AP Statistics 3.9 Sampling Distributions for the Difference Between Sample Proportions Study Notes - New Syllabus

3.9.B.1 Conditions for the Sampling Distribution of the Difference Between Two Sample Proportions

Before using the sampling distribution of the difference between two sample proportions, certain conditions must be satisfied.

When sampling without replacement, two important conditions must be verified to ensure that the observations are independent and that the sampling distribution is valid.

Condition 1: Randomization Condition

The data should be collected using two independent random samples.

This ensures that:

• Each sample is randomly selected from its population.
• The two samples do not influence each other.
• The observations within each sample are unbiased.

Requirement

Two independent random samples must be used.

Condition 2: 10% Condition

If sampling is performed without replacement, each sample must represent no more than 10% of its respective population.

This condition allows observations within each sample to be treated as approximately independent.

Requirement

\(\boxed{n_1\le0.10N_1 \quad\text{and}\quad n_2\le0.10N_2}\)

Where:

• \(n_1\) = Sample size from Population 1
• \(N_1\) = Size of Population 1
• \(n_2\) = Sample size from Population 2
• \(N_2\) = Size of Population 2

Randomization Condition

Two independent random samples.

10% Condition

\(n_1\le0.10N_1\)

\(n_2\le0.10N_2\)

Example 1: Conditions Satisfied

A researcher randomly selects:

• 120 students from a university with 5,000 students.
• 100 students from another university with 4,200 students.

Check the Conditions

Randomization

Both samples were selected randomly and independently.

✔ Condition satisfied.

10% Condition

\(120\le0.10(5000)=500\)

\(100\le0.10(4200)=420\)

✔ Both conditions are satisfied.

Example 2: 10% Condition Not Met

A sample of 300 people is selected from a town containing only 2,000 residents without replacement.

Check the 10% Condition

\(0.10(2000)=200\)

Since

\(300>200\)

the 10% condition is not satisfied.

The observations may no longer be approximately independent.

Summary Table

Important AP Exam Notes

• These conditions apply when sampling without replacement.
• The two samples must be independent random samples.
• Each sample size must be no more than 10% of its population.
• The 10% condition helps justify treating observations as approximately independent.
• Always verify these conditions before using inference procedures for two population proportions.

Common AP Exam Mistakes

3.9.B.2 Conditions for Experiments Involving Two Population Proportions

When data come from a randomized experiment, the conditions required for inference are different from those for random samples.

In a randomized experiment, the randomization condition is the primary requirement.

The 10% condition is not required because participants are randomly assigned to treatments rather than randomly sampled from a population.

Key Idea

If the data are collected through a randomized experiment, the treatments must be randomly assigned to the experimental units.

Random assignment helps create comparable treatment groups and supports the assumption that the groups are independent.

Randomization Condition

Experimental units must be randomly assigned to the treatment groups.

Why Is Random Assignment Important?

• Helps create treatment groups that are similar before the treatment is applied.
• Reduces the effects of confounding variables.
• Allows differences in sample proportions to be attributed to the treatments rather than pre-existing differences.
• Supports cause-and-effect conclusions when other experimental principles are also satisfied.

Random Sampling vs. Random Assignment

Example 1

A researcher wants to compare two teaching methods.

One hundred students volunteer for the study.

The researcher randomly assigns:

• 50 students to Method A.
• 50 students to Method B.

Check the Condition

• Students were randomly assigned to treatments.
• ✔ Randomization condition is satisfied.
• The 10% condition is not required because this is a randomized experiment.

Example 2

A pharmaceutical company randomly assigns patients to receive either:

• A new medication.
• A placebo.

Because treatment assignment is random, the randomization condition is satisfied.

The experiment may proceed without checking the 10% condition.

Summary Table

Important AP Exam Notes

• For randomized experiments, verify that treatments were randomly assigned.
• The 10% condition is unnecessary for randomized experiments.
• Random assignment helps create independent treatment groups.
• Random assignment supports cause-and-effect conclusions when the experiment is well designed.
• Do not confuse random sampling with random assignment.

Common AP Exam Mistakes

3.9.B.3 Large Counts Condition for the Sampling Distribution of the Difference Between Two Sample Proportions

To use a normal distribution to model the sampling distribution of the difference between two sample proportions, both samples must be large enough.

This is verified using the Large Counts Condition, which checks that the expected numbers of successes and failures in each sample are sufficiently large.

Key Idea

The sampling distribution of

\(\hat{p}_1-\hat{p}_2\)

can be approximated by a normal distribution if each sample has at least 10 expected successes and 10 expected failures.

Large Counts Condition

Both samples must satisfy all four conditions:

• \(\boxed{n_1p_1\ge10}\)
• \(\boxed{n_1(1-p_1)\ge10}\)
• \(\boxed{n_2p_2\ge10}\)
• \(\boxed{n_2(1-p_2)\ge10}\)

Meaning of Each Quantity

Why Is This Condition Important?

• It ensures that each sample contains enough expected successes and failures.
• This allows the sampling distribution of the difference between sample proportions to be well approximated by a normal distribution.
• If any of the four quantities is less than 10, the normal approximation may not be appropriate.

Example 1: Condition Satisfied

A study compares two populations with:

• \(n_1=200,\; p_1=0.40\)
• \(n_2=150,\; p_2=0.70\)

Check the Conditions

\(n_1p_1=200(0.40)=80\)

\(n_1(1-p_1)=200(0.60)=120\)

\(n_2p_2=150(0.70)=105\)

\(n_2(1-p_2)=150(0.30)=45\)

All four values are at least 10.

✔ The Large Counts Condition is satisfied.

Example 2: Condition Not Satisfied

Suppose

• \(n_1=20,\; p_1=0.15\)
• \(n_2=25,\; p_2=0.60\)

Check the Conditions

\(n_1p_1=20(0.15)=3\)

Since

\(3<10\)

the Large Counts Condition is not satisfied.

The sampling distribution should not be modeled using a normal distribution.

Summary Table

Important AP Exam Notes

• All four expected counts must be at least 10.
• The Large Counts Condition justifies using a normal approximation for the sampling distribution.
• If any one of the four values is less than 10, the condition is not satisfied.
• Always check this condition before performing inference for two population proportions.
• Keep the order of the populations consistent throughout the analysis.

Common AP Exam Mistakes