AP Statistics Course and Exam Description

4UNIT Probability, Random Variables, and Probability Distributions

SKILL TOPIC 4.4

Statistical Mutually Exclusive
Argumentation Events

4.B

Interpret statistical
calculations and findings
to assign meaning or
assess a claim.

Required Course Content

ENDURING UNDERSTANDING

VAR-4

The likelihood of a random event can be quantified.

LEARNING OBJECTIVE ESSENTIAL KNOWLEDGE

VAR-4.C VAR-4.C.1

Explain why two events are The probability that events A and B both will
(or are not) mutually exclusive.
[Skill 4.B] occur, sometimes called the joint probability,

is the probability of the intersection of A and B,
denoted P(A ∩ B).

VAR-4.C.2

Two events are mutually exclusive or
disjoint if they cannot occur at the same time.

So P(A ∩ B) = 0.

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Probability, Random Variables, and Probability Distributions 4UNIT

TOPIC 4.5 SKILL

Conditional Using Probability
Probability and Simulation

3.A

Determine relative
frequencies, proportions,
or probabilities using
simulation or calculations.

Required Course Content

ENDURING UNDERSTANDING

VAR-4

The likelihood of a random event can be quantified.

LEARNING OBJECTIVE ESSENTIAL KNOWLEDGE

VAR-4.D VAR-4.D.1

Calculate conditional The probability that event A will occur
probabilities. [Skill 3.A] given that event B has occurred is called

a conditional probability and denoted

P(A | B) = P(A ∩ B) .
P(B)  

VAR-4.D.2

The multiplication rule states that the

probability that events A and B both will occur
is equal to the probability that event A will
occur multiplied by the probability that event B
will occur, given that A has occurred. This is
denoted P(A ∩ B) = P(A)⋅ P(B | A).

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4UNIT Probability, Random Variables, and Probability Distributions

SKILL TOPIC 4.6

Using Probability Independent Events
and Simulation and Unions of Events

3.A

Determine relative
frequencies, proportions,
or probabilities using
simulation or calculations.

Required Course Content

ENDURING UNDERSTANDING

VAR-4

The likelihood of a random event can be quantified.

LEARNING OBJECTIVE ESSENTIAL KNOWLEDGE

VAR-4.E VAR-4.E.1

Calculate probabilities for Events A and B are independent if, and only if,
independent events and knowing whether event A has occurred (or will
for the union of two events.
[Skill 3.A] occur) does not change the probability that

event B will occur.

VAR-4.E.2

If, and only if, events A and B are independent,
then P(A | B) = P(A), P(B | A) = P(B), and
P(A ∩ B) = P(A)⋅ P(B).

VAR-4.E.3

The probability that event A or event B (or both)

will occur is the probability of the union of

A and B, denoted P(A ∪ B).

VAR-4.E.4

The addition rule states that the probability

that event A or event B or both will occur
is equal to the probability that event A will
occur plus the probability that event B

will occur minus the probability that both

events A and B will occur. This is denoted
P(A ∪ B) = P(A) + P(B) − P(A ∩ B).

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Probability, Random Variables, and Probability Distributions 4UNIT

TOPIC 4.7 SKILLS

Introduction to Data Analysis
Random Variables
and Probability 2.B
Distributions Construct numerical or
graphical representations
Required Course Content of distributions.

ENDURING UNDERSTANDING Statistical
Argumentation
VAR-5
4.B
Probability distributions may be used to model variation in populations. Interpret statistical
calculations and findings to
LEARNING OBJECTIVE ESSENTIAL KNOWLEDGE assign meaning or assess
a claim.
VAR-5.A VAR-5.A.1
ILLUSTRATIVE EXAMPLES
Represent the probability The values of a random variable are the Outcomes of trials of a
distribution for a discrete numerical outcomes of random behavior. random process:
random variable. [Skill 2.B]
VAR-5.A.2 §§ The sum of the
outcomes for rolling
A discrete random variable is a variable that two dice
can only take a countable number of values.
Each value has a probability associated with it. §§ The number of puppies
The sum of the probabilities over all of the in a randomly selected
possible values must be 1. litter for a certain breed
of dog
VAR-5.A.3

A probability distribution can be represented
as a graph, table, or function showing the
probabilities associated with values of a
random variable.

VAR-5.A.4

A cumulative probability distribution can be
represented as a table or function showing the
probability of being less than or equal to each
value of the random variable.

VAR-5.B VAR-5.B.1

Interpret a probability An interpretation of a probability distribution
distribution. [Skill 4.B] provides information about the shape, center, and
spread of a population and allows one to make
conclusions about the population of interest.

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4UNIT Probability, Random Variables, and Probability Distributions

SKILLS TOPIC 4.8

Using Probability Mean and Standard
and Simulation Deviation of
Random Variables
3.B
Determine parameters for Required Course Content
probability distributions.

Statistical
Argumentation

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

ENDURING UNDERSTANDING

VAR-5

Probability distributions may be used to model variation in populations.

LEARNING OBJECTIVE ESSENTIAL KNOWLEDGE

VAR-5.C VAR-5.C.1

Calculate parameters for a A numerical value measuring a characteristic
discrete random variable. of a population or the distribution of a random
[Skill 3.B] variable is known as a parameter, which is a
single, fixed value.
VAR-5.D
VAR-5.C.2
Interpret parameters for a
discrete random variable. The mean, or expected value, for a discrete
[Skill 4.B]
 ( )random variable X is μX = xi ⋅ P xi .

VAR-5.C.3

The standard deviation for a discrete random

variable X isσ X = (xi − μX )2 ⋅ P (xi ).

VAR-5.D.1

Parameters for a discrete random variable
should be interpreted using appropriate units
and within the context of a specific population.

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Probability, Random Variables, and Probability Distributions 4UNIT

TOPIC 4.9 SKILLS

Combining Random Using Probability
Variables and Simulation

3.B
Determine parameters for
probability distributions.

3.C
Describe probability
distributions.

Required Course Content

ENDURING UNDERSTANDING

VAR-5

Probability distributions may be used to model variation in populations.

LEARNING OBJECTIVE ESSENTIAL KNOWLEDGE

VAR-5.E VAR-5.E.1

Calculate parameters for For random variables X and Y and real
linear combinations of numbers a and b, the mean of aX + bY is
random variables. [Skill 3.B] aμx + bμy.

VAR-5.F VAR-5.E.2

Describe the effects of Two random variables are independent if
linear transformations knowing information about one of them does not
of parameters of change the probability distribution of the other.
random variables. [Skill 3.C]
VAR-5.E.3

For independent random variables X and Y and
real numbers a and b, the mean of aX + bY
is aμx + bμy, and the variance of aX + bY is
a2 2 b2 2
x  y .

VAR-5.F.1

For Y = a + bX, the probability distribution of
the transformed random variable, Y, has the

same shape as the probability distribution for

X, so long as a > 0 and b > 0. The mean of Y is
μy = a + bμx. The standard deviation of Y is
σ y = b σx.

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4UNIT Probability, Random Variables, and Probability Distributions

SKILL TOPIC 4.10

Using Probability Introduction to the
and Simulation Binomial Distribution

3.A

Determine relative
frequencies, proportions,
or probabilities using
simulation or calculations.

Required Course Content

ENDURING UNDERSTANDING

UNC-3

Probabilistic reasoning allows us to anticipate patterns in data.

LEARNING OBJECTIVE ESSENTIAL KNOWLEDGE

UNC-3.A UNC-3.A.1

Estimate probabilities of A probability distribution can be constructed
binomial random variables using the rules of probability or estimated with
using data from a simulation. a simulation using random number generators.
[Skill 3.A]
UNC-3.A.2

A binomial random variable, X, counts
the number of successes in n repeated

independent trials, each trial having two
possible outcomes (success or failure), with the

probability of success p and the probability of
failure 1 – p.

