unusual when the machine is working properly

as discussed in part (d-i)

Partially correct (P) if the response does not meet

the requirements for E and satisfies at least three of

the five components

the response satisfies components 1 and 2

the response satisfies component 3 and either

component 4 or 5.

Incorrect (I) if the response does not meet the

criteria for E or P.

Additional Notes:

• An argument that a sample range of 10 mg or more is more likely to occur (or closer to the center of the

sampling distribution of the range) when

mg or

mg is not required for component 2, but by

itself does not satisfy component 2. Inclusion of this argument should be considered if holistic scoring is

required.

AP® Statistics 2023 Scoring Guidelines

• Other reasonable justifications to satisfy component 4 include arguments based on a z-value close to zero, a

one-sample z-test or one-sample t-test for a population mean, or a one sample z-interval or t-interval for a

population mean.

Alternate Solutions:

• A response that uses the intersection (or union) of the event ‘the sample mean is greater than or equal to

303 mg’ and the event ‘the sample range is greater than or equal to 10 mg’ can satisfy components 3, 4,

and 5.

o Components 3, 4, and 5 are satisfied if the response shows, for example:

 Let

( ) ( )

mean 303 0.198PM P == ≥

(from part b-ii).

 Let

( ) (

)

range 10 if 5 0.2

PR P

= ≥=≈

(from part d-i).

 For Intersection:

( ) ( ) ( ) ( )( )

0.198 0.2 0.0396,P M and R P M P R

= = =

because this probability

is small, this is evidence that the machine is not working properly.

 For Union:

( ) (

)

( ) (

) ( )

( )( )

0.198 0.2 0.198 0.2 0.3584,

PMorR PM PR PM PR=+=− = +−

because this probability is large, this is not evidence that the machine is not working properly.

General Notes:

• Context (amount of gold in necklaces) is not required in parts (a), (b), (c), or (d) of the response. However,

including context is considered good communication and may be considered if holistic scoring is required.

AP® Statistics 2023 Scoring Guidelines

Scoring for Question 6

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 6A

1 of 5

Sample 6A

2 of 5

Sample 6A

3 of 5

Sample 6A

4 of 5

Sample 6A

5 of 5

Sample 6B

1 of 5

Sample 6B

2 of 5

Sample 6B

3 of 5

Sample 6B

4 of 5

Sample 6B

5 of 5

Sample 6C

1 of 5

Sample 6C

2 of 5

Sample 6C

3 of 5

Sample 6C

4 of 5

Sample 6C

5 of 5

Statistics 2023 Scoring Commentary

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Question 6

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

Overview

The primary goals of the question were to assess a student’s ability to (1) calculate the probability a normally

distributed random variable is between two values; (2) calculate the probability a sample mean is greater than a

value using the distribution of the sample mean; (3) interpret a probability to determine whether it implies the

population mean is a different value than what was assumed; (4) describe the sampling distribution of the sample

range, when provided a graph of the results of a simulation; (5) describe how the sampling distribution of the

sample range changes as the population standard deviation increases, based on simulated graphs; (6) determine if

a value for the sample range is unlikely based on the results of a simulation; and (7) determine whether a machine

is working properly based on the previously calculated probability for a mean and the simulated sampling

distribution for the sample range.

This question primarily assesses skills in skill category 2: Data Analysis, skill category 3: Using Probability and

Simulation, and skill category 4: Statistical Argumentation. Skills required for responding to this question include

(2.A) Describe data presented numerically or graphically, (2.D) Compare distributions or relative positions of

points within a distribution, (3.A) Determine relative frequencies, proportions, or probabilities using simulation or

calculations, and (4.B) Interpret statistical calculations and findings to assign meaning or assess a claim.

This question covers content from Unit 1: Exploring One-Variable Data, Unit 4: Probability, Random Variables,

and Probability Distributions, and Unit 5: Sampling Distributions of the course framework in the AP Statistics

Course and Exam Description. Refer to topics 1.8, 1.9, 4.7 and 5.2, and learning objectives UNC-1.M, UNC-1.N,

VAR-5.B, and VAR-6.A.

Sample: 6A

Score: 4

The response earned the following: Part (a) – E; Part (b) – E; Part (c) – E; Part (d) – E.

In part (a) the response shows a sketch of a normal curve. To the right of the sketch, the mean and standard

deviation are satisfied. Therefore, the response satisfies component 1. The boundary values are stated, and

standard probability notation is used, satisfying component 2. The response reports the correct probability,

satisfying component 3. Part (a) was scored essentially correct (E). In part (b-i) the response shows correct work

in computing the standard deviation, shows a sketch, and uses standard random variable notation, satisfying

component 1. The response indicates the correct boundary with a probability statement, satisfying component 2.