UNC-3.B UNC-3.B.1

Calculate probabilities for The probability that a binomial random variable, X,
a binomial distribution. has exactly x successes for n independent trials,
[Skill 3.A] when the probability of success is p, is calculated

as P(X = x) =  n  px (1 − p)n−x, x = 0, 1, 2, . . . , n.
 x 
 

This is the binomial probability function.

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Probability, Random Variables, and Probability Distributions 4UNIT

TOPIC 4.11 SKILLS

Parameters for a Using Probability
Binomial Distribution and Simulation

Required Course Content 3.B
Determine parameters for
probability distributions.

Statistical
Argumentation

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

ENDURING UNDERSTANDING

UNC-3

Probabilistic reasoning allows us to anticipate patterns in data.

LEARNING OBJECTIVE ESSENTIAL KNOWLEDGE

UNC-3.C UNC-3.C.1

Calculate parameters for a If a random variable is binomial, its mean,
binomial distribution.
[Skill 3.B] μx, is np and its standard deviation, σ x, is
np(1− p).

UNC-3.D UNC-3.D.1

Interpret probabilities and Probabilities and parameters for a binomial
parameters for a binomial distribution should be interpreted using
distribution. [Skill 4.B] appropriate units and within the context of a
specific population or situation.

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4UNIT Probability, Random Variables, and Probability Distributions

SKILLS TOPIC 4.12

Using Probability The Geometric
and Simulation Distribution

3.A Required Course Content
Determine relative
frequencies, proportions, ENDURING UNDERSTANDING
or probabilities using
simulation or calculations. UNC-3

3.B Probabilistic reasoning allows us to anticipate patterns in data.
Determine parameters for
probability distributions.

Statistical
Argumentation

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

LEARNING OBJECTIVE ESSENTIAL KNOWLEDGE

UNC-3.E UNC-3.E.1

Calculate probabilities for For a sequence of independent trials, a
geometric random variables.
[Skill 3.A] geometric random variable, X, gives the

number of the trial on which the first success
occurs. Each trial has two possible outcomes
(success or failure) with the probability of

success p and the probability of failure 1 – p.

UNC-3.E.2

The probability that the first success for
repeated independent trials with probability

of success p occurs on trial x is calculated as
P(X = x) = (1− p)x−1 p, x = 1,2,3,.... This is the

geometric probability function.

UNC-3.F UNC-3.F.1

Calculate parameters of a If a random variable is geometric, its mean,
geometric distribution. 1
[Skill 3.B] μx, is p and its standard deviation,  x , is

(1− p) .
p

UNC-3.G UNC-3.G.1

Interpret probabilities and Probabilities and parameters for a geometric
parameters for a geometric distribution should be interpreted using
distribution. [Skill 4.B] appropriate units and within the context of a
specific population or situation.

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AP Statistics Course and Exam Description QUICK REFERENCE FOR NOTATION AND Probability, Random Variables, and Probability Distributions
FORMULAS FOR PROBABILITY DISTRIBUTIONS

Random Mean for Standard Deviation
Parameter(s) Variable Conditions
Distribution Notes Distribution for Distribution

Probability §§ Represent discrete μx, x X §§ All probabilities  μX = xi ⋅ P (xi ) σ X = (xi − μX )2 ⋅ P (xi )
distribution for a random variables using must be between 0
random variable frequency/ relative and 1. “expected value”
frequency tables or
histograms §§ Probabilities = 1.

§§ Represent continuous
random variables with
density functions.

Sum or difference See Unit 5 for distributions of µX , σX , µY , σY X +Y To calculate the variance μX+Y = μX + μY Variance,
of independent other linear transformations or X − Y or standard deviation or μX−Y = μX − μY
random variables of random variables. of the difference, the σ =2 σ 2 + σ 2
random variables must X Y
be independent. X +Y

Variance,

σ =2 σ 2 + σ 2
X Y
X −Y

n and p §§ n is predetermined. μx = np
Binomial Binomial probability function: ( )X
probability σ X = np 1− p
distribution ( )P(X = x) =n px (1 − p)n−x §§ Binary
x
x = 0,1,2,3,...,n.
§§ Independent

§§ p is the same for

each trial.

Geometric Geometric p X §§ n is not μ = 1 1− p
probability p p
Course Framework V.1 | 103 distribution probability formula: predetermined. σ=

Return to Table of Contents P(X = x) = (1 − p)x−1 p, §§ Binary expected number
x = 1,2,3,....
© 2020 College Board §§ Independent of trials to get the
first success
§§ p is the same for
4UNIT
each repetition

(random).

Note: Other notation could also be correct if properly defined. Incorrect notation will result in lost points on the AP exam.

THIS PAGE IS INTENTIONALLY LEFT BLANK.

AP STATISTICS

UNIT 5

Sampling
Distributions

7–12%

AP EXAM WEIGHTING

~10–12

CLASS PERIODS

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Remember to go to AP Classroom
to assign students the online
Personal Progress Check for
this unit.

Whether assigned as homework or
completed in class, the Personal
Progress Check provides each
student with immediate feedback
related to this unit’s topics and skills.

Personal Progress Check 5
Multiple-choice: ~35 questions
Free-response: 2 questions

§§ Probability and Sampling
Distributions

§§ Investigative Task

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5UNIT 7–12% ~10–12 CLASS PERIODS
  AP EXAM WEIGHTING

Sampling
Distributions

Developing Understanding

BIG IDEA 1 This unit applies probabilistic reasoning to sampling, introducing students to sampling
Variation and distributions of statistics they will use when performing inference in Units 6 and 7. Students
Distribution  VAR should understand that sample statistics can be used to estimate corresponding population
parameters and that measures of center (mean) and variability (standard deviation) for these
§§ How likely is it to get sampling distributions can be determined directly from the population parameters when
a value this large just certain sampling criteria are met. For large enough samples from any population, these
by chance? sampling distributions can be approximated by a normal distribution. Simulating sampling
distributions helps students to understand how the values of statistics vary in repeated
BIG IDEA 2 random sampling from populations with known parameters.
Patterns and
Uncertainty  UNC Building Course Skills like “it” in their interpretations. Using an error
analysis strategy with sample responses can
§§ How can we anticipate 3.B 3.C 4.B help familiarize students with these issues
patterns in the values before they make similar mistakes.
of a statistic from one The probabilities associated with the normal
sample to another? distribution are what statisticians use to Preparing for the AP Exam
justify claims about populations they’ll never
be able to measure directly. Revisiting these Responses on the AP Exam often uncover
properties early in Unit 5 will reinforce why gaps in understanding of sampling
sampling distributions allow statisticians to distributions. Students must clearly
approximate parameters for the population communicate whether they are talking about
of interest. Sketching, shading, and labeling the distribution of a population, a sample of
a normal distribution aids in understanding values (heights, for example), or a sample
the probability being calculated. Students statistic from repeated samples (mean
should practice creating graphical heights, for example). Broad generalizations,
representations, labeling the mean, and such as “larger samples have less variability,”
marking off values 1, 2, and 3 standard leave the exam reader unsure of whether
deviations from the mean. the student is referring to variability within
a sample (for which the statement would be
Students often struggle to interpret false) or a sampling distribution. The word “it”
parameters of probability distributions in often introduces ambiguity to a response.
context, simply describing features of the Students frequently show confusion about
graph rather than explicitly connecting those what condition to check when asserting that
features to the situation described in the the sampling distribution of a given statistic
problem. Teachers can remind students that is approximately normal. Students should
context is about a variable (“tip amounts,” for support normal probability calculations with
example), not just the units (dollars). It’s also a sketch or a calculation of a standardized
critical that students explicitly show that the
appropriate conditions have been verified, and score (z-score), rather than relying on
that they avoid using nonspecific language
calculator syntax.

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5UNIT Sampling Distributions

UNIT AT A GLANCE

Enduring Topic Skills Class Periods
Understanding 5.1 I ntroducing Statistics: ~10–12 CLASS PERIODS

VAR-1 Why Is My Sample Not 1.A Identify the question to be answered or problem
Like Yours? to be solved (not assessed).

VAR-6 5.2 T he Normal Distribution, 3.A Determine relative frequencies, proportions, or
Revisited probabilities using simulation or calculations.

3.C Describe probability distributions.

5.3 T he Central Limit Theorem 3.C Describe probability distributions.

5.4 B iased and Unbiased 4.B Interpret statistical calculations and findings to
Point Estimates assign meaning or assess a claim.

3.B Determine parameters for probability distributions.

5.5 S ampling Distributions for 3.B Determine parameters for probability distributions.
Sample Proportions 3.C Describe probability distributions.

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

UNC-3 5.6 S ampling Distributions 3.B Determine parameters for probability distributions.
for Differences in
Sample Proportions 3.C Describe probability distributions.

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

5.7 S ampling Distributions for 3.B Determine parameters for probability distributions.
Sample Means 3.C Describe probability distributions.

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

5.8 S ampling Distributions 3.B Determine parameters for probability distributions.
for Differences in
Sample Means 3.C Describe probability distributions.

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

Go to AP Classroom to assign the Personal Progress Check for Unit 5.
Review the results in class to identify and address any student misunderstandings.

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Sampling Distributions 5UNIT

SAMPLE INSTRUCTIONAL ACTIVITIES

The sample activities on this page are optional and are offered to provide possible ways to
incorporate various instructional approaches into the classroom. They were developed in
partnership with teachers from the AP community to share ways that they approach teaching
some of the topics in this unit. Please refer to the Instructional Approaches section beginning
on p. 207 for more examples of activities and strategies.

Activity Topic Sample Activity
1 5.2
Think Aloud
2 5.3 Group students into pairs within a larger group of four. Have each student individually
read 2014 FRQ 3 and think aloud with their partner, brainstorming ways to begin each
3 5.5 part of the question. Each student then independently completes all parts. Have the pairs
5.7 compare answers within their groups, improving their individual responses as necessary.
Groups can then compare their responses with other groups. Finally, have students score
their responses according to the rubric.

Use Manipulatives
From a large container of pennies, have each student take two random samples of size 5,
two of size 10, and two of size 25, and record the dates on those pennies. Have students
calculate the mean of the dates in each sample and then construct four “dotplots” on the
floor: one using the pennies, one using nickels placed at the mean of the student’s sample
size 5, one using dimes placed at the mean of the sample size 10, and one using quarters
placed at the mean of the sample size 25.

Password-Style Games
Have partners sit facing opposite sides of the room. Display vocabulary terms from the
unit on the classroom screen. Have the students facing the screen describe the terms
to their partner who then tries to guess the terms described. After half of the terms have
been used, have students switch roles. Terms to include: parameter, statistic, sampling
distribution, distribution of sample data, sample distribution, unbiased estimator, sampling

variability of a statistic, bias, sample proportion, sample mean, μpˆ , σ pˆ , μx , σ x , and central

limit theorem.

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5UNIT Sampling Distributions

SKILL TOPIC 5.1

Selecting Statistical Introducing Statistics:
Methods Why Is My Sample
Not Like Yours?
1.A
Identify the question to be
answered or problem to
be solved.

Required Course Content

ENDURING UNDERSTANDING

VAR-1

Given that variation may be random or not, conclusions are uncertain.

LEARNING OBJECTIVE ESSENTIAL KNOWLEDGE

VAR-1.G VAR-1.G.1

Identify questions suggested Variation in statistics for samples taken from
by variation in statistics for the same population may be random or not.
samples collected from the
same population. [Skill 1.A]

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Sampling Distributions 5UNIT

TOPIC 5.2 SKILLS

The Normal Using Probability
Distribution, Revisited and Simulation

Required Course Content 3.A
Determine relative
ENDURING UNDERSTANDING frequencies, proportions,
or probabilities using
VAR-6 simulation or calculations.
3.C
The normal distribution may be used to model variation. Describe probability
distributions.

ILLUSTRATIVE EXAMPLE
Continuous random variable:
If one looks at a clock at a
random time, the probability
that the minute hand is
between the 3 and the 6 is
one fourth.

LEARNING OBJECTIVE ESSENTIAL KNOWLEDGE

VAR-6.A VAR-6.A.1

Calculate the probability A continuous random variable is a variable
that a particular value that can take on any value within a specified
lies in a given interval of a domain. Every interval within the domain has a
normal distribution. [Skill 3.A] probability associated with it.

VAR-6.A.2

A continuous random variable with a normal
distribution is commonly used to describe
populations. The distribution of a normal
random variable can be described by a normal,
or “bell-shaped,” curve.

VAR-6.A.3

The area under a normal curve over a given
interval represents the probability that a
particular value lies in that interval.

VAR-6.B VAR-6.B.1

Determine the interval The boundaries of an interval associated with
associated with a given area a given area in a normal distribution can be
in a normal distribution.
[Skill 3.A] determined using z-scores or technology, such

as a calculator, a standard normal table, or
computer-generated output.

continued on next page

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5UNIT Sampling Distributions

LEARNING OBJECTIVE ESSENTIAL KNOWLEDGE

VAR-6.B VAR-6.B.2

Determine the interval Intervals associated with a given area in a
associated with a given area
in a normal distribution. normal distribution can be determined by
[Skill 3.A]
assigning appropriate inequalities to the

boundaries of the intervals:

a.  P(X < xa ) = p means that the lowest p%
100
of values l<iextbo)t=he10ple0ftmoef axnas.
that p% of
b. P (xa < X

values xlieb )b=e1tw0pe0emnexaanasntdhaxtb.the highest p%

c.  P(X >

of values lie to the right of xb.
d. T o determine the most extreme p%

of values requires dividing the area

associated with p% into two equal areas

on either )e=xtr21e1m0pe0 oafntdheP(dXist>ribxub )tio=n12: p
100
P(X < xa

means that half of the p% most extreme
values lie to the left of xa and half of the p%
most extreme values lie to the right of xb.

VAR-6.C xa xb

Determine the VAR-6.C.1
appropriateness of using
the normal distribution to Normal distributions are symmetrical and
approximate probabilities for “bell-shaped.” As a result, normal distributions
unknown distributions. can be used to approximate distributions with
[Skill 3.C] similar characteristics.

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Sampling Distributions 5UNIT

TOPIC 5.3 SKILL

The Central Using Probability
Limit Theorem and Simulation

3.C
Describe probability
distributions.

Required Course Content

ENDURING UNDERSTANDING

UNC-3

Probabilistic reasoning allows us to anticipate patterns in data.

LEARNING OBJECTIVE ESSENTIAL KNOWLEDGE

UNC-3.H UNC-3.H.1

Estimate sampling A sampling distribution of a statistic is the
distributions using simulation. distribution of values for the statistic for
[Skill 3.C] all possible samples of a given size from a
given population.

UNC-3.H.2

The central limit theorem (CLT) states that
when the sample size is sufficiently large,
a sampling distribution of the mean of
a random variable will be approximately
normally distributed.

UNC-3.H.3

The central limit theorem requires that the
sample values are independent of each other

and that n is sufficiently large.

UNC-3.H.4

A randomization distribution is a collection of
statistics generated by simulation assuming
known values for the parameters. For a
randomized experiment, this means repeatedly
randomly reallocating/reassigning the
response values to treatment groups.

UNC-3.H.5

The sampling distribution of a statistic can
be simulated by generating repeated random
samples from a population.

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5UNIT Sampling Distributions

SKILLS TOPIC 5.4

Statistical Biased and Unbiased
Argumentation Point Estimates

4.B Required Course Content
Interpret statistical
calculations and findings to
assign meaning or assess
a claim.

Using Probability
and Simulation

3.B
Determine parameters for
probability distributions.

ENDURING UNDERSTANDING

UNC-3

Probabilistic reasoning allows us to anticipate patterns in data.