The response reports the correct probability, satisfying component 3. Although there is a small notational error,

this was ignored in scoring the response. In part (b-ii) the response states, “No,” satisfying component 4. The

response interprets the probability from part (b-i) correctly, by stating, “it is not unlikely to get that extreme of an

observation,” satisfying component 5. The response satisfies all five components. Part (b) was scored essentially

correct (E). In part (c-i) the response states, “The distribution is skewed to the right,” satisfying component 1. The

response indicates the median sample range is around 4–5. Because

5 mg

is within the range of acceptable

values, component 2 is satisfied. Finally, the response reports that the sample ranges have a range of about 23,

satisfying component 3. In part (c-ii) the response states, “The sampling distributions become less skewed to the

right,” which is extraneous. The response goes on to indicate that the range and median of the sample ranges

increase, satisfying components 4 and 5. Because all five components are satisfied, part (c) was scored essentially

correct (E). In part (d-i) the response concludes the sample range is not unusual at the end of the response,

satisfying component 1. The response indicates using the area under Graph I for ranges of 10 or greater. The

response attempts to estimate the average height of 12 bars at 10 or greater to estimate the simulated p-value.

Finally, the response uses this approximate probability to justify the response, satisfying component 2. 

Statistics 2023 Scoring Commentary

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Question 6 (continued)

In part (d-ii) the response states, “No,” satisfying component 3. The response then references the sample mean

and range having already been deemed not to be unusual. Because the response made correct numerical

arguments based on the probabilities in parts (b-i) and (d-i), the response satisfies components 4 and 5. Part (d)

was scored essentially correct (E).

Sample: 6B

Score: 2

The response earned the following: Part (a) – P; Part (b) – P; Part (c) – E; Part (d) – I.

In part (a) the response uses a sketch of a normal distribution with the mean and one standard deviation, noted

under the graph, satisfying component 1. The response does not label the graph, or use labeled calculator syntax,

standard notation, or a statement in words to establish correct boundaries, and, therefore, does not satisfy

component 2. Finally, the response reports a correct probability, satisfying component 3. Part (a) was scored

partially correct (P). In part (b-i) the response indicates normality by calculator syntax and reports the correct

standard error. However, the response does not identify the mean of the sampling distribution. Therefore

component 1 is not satisfied. The unlabeled calculator boundaries do not satisfy component 2. The response goes

on to calculate a correct probability, satisfying component 3. In part (b-ii) the response states, “No,” satisfying

component 4. The response justifies the answer by indicating that 303 is within one standard deviation of the

mean, satisfying component 5. Part (b) was scored partially correct (P). In part (c-i) the response states, “Graph I

is strongly skewed right,” satisfying component 1. The response also reports a center of about 5, satisfying

component 2. There is no acceptable value given for spread, failing to satisfy component 3. In part (c-i) the last

two sentences of the response are considered extraneous. In part (c-ii) the response goes on to state, “the sampling

distribution increases in spread,” and “the centers are also moving to the right and getting higher,” satisfying

components 4 and 5. Because the response satisfies four of the five components, part (c) was scored essentially

correct (E). In part (d-i) the response includes the incorrect answer of “Yes,” failing to satisfy component 1, with

incorrect justification. In part (d-ii) the response reiterates an incorrect conclusion and fails to correctly address

the mean or the range. Part (d) was scored incorrect (I).

Sample: 6C

Score: 1

The response earned the following: Part (a) – E; Part (b) – P; Part (c) – I; Part (d) – I.

In part (a) the response uses a sketch of a normal curve, with mean, standard deviation, and boundary values

labeled. The response also includes a correct probability. Part (a) was scored essentially correct (E). In part (b-i)

the response uses the incorrect standard deviation, failing to satisfy component 1. The sketch has the correct

boundary and direction, satisfying component 2. The response indicates a probability consistent with components

1 and 2, satisfying component 3. In part (b-ii) the response states, “No,” satisfying component 4. However, the

justification is incorrect, failing to satisfy component 5. Three of the five components are satisfied. Part (b) was

scored partially correct (P). In part (c-i) the response indicates skewness to the right, satisfying component 1. The

response fails to address components 2 or 3. In part (c-ii) the response states, “As the s.d. increases, the sample

distribution for sample ranges also increases.” However, the response does not indicate what it is about the

sampling distribution that increases. Therefore, neither components 4 nor 5 are satisfied. Part (c) was scored

incorrect (I). In part (d-i) the response includes the incorrect answer of “Yes,” failing to satisfy component 1 with

incorrect justification. In part (d-ii) the response reiterates an incorrect conclusion and fails to correctly address

the mean or the range. Part (d) was scored incorrect (I). Because the response earned 1.5 points, holistic scoring

must be used. In part (a), standard notation is not used, and the response contains an incorrect string of equalities.

In part (b-i) the response includes an incorrect string of equalities. In part (b-ii) the response satisfies component 4

by stating, “No.” However, the response is based on incorrect justification. In parts (c) and (d), the

communication is weak. The response was scored 1.