LEARNING OBJECTIVE ESSENTIAL KNOWLEDGE

UNC-3.I UNC-3.I.1

Explain why an estimator is or When estimating a population parameter,
is not unbiased. [Skill 4.B] an estimator is unbiased if, on average,
the value of the estimator is equal to the
population parameter.

UNC-3.J UNC-3.J.1

Calculate estimates for a When estimating a population parameter,
population parameter. an estimator exhibits variability that can be
[Skill 3.B] modeled using probability.

UNC-3.J.2

A sample statistic is a point estimator of the
corresponding population parameter.

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Sampling Distributions 5UNIT

TOPIC 5.5 SKILLS

Sampling Using Probability
Distributions for and Simulation
Sample Proportions
3.B
Required Course Content Determine parameters for
probability distributions.
ENDURING UNDERSTANDING 3.C
Describe probability
ULONRC--13 distributions.

Probabilistic reasoning allows us to anticipate patterns in data. Statistical
Argumentation
LEARNING OBJECTIVE ESSENTIAL KNOWLEDGE
LUENCA-3R.KNING OBJECTIVE 4.B
EUSNCS-3E.KN.1TIAL KNOWLEDGE Interpret statistical
LDOeRt-e1r.mA ine parameters of calculations and findings to
a sampling distribution FLOoRr-i1n.dAe.1pendent samples (sampling with assign meaning or assess
for sample proportions. a claim.
[Skill 3.B] replacement) of a categorical variable from a
AVAILABLE RESOURCES
population with population proportion, p, the §§ Classroom Resources >
◆◆ Sampling
spˆa, mhapsliangmdeiasntr,ibμupˆti=onpoafntdhea sample proportion, Distributions
standard deviation, ◆◆ Calculations
Aren’t Enough!
The Importance of
Communication in
AP Statistics

σ pˆ = p(1− p)

n.

UNC-3.K.2

If sampling without replacement, the standard
deviation of the sample proportion is smaller
than what is given by the formula above. If the
sample size is less than 10% of the population
size, the difference is negligible.

UNC-3.L UNC-3.L.1

Determine whether a sampling For a categorical variable, tphreopsoarmtipolnin, gpˆ, will
distribution for a sample distribution of the sample
proportion can be described
as approximately normal. have an approximate normal distribution,
[Skill 3.C]
provided the sample size is large enough:

np ≥ 10 and n(1− p) ≥ 10

UNC-3.M UNC-3.M.1

Interpret probabilities Probabilities and parameters for a sampling
and parameters for a distribution for a sample proportion should be
sampling distribution for a interpreted using appropriate units and within
sample proportion. [Skill 4.B] the context of a specific population.

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5UNIT Sampling Distributions

SKILLS TOPIC 5.6

Using Probability Sampling
and Simulation Distributions for
Differences in Sample
3.B Proportions
Determine parameters for
probability distributions. Required Course Content
3.C
Describe probability
distributions.

Statistical
Argumentation

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

AVAILABLE RESOURCES ENDURING UNDERSTANDING
§§ Classroom Resource >
UNC-3
◆◆ Sampling
Distributions Probabilistic reasoning allows us to anticipate patterns in data.

◆◆ Calculations LEARNING OBJECTIVE ESSENTIAL KNOWLEDGE
Aren’t Enough!
The Importance of UNC-3.N UNC-3.N.1
Communication in
AP Statistics Determine parameters of For a categorical variable, when randomly
a sampling distribution
for a difference in sample sampling with replacement from two
proportions. [Skill 3.B]
independent populations with population

proportions p1 and spa2,mthpelesparmoppolinrtgiodnisstrpˆi1b−utpiˆo2n

of the difference in

has mean, μpˆ1 − pˆ2 = p1 − p2 and standard

deviation, σ pˆ1 − pˆ2 = p1(1 − p1 ) + p2 (1 − p2 ) .
n1 n2

UNC-3.N.2

If sampling without replacement, the standard
deviation of the difference in sample proportions
is smaller than what is given by the formula
above. If the sample sizes are less than 10% of
the population sizes, the difference is negligible.

continued on next page

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Sampling Distributions 5UNIT

LEARNING OBJECTIVE ESSENTIAL KNOWLEDGE

UNC-3.O UNC-3.O.1

Determine whether a iTnhseasmapmleplpinrogpdoisrttrioibnustipˆo1n−opfˆ2thweilldhifafevereannce
sampling distribution for
a difference of sample approximate normal distribution provided
proportions can be described the sample sizes are large enough:
as approximately normal.
[Skill 3.C] n1 p1 ≥ 10,n1(1 − p1) ≥ 10,n2 p2 ≥ 10,n2 (1 − p2 ) ≥ 10.

UNC-3.P UNC-3.P.1

Interpret probabilities and Parameters for a sampling distribution for a
parameters for a sampling difference of proportions should be interpreted
distribution for a difference using appropriate units and within the context
in proportions. [Skill 4.B] of a specific populations.

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5UNIT Sampling Distributions

SKILLS TOPIC 5.7

Using Probability Sampling
and Simulation Distributions for
Sample Means
3.B
Determine parameters for Required Course Content
probability distributions.
3.C ENDURING UNDERSTANDING
Describe probability
distributions. LUONRC--13

Statistical Probabilistic reasoning allows us to anticipate patterns in data.
Argumentation
LEARNING OBJECTIVE ESSENTIAL KNOWLEDGE
4.B LUENCA-3R.QNING OBJECTIVE EUSNCS-3E.QN.1TIAL KNOWLEDGE
Interpret statistical
calculations and findings to DLOeRt-e1r.mA ine parameters for FLOoRr-a1.nAu.1merical variable, when random
assign meaning or assess a sampling distribution for
a claim. sample means. [Skill 3.B] sampling with replacement from a population

AVAILABLE RESOURCES with mean μ and standard deviation, σ pˆ, the
§§ Classroom Resources > sampling distribution of the sample mean hσas
◆◆ Sampling n
Distributions mean μx = μ and standard deviation σ x =
◆◆ Calculations
Aren’t Enough!
The Importance of
Communication in
AP Statistics

.

UNC-3.Q.2

If sampling without replacement, the standard
deviation of the sample mean is smaller than
what is given by the formula above. If the
sample size is less than 10% of the population
size, the difference is negligible.

UNC-3.R UNC-3.R.1

Determine whether a sampling For a numerical variable, if the population
distribution of a sample distribution can be modeled with a normal
mean can be described as distribution, the sampling distribution of
approximately normal.
[Skill 3.C] the sample mean, x , can be modeled with a

normal distribution.

continued on next page

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Sampling Distributions 5UNIT

LEARNING OBJECTIVE ESSENTIAL KNOWLEDGE

UNC-3.R UNC-3.R.2

Determine whether a sampling For a numerical variable, if the population
distribution of a sample distribution cannot be modeled with a
mean can be described as normal distribution, the sampling distribution
approximately normal.
[Skill 3.C] of the sample mean, x , can be modeled

UNC-3.S approximately by a normal distribution,
provided the sample size is large enough,
Interpret probabilities e.g., greater than or equal to 30.
and parameters for a
sampling distribution for a UNC-3.S.1
sample mean. [Skill 4.B]
Probabilities and parameters for a sampling
distribution for a sample mean should be
interpreted using appropriate units and within
the context of a specific population.

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5UNIT Sampling Distributions

SKILLS TOPIC 5.8

Using Probability Sampling
and Simulation Distributions for
Differences in
3.B Sample Means
Determine parameters for
probability distributions. Required Course Content
3.C
Describe probability
distributions.

Statistical
Argumentation

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

AVAILABLE RESOURCES ENDURING UNDERSTANDING
§§ Classroom Resources >
UNC-3
◆◆ Sampling
Distributions Probabilistic reasoning allows us to anticipate patterns in data.

◆◆ Calculations LEARNING OBJECTIVE ESSENTIAL KNOWLEDGE
Aren’t Enough!
The Importance of UNC-3.T UNC-3.T.1
Communication in
AP Statistics Determine parameters of a For a numerical variable, when randomly
sampling distribution for a
difference in sample means. sampling with replacement from two
[Skill 3.B]
independent populations with population

means μ1 and μ2 and population standard
deviations σ1 and σ 2, the sampling distribution
of the difference in sample means x1 − x2 has
mean μ(x1−x2) = μ1 − μ2 and standard deviation,

σ (x1−x2 ) = σ 12 + σ 2 .
n1 2

n2

UNC-3.T.2

If sampling without replacement, the standard
deviation of the difference in sample means
is smaller than what is given by the formula
above. If the sample sizes are less than
10% of the population sizes, the difference
is negligible.

continued on next page

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Sampling Distributions 5UNIT

LEARNING OBJECTIVE ESSENTIAL KNOWLEDGE

UNC-3.U UNC-3.U.1

Determine whether a The sampling distribution of the difference in
sampling distribution of
a difference in sample sample means x1 − x2 can be modeled with
means can be described
as approximately normal. a normal distribution if the two population
[Skill 3.C] distributions can be modeled with a
normal distribution.
UNC-3.V
UNC-3.U.2
Interpret probabilities and
parameters for a sampling The sampling distribution of the difference
distribution for a difference in
sample means. [Skill 4.B] in sample means x 1 − x 2 can be modeled

approximately by a normal distribution if
the two population distributions cannot be
modeled with a normal distribution but both
sample sizes are greater than or equal to 30.

UNC-3.V.1

Probabilities and parameters for a sampling
distribution for a difference of sample
means should be interpreted using
appropriate units and within the context of a
specific populations.

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AP Statistics Course and Exam Description QUICK REFERENCE FOR NOTATION AND 5UNIT Sampling Distributions
FORMULAS FOR SAMPLING DISTRIBUTIONS

Mean for Standard Deviation

Distribution Notes Parameter(s) Statistic Conditions Distribution for Distribution

Normal distribution A continuous μ and σ pˆ μ σ pˆ
random probability p
distribution

Sampling Compare to the pˆ §§ Simple random sample μpˆ = p σ pˆ = p(1− p)
distribution for a mean and standard n
sample proportion deviation of a (Random)
binomial random
§§ Normal or np ≥ 10 and
variable, X n(1 − p) ≥ 10, (Large counts)

§§ For standard deviations:

population  10n (10% rule)

Sampling p1 − p2 p1 − p2 §§ Simple random samples μ( pˆ1− pˆ2 ) = p1 − p2 p1(1 − p1) + p2 (1 − p2 )
distribution for (Random) n1 n2
a difference in σ =pˆ1 − pˆ2
sample proportions §§ Large counts
§§ 10% rule

Sampling μ x §§ SRS (Random) μx = μ σx = σ
distribution for the μ1 − μ2 x1 − x2 §§ Normal or sample size ≥30 n
sample mean §§ 10% rule

Sampling §§ SRS (Random) μ(x1−x2 ) = μ1 − μ2 σ (x1−x2 ) = σ 12 + σ 2
distribution for §§ Normal or sample sizes ≥30 2
the difference in §§ 10% rule
sample means n1 n2

Course Framework V.1 | 122 Standard deviation σ pˆ s

Return to Table of Contents Note: Other notation could also be correct if properly defined. Incorrect notation will result in lost points on the AP exam.

© 2020 College Board

AP STATISTICS

UNIT 6

Inference for
Categorical
Data:
Proportions

12–15%

AP EXAM WEIGHTING

~16–18

CLASS PERIODS

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Remember to go to AP Classroom
to assign students the online
Personal Progress Check for
this unit.

Whether assigned as homework or
completed in class, the Personal
Progress Check provides each
student with immediate feedback
related to this unit’s topics and skills.

Personal Progress Check 6
Multiple-choice: ~55 questions
Free-response: 2 questions

§§ Inference

§§ Investigative Task

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6UNIT 12–15% ~16–18 CLASS PERIODS
  AP EXAM WEIGHTING

Inference for
Categorical Data:
Proportions

Developing Understanding

BIG IDEA 1 This unit introduces statistical inference, which will continue through the end of the course.
Variation and Students will analyze categorical data to make inferences about binomial population
Distribution  VAR proportions. Provided conditions are met, students will use statistical inference to construct
and interpret confidence intervals to estimate population proportions and perform
§§ When can we use a significance tests to evaluate claims about population proportions. Students begin by
normal distribution learning inference procedures for one proportion and then examine inference methods for a
to perform inference difference between two proportions. They will also interpret the two types of errors that can
calculations involving be made in a significance test, their probabilities, and possible consequences in context.
population proportions?
Building Course Skills early and often that statistical tests do not
BIG IDEA 2 provide evidence for what can be accepted
Patterns and 1.D 3.D 4.D or proved; they only provide evidence for
Uncertainty  UNC “rejecting” or “failing to reject” the null.
Unit 6 is a critical transition point in the
§§ How can we narrow course, as students begin learning skills that Preparing for the AP Exam
the width of a will be applied repeatedly in subsequent
confidence interval? units. Students need to familiarize When using statistical inference to
themselves with these procedures so they construct confidence intervals or perform
BIG IDEA 3 can build proficiency over time. Applying significance tests, students should identify
Data-Based different inference methods requires fluency the appropriate inference method by name
Predictions, Decisions, with verifying conditions. Students often or formula. For inference with population
and Conclusions  DAT check conditions superficially (e.g., just listing proportions, students should verify that the
“SRS”) without explicitly connecting them to following conditions are met: (1) random
§§ If the proportion the problem. Teachers can make sure
of subjects who students practice verifying conditions in sample and (2) large sample (e.g., npˆ ≥ 10
experience serious side context by providing numerical calculations and n(1 − pˆ) ≥ 10). When sampling without
effects when taking and explaining how each condition is met.
a new drug is smaller replacement, students should also verify
than the proportion Precision of language is key. Students that the sample size is at most 10% of the
of subjects who often interpret confidence intervals and population. Verification should be simple
experience serious confidence levels incorrectly. Providing and specific.
side effects when students with sentence starters or
taking a placebo, how templates can help them learn to generate Next, students should present calculations
can we determine appropriate responses (e.g., Confidence and then interpret results in the context of
if the difference is interval: “We are 95% confident that the problem. Students often find it beneficial
statistically significant? the interval from to captures the to use language provided in the question.
[parameter in context].”). For decisions In 2017 FRQ 2, for example, the response
based on a hypothesis test, students may might read “We can be 95% confident that
incorrectly claim that “we can accept” or the proportion of all customers who, having
“have proven” the null. Teachers can reinforce asked for a cup of water when placing an
order, will fill the cup with a soft drink is
between 0.1883 and 0.3867.”

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6UNIT Inference for Categorical Data: Proportions

UNIT AT A GLANCE

Enduring Topic Skills Class Periods
Understanding 6.1 I ntroducing Statistics: ~16–18 CLASS PERIODS

VAR-1 Why Be Normal? 1.A Identify the question to be answered or problem
to be solved (not assessed).

UNC-4 6.2 C onstructing a 1.D Identify an appropriate inference method for
Confidence Interval for a confidence intervals.
VAR-6 Population Proportion
4.C Verify that inference procedures apply in a
VAR-6, 6.3 J ustifying a Claim given situation.
DAT-3 Based on a Confidence
Interval for a Population 3.D Construct a confidence interval, provided
DAT-3 Proportion conditions for inference are met.

6.4 S etting Up a Test for a 4.B Interpret statistical calculations and findings to
Population Proportion assign meaning or assess a claim.

6.5 I nterpreting p-Values 4.D Justify a claim based on a confidence interval.
6.6 C oncluding a Test for a
4.A Make an appropriate claim or draw an
Population Proportion appropriate conclusion.

1.F Identify null and alternative hypotheses.

1.E Identify an appropriate inference method for
significance tests.

4.C Verify that inference procedures apply in a
given situation.

3.E Calculate a test statistic and find a p-value,

provided conditions for inference are met.

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

4.E Justify a claim using a decision based on
significance tests.

continued on next page

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Inference for Categorical Data: Proportions 6UNIT

UNIT AT A GLANCE (cont’d)

Enduring Topic Skills Class Periods
Understanding 6.7 P otential Errors When ~16–18 CLASS PERIODS

Performing Tests 1.B Identify key and relevant information to answer a
question or solve a problem.
UNC-5
3.A Determine relative frequencies, proportions, or
probabilities using simulation or calculations.

4.A Make an appropriate claim or draw an
appropriate conclusion.

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

6.8 C onfidence Intervals 1.D Identify an appropriate inference method for
for the Difference of confidence intervals.
Two Proportions
4.C Verify that inference procedures apply in a
UNC-4 given situation.

3.D Construct a confidence interval, provided
conditions for inference are met.

6.9 J ustifying a Claim 4.B Interpret statistical calculations and findings to
Based on a Confidence assign meaning or assess a claim.
Interval for a Difference of
Population Proportions 4.D Justify a claim based on a confidence interval.

VAR-6 6.10 S etting Up a Test 1.F Identify null and alternative hypotheses.
for the Difference
of Two Population 1.E Identify an appropriate inference method for
Proportions significance tests.

4.C Verify that inference procedures apply in a
given situation.

VAR-6, DAT-3 6.11 C arrying Out a Test 3.E Calculate a test statistic and find a p-value,
for the Difference
of Two Population provided conditions for inference are met.
Proportions
4.B Interpret statistical calculations and findings to
assign meaning or assess a claim.

4.E Justify a claim using a decision based on
significance tests.

Go to AP Classroom to assign the Personal Progress Check for Unit 6.
Review the results in class to identify and address any student misunderstandings.

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6UNIT Inference for Categorical Data: Proportions

SAMPLE INSTRUCTIONAL ACTIVITIES

The sample activities on this page are optional and are offered to provide possible ways to
incorporate various instructional approaches into the classroom. They were developed in
partnership with teachers from the AP community to share ways that they approach teaching
some of the topics in this unit. Please refer to the Instructional Approaches section beginning
on p. 207 for more examples of activities and strategies.

Activity Topic Sample Activity
1 6.4
6.7 Error Analysis
2 6.8 Give student pairs a worksheet with 20 sets of hypotheses (including hypotheses for a
population proportion and for the difference of two proportions), each with a common
3 6.5 student mistake. Have students circle the incorrect part, write why the circled component
6.6 is incorrect, and then write the correct hypotheses. Include errors such as using statistics
6.11 instead of parameters, and interchanging the = and > in the two hypotheses.

6.2 Sentence Starters
6.8
For a given question, provide students with a set of hypotheses, p-value, significance
level, and context. Have them compare the p-value to the significance level to determine

whether or not to reject the null hypothesis. Using a given sentence starter with blanks to
fill in, have students write a sentence in context explaining if they have enough evidence to

“reject H0”, or if they will “fail to reject H0.” Make sure students avoid the common mistake
of implying that evidence supports an “accept H0” conclusion or a “reject Ha” conclusion.

The Scribe and the Calculator
Have students work with a partner to construct and interpret a confidence interval for
a population proportion. Only one partner is allowed to use the calculator, and only the
other partner is allowed to write. When a calculation needs to be made, the scribe can only
describe to the calculator operator which buttons to push; when writing needs to be done,
the calculator operator can only describe to the scribe what needs to be written. Have
students switch roles when constructing and interpreting a confidence interval for the
difference of two population proportions.

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Inference for Categorical Data: Proportions 6UNIT

TOPIC 6.1 SKILL

Introducing Statistics: Selecting Statistical
Why Be Normal? Methods

1.A
Identify the question to be
answered or problem to
be solved.

Required Course Content

ENDURING UNDERSTANDING

VAR-1

Given that variation may be random or not, conclusions are uncertain.

LEARNING OBJECTIVE ESSENTIAL KNOWLEDGE

VAR-1.H VAR-1.H.1

Identify questions suggested Variation in shapes of data distributions may be
by variation in the shapes of random or not.
distributions of samples taken
from the same population.
[Skill 1.A]

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6UNIT Inference for Categorical Data: Proportions

SKILLS TOPIC 6.2

Selecting Statistical Constructing a
Methods Confidence Interval
for a Population
1.D Proportion
Identify an appropriate
inference method for Required Course Content
confidence intervals.
ENDURING UNDERSTANDING
Statistical
Argumentation UNC-4

4.C An interval of values should be used to estimate parameters, in order to account
Verify that inference for uncertainty.
procedures apply in a
given situation.

Using Probability
and Simulation

3.D
Construct a confidence
interval, provided conditions
for inference are met.

LEARNING OBJECTIVE ESSENTIAL KNOWLEDGE

UNC-4.A UNC-4.A.1

Identify an appropriate The appropriate confidence interval procedure
confidence interval for a one-sample proportion for one
procedure for a
population proportion. categorical variable is a one sample z-interval
[Skill 1.D]
for a proportion.

UNC-4.B UNC-4.B.1

Verify the conditions In order to make assumptions necessary for
for calculating inference on population proportions, means,
confidence intervals for a and slopes, we must check for independence
population proportion. in data collection methods and for selection of
[Skill 4.C] the appropriate sampling distribution.

continued on next page

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Inference for Categorical Data: Proportions 6UNIT

LEARNING OBJECTIVE ESSENTIAL KNOWLEDGE

UNC-4.B UNC-4.B.2

Verify the conditions In order to calculate a confidence interval to
for calculating
confidence intervals for a estimate a population proportion, p, we must
population proportion.
[Skill 4.C] check for independence and that the sampling
distribution is approximately normal.
UNC-4.C
a. To check for independence:
Determine the margin of
error for a given sample i. Data should be collected using a random
size and an estimate for the sample or a randomized experiment.
sample size that will result in
a given margin of error for a ii. When sampling without replacement,
population proportion.
[Skill 3.D] check that n ≤ 10%N, where N is the size

of the population.

b. To check that the sampling distribution of pˆ

is approximately normal (shape):

i. tFnhouemr ncbuaetmer bgoeof rfraiocilfaulsrveuascr,cinae(bs1lse−esspˆ, ,c)nhapˆer,ecakanttdhletahatsebto1t0h

so that the sample size is large enough to
support an assumption of normality.

UNC-4.C.1

Based on sample data, the standard error

of a statistic is an estimate for the standard

deviation for the statistic. The standard error
pˆ(1− pˆ) .
of pˆ is SEpˆ = n

UNC-4.C.2

A margin of error gives how much a value of a
sample statistic is likely to vary from the value
of the corresponding population parameter.

UNC-4.C.3

For categorical variables, the margin of error

is the critical value (z *) times the standard

error (SE) of the relevant statistic, which equals

z* pˆ(1 − pˆ) for a one sample proportion.
n

UNC-4.C.4

The formula for margin of error can be

rearranged to solve for n, the minimum sample

size needed to achieve gaugeivsesnfomr apˆrgoirnuosfeepˆrro=r.0.5
For this purpose, use a

in order to find an upper bound for the sample

size that will result in a given margin of error.

continued on next page

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6UNIT Inference for Categorical Data: Proportions

LEARNING OBJECTIVE ESSENTIAL KNOWLEDGE

UNC-4.D UNC-4.D.1

Calculate an appropriate In general, an interval estimate can be
confidence interval for a constructed as point estimate ± (margin
population proportion. of error). For a one-sample proportion, the
[Skill 3.D]
interval estimate is pˆ ± z* pˆ(1 − pˆ) .
UNC-4.E n

Calculate an interval CLARIFYING STATEMENT:
estimate based on a Formulas for interval estimates do not
confidence interval for a appear explicitly on the AP Statistics Formula
population proportion. Sheet provided with the AP Statistics Exam.
[Skill 3.D] However, these formulas do not need to be
memorized, as they can be constructed
based on the general test statistic formula
and the relevant standard error formulas that
are provided on the formula sheet.

UNC-4.D.2

Critical values represent the boundaries
encompassing the middle C% of the
standard normal distribution, where C% is an
approximate confidence level for a proportion.

UNC-4.E.1

Confidence intervals for population
proportions can be used to calculate interval
estimates with specified units.

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Inference for Categorical Data: Proportions 6UNIT

TOPIC 6.3 SKILLS

Justifying a Claim Statistical
Based on a Confidence Argumentation
Interval for a
Population Proportion 4.B
Interpret statistical
Required Course Content calculations and findings to
assign meaning or assess
ENDURING UNDERSTANDING a claim.

UNC-4 4.D
Justify a claim based on a
An interval of values should be used to estimate parameters, in order to account confidence interval.
for uncertainty.
4.A
LEARNING OBJECTIVE ESSENTIAL KNOWLEDGE Make an appropriate
claim or draw an
UNC-4.F UNC-4.F.1 appropriate conclusion.

Interpret a confidence A confidence interval for a population AVAILABLE RESOURCE
interval for a proportion either contains the population §§ Classroom Resource >
population proportion. proportion or it does not, because each Calculations
[Skill 4.B] interval is based on random sample data, which Aren’t Enough!
varies from sample to sample. The Importance of
Communication in
UNC-4.F.2 AP Statistics

We are C% confident that the confidence ILLUSTRATIVE EXAMPLE
interval for a population proportion captures UNC-4.F.4:
the population proportion. For interpreting a 99%
confidence interval of
UNC-4.F.3 (0.268, 0.292), based on the
proportion of a nationally
In repeated random sampling with the representative sample of
same sample size, approximately C% of twelfth-grade students
confidence intervals created will capture the who answered a particular
population proportion. multiple choice question
correctly:
UNC-4.F.4 “We are 99 percent
confident that the interval
Interpreting a confidence interval for a one- from 0.268 to 0.292
sample proportion should include a reference contains the population
to the sample taken and details about the proportion of all United
population it represents. States twelfth-grade
students who would answer
continued on next page this question correctly”
(2011 FRQ 6(a)).

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6UNIT Inference for Categorical Data: Proportions

LEARNING OBJECTIVE ESSENTIAL KNOWLEDGE

UNC-4.G UNC-4.G.1

Justify a claim based on a A confidence interval for a population
confidence interval for a proportion provides an interval of values that
population proportion. may provide sufficient evidence to support a
[Skill 4.D] particular claim in context.

UNC-4.H UNC-4.H.1

Identify the relationships When all other things remain the same,
between sample size, width
of a confidence interval, the width of the confidence interval for a
confidence level, and margin
of error for a population population proportion tends to decrease as
proportion. [Skill 4.A]
the sample size increases. For a population

proportion, the w1idth. of the interval is
proportional to
n

UNC-4.H.2

For a given sample, the width of the confidence
interval for a population proportion increases
as the confidence level increases.

UNC-4.H.3

The width of a confidence interval for a
population proportion is exactly twice the
margin of error.

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Inference for Categorical Data: Proportions 6UNIT

TOPIC 6.4 SKILLS

Setting Up a Test for a Selecting Statistical
Population Proportion Methods

Required Course Content 1.F
Identify null and alternative
ENDURING UNDERSTANDING hypotheses.

VAR-6 1.E
Identify an appropriate
The normal distribution may be used to model variation. inference method for
significance tests.
LEARNING OBJECTIVE ESSENTIAL KNOWLEDGE
Statistical
VAR-6.D VAR-6.D.1 Argumentation

Identify the null and The null hypothesis is the situation that is 4.C
alternative hypotheses for a assumed to be correct unless evidence Verify that inference
population proportion. suggests otherwise, and the alternative procedures apply in a
[Skill 1.F] hypothesis is the situation for which evidence given situation.
is being collected.
AVAILABLE RESOURCES
§§ Classroom Resources >
◆◆ Inference
◆◆ Coke® Versus Pepsi®:
An Introductory
Activity for Test of
Significance

VAR-6.D.2

For hypotheses about parameters, the null
hypothesis contains an equality reference
(=, ≥, or ≤), while the alternative hypothesis
contains a strict inequality (<, >, or ≠). The type
of inequality in the alternative hypothesis is
based on the question of interest. Alternative
hypotheses with < or > are called one-sided, and
alternative hypotheses with ≠ are called two-
sided. Although the null hypothesis for a one-
sided test may include an inequality symbol, it is
still tested at the boundary of equality.

VAR-6.D.3

The null hypothesis for a population proportion is:

H0 : p = p0, where p0 is the null hypothesized

value for the population proportion.

VAR-6.D.4

A one-sided alternative hypothesis for a
a : p < p0 or a:
proportion is either H H H p > p0 . p2 .
hypothesis a: p1 ≠
A two-sided alternate is

continued on next page

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6UNIT Inference for Categorical Data: Proportions

LEARNING OBJECTIVE ESSENTIAL KNOWLEDGE

VAR-6.D VAR-6.D.5

Identify the null and For a one-sample z-test for a population
alternative hypotheses for a
population proportion. proportion, the null hypothesis specifies a
[Skill 1.F] value for the population proportion, usually one
indicating no difference or effect.
VAR-6.E
VAR-6.E.1
Identify an appropriate
testing method for a For a single categorical variable, the
population proportion. appropriate testing method for a population
[Skill 1.E]
proportion is a one-sample z-test for a
VAR-6.F
population proportion.
Verify the conditions
for making statistical VAR-6.F.1
inferences when testing a
population proportion. In order to make statistical inferences when
[Skill 4.C] testing a population proportion, we must
check for independence and that the sampling
distribution is approximately normal:

a. To check for independence:

i. Data should be collected using a random
sample or a randomized experiment.

ii. When sampling without replacement,

check that n ≤ 10%N.
b. To check that the sampling distribution of pˆ

is approximately normal (shape):

i. Assuming that H0 is true ( p = p0 ), verify
that both the number of successes, np0 ,
and
the number of failures, n(1 − p0 ) are

at least 10 so that that the sample size is

large enough to support an assumption

of normality.

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Inference for Categorical Data: Proportions 6UNIT

TOPIC 6.5 SKILLS

Interpreting p-Values Using Probability
and Simulation
Required Course Content
3.E
ENDURING UNDERSTANDING Calculate a test statistic

VAR-6 and find a p-value, provided

The normal distribution may be used to model variation. conditions for inference
are met.

Statistical
Argumentation

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

AVAILABLE RESOURCE
§§ Classroom Resource >
Inference

LEARNING OBJECTIVE ESSENTIAL KNOWLEDGE

VAR-6.G VAR-6.G.1

Calculate an appropriate test The distribution of the test statistic assuming
the null hypothesis is true (null distribution) can
statistic and p-value for a be either a randomization distribution or when
a probability model is assumed to be true, a
population proportion.
[Skill 3.E] theoretical distribution (z).

VAR-6.G.2

When using a z-test, the standardized test

statistic can be written:

test statistic = sample statistic-null value of the parameter .
standard deviation of the statistic

This is called a z-statistic for proportions.

VAR-6.G.3

The test statistic for a population proportion is:
p  p0 .
z p0(1  p0 )

n

CLARIFYING STATEMENT:

The formulas for test statistics do not appear
explicitly on the AP Statistics Formula
Sheet provided with the AP Statistics Exam.
However, these formulas do not need to
be memorized, as they can be constructed
based on the general test statistic formula
and the relevant standard error formulas that
are provided on the formula sheet.

continued on next page

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6UNIT Inference for Categorical Data: Proportions

LEARNING OBJECTIVE ESSENTIAL KNOWLEDGE

VAR-6.G VAR-6.G.4

Calculate an appropriate test A p-value is the probability of obtaining a

statistic and p-value for a test statistic as extreme or more extreme
than the observed test statistic when the null
population proportion. hypothesis and probability model are assumed
[Skill 3.E] to be true. The significance level may be given
or determined by the researcher.

ENDURING UNDERSTANDING

DAT-3

Significance testing allows us to make decisions about hypotheses within a
particular context.

LEARNING OBJECTIVE ESSENTIAL KNOWLEDGE

DAT-3.A DAT-3.A.1

Interpret the p-value of The p-value is the proportion of values for the

a significance test for a null distribution that are as extreme or more
population proportion. extreme than the observed value of the test
[Skill 4.B] statistic. This is:

a. The proportion at or above the
observed value of the test statistic, if the
alternative is >.

b. The proportion at or below the observed
value of the test statistic, if the
alternative is <.

c. The proportion less than or equal to the
negative of the absolute value of the test
statistic plus the proportion greater than
or equal to the absolute value of the test
statistic, if the alternative is ≠.

DAT-3.A.2

An interpretation of the p-value of a

significance test for a one-sample proportion

should recognize that the p-value is computed

by assuming that the probability model and
null hypothesis are true, i.e., by assuming that
the true population proportion is equal to the
particular value stated in the null hypothesis.

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Inference for Categorical Data: Proportions 6UNIT

TOPIC 6.6 SKILL

Concluding a Test for a Statistical
Population Proportion Argumentation

4.E
Justify a claim using
a decision based on
significance tests.

Required Course Content AVAILABLE RESOURCES
§§ Classroom Resource >
ENDURING UNDERSTANDING
◆◆ Inference
DAT-3
◆◆ Calculations
Significance testing allows us to make decisions about hypotheses within a Aren’t Enough!
particular context. The Importance of
Communication in
AP Statistics

LEARNING OBJECTIVE ESSENTIAL KNOWLEDGE

DAT-3.B DAT-3.B.1

Justify a claim about the The significance level, , is the predetermined
population based on the
results of a significance test probability of rejecting the null hypothesis
for a population proportion. given that it is true.
[Skill 4.E]
DAT-3.B.2

A formal decision explicitly compares the

p-value to the significance level, . If the
p-value ≤ , reject the null hypothesis. If
the p-value > , fail to reject the null hypothesis.

DAT-3.B.3

Rejecting the null hypothesis means there is
sufficient statistical evidence to support the
alternative hypothesis. Failing to reject the null
means there is insufficient statistical evidence
to support the alternative hypothesis.

DAT-3.B.4

The conclusion about the alternative
hypothesis must be stated in context.

DAT-3.B.5

A significance test can lead to rejecting or not
rejecting the null hypothesis, but can never
lead to concluding or proving that the null
hypothesis is true. Lack of statistical evidence
for the alternative hypothesis is not the same
as evidence for the null hypothesis.

continued on next page

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6UNIT Inference for Categorical Data: Proportions

LEARNING OBJECTIVE ESSENTIAL KNOWLEDGE

DAT-3.B DAT-3.B.6

Justify a claim about the Small p-values indicate that the observed
population based on the
results of a significance test value of the test statistic would be unusual
for a population proportion. if the null hypothesis and probability model
[Skill 4.E] were true, and so provide evidence for the

alternative. The lower the p-value, the more

convincing the statistical evidence for the
alternative hypothesis.

DAT-3.B.7

p-values that are not small indicate that the

observed value of the test statistic would not
be unusual if the null hypothesis and probability
model were true, so do not provide convincing
statistical evidence for the alternative
hypothesis nor do they provide evidence that
the null hypothesis is true.

DAT-3.B.8

A formal decision explicitly compares

the p-value to the significance . If the
p-value ≤ , then reject the null hypothesis,
H0 : p = p0. If the p-value > , then fail to reject

the null hypothesis.

DAT-3.B.9

The results of a significance test for a
population proportion can serve as the
statistical reasoning to support the answer to
a research question about the population that
was sampled.

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Inference for Categorical Data: Proportions 6UNIT

TOPIC 6.7 SKILLS

Potential Errors When Selecting Statistical
Performing Tests Methods

Required Course Content 1.B
Identify key and relevant
ENDURING UNDERSTANDING information to answer
a question or solve
UNC-5 a problem.

Probabilities of Type I and Type II errors influence inference. Using Probability
and Simulation
LEARNING OBJECTIVE ESSENTIAL KNOWLEDGE
3.A
UNC-5.A UNC-5.A.1 Determine relative
frequencies, proportions,
Identify Type I and Type II A Type I error occurs when the null hypothesis or probabilities using
errors. [Skill 1.B] is true and is rejected (false positive). simulation or calculations.

UNC-5.A.2 Statistical
Argumentation
A Type II error occurs when the null hypothesis
is false and is not rejected (false negative). 4.A
Make an appropriate
Table of Errors claim or draw an
appropriate conclusion.

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

Actual Population Value

Decision Reject H0 H0 true Ha true
Type I Error
Correct
Fail to Correct Decision
Reject H0 Decision
Type II Error

continued on next page

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6UNIT Inference for Categorical Data: Proportions

LEARNING OBJECTIVE ESSENTIAL KNOWLEDGE

UNC-5.B UNC-5.B.1

Calculate the probability of The significance level, , is the probability of
a Type I and Type II errors.
[Skill 3.A] making a Type I error, if the null hypothesis
is true.
UNC-5.C
UNC-5.B.2
Identify factors that affect
the probability of errors in The power of a test is the probability that a test
significance testing. will correctly reject a false null hypothesis.
[Skill 4.A]
UNC-5.B.3
UNC-5.D
The probability of making a Type II error
Interpret Type I and
Type II errors. [Skill 4.B] =1− power.

UNC-5.C.1

The probability of a Type II error decreases
when any of the following occurs, provided the
others do not change:
i. Sample size(s) increases.

ii. Significance level () of a test increases.

iii. Standard error decreases.
iv. True parameter value is farther from the null.

UNC-5.D.1

Whether a Type I or a Type II error is more
consequential depends upon the situation.

UNC-5.D.2

Since the significance level, , is the probability

of a Type I error, the consequences of a
Type I error influence decisions about a
significance level.

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Inference for Categorical Data: Proportions 6UNIT

TOPIC 6.8 SKILLS

Confidence Intervals Selecting Statistical
for the Difference of Methods
Two Proportions
1.D
Required Course Content Identify an appropriate
inference method for
ENDURING UNDERSTANDING confidence intervals.

UNC-4 Statistical
Argumentation
An interval of values should be used to estimate parameters, in order to account
for uncertainty. 4.C
Verify that inference
procedures apply in a
given situation.

Using Probability
and Simulation

3.D
Construct a confidence
interval, provided
conditions for inference
are met.

LEARNING OBJECTIVE ESSENTIAL KNOWLEDGE

UNC-4.I UNC-4.I.1

Identify an appropriate The appropriate confidence interval procedure
confidence interval for a two-sample comparison of proportions
procedure for a comparison for one categorical variable is a two-sample
of population proportions.
[Skill 1.D] z-interval for a difference between population

proportions.

UNC-4.J UNC-4.J.1

Verify the conditions In order to calculate confidence intervals to
for calculating estimate a difference between proportions,
confidence intervals for we must check for independence and that the
a difference between sampling distribution is approximately normal:
population proportions.
[Skill 4.C] a. To check for independence:

i. Data should be collected using two
independent, random samples or a
randomized experiment.

ii. When sampling without replacement,
n1 ≤ 10%N1 and n2 ≤ 1o0f%pˆ1N−2
b. To check that .
check that sampling distribution
pˆ2
is approximately normal (shape).

i. For categorical variables, check that

 n1 pˆ1 , n1(1 − pˆ1), n2 pˆ2 , and n2 1 pˆ2

are all greater than or equal to some
predetermined value, typically either 5
or 10.

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