Sampling Distributions for Differences in Sample Proportions - AP Statistics Study Guide

Quick Summary

This guide covers the sampling distribution of the difference between two sample proportions (${\hat{p}}_1-{\hat{p}}_2$). You will learn how to describe the shape, center, and spread of this distribution and how to verify the necessary conditions—Random, 10% Condition, and Large Counts Condition—for both samples. Mastering this topic is essential for comparing two populations based on categorical data and forms the foundation for two-sample confidence intervals and significance tests for proportions.

Key Concepts

When we want to compare the proportions of successes in two different populations (e.g., the proportion of voters favoring a candidate in two different states), we take independent random samples from each. We then calculate the difference between the sample proportions, ${\hat{p}}_1-{\hat{p}}_2$. The sampling distribution of the difference in sample proportions describes the distribution of this statistic if we were to take every possible pair of samples and calculate the difference.

To properly use this model, we must check three conditions and then describe its shape, center, and spread.

[Image: A normal curve labeled "Sampling Distribution of p̂₁ - p̂₂". The center of the curve is labeled with the mean, μ = p₁ - p₂. The spread is indicated by the standard deviation, σ = √[ (p₁(1-p₁)/n₁) + (p₂(1-p₂)/n₂) ].]

1. Conditions for the Sampling Distribution of ${\hat{p}}_1-{\hat{p}}_2$

Before we can describe the distribution, we must verify three critical conditions:

• Random Condition: The data must come from two independent random samples or two groups in a randomized experiment. The "independent" part is crucial; the selection of one sample cannot influence the selection of the other.

• 10% Condition (for Independence within samples): When sampling without replacement, the sample size should be no more than 10% of the population size for each sample. This ensures that individual observations within each sample can be treated as independent.

  • $n_1\le (1/10)N_1$ and $n_2\le (1/10)N_2$

• Large Counts Condition (for Normality): The number of expected successes and failures must be at least 10 in each sample. This condition allows us to assume the sampling distribution of ${\hat{p}}_1-{\hat{p}}_2$ is approximately Normal.

  • $n_1{p}_1\ge 10$, $n_1(1-p_1)\ge 10$, $n_2{p}_2\ge 10$, and $n_2(1-p_2)\ge 10$

  • In practice, if the true population proportions $p_1$ and $p_2$ are unknown, we check this condition using the sample proportions: $n_1{\hat{p}}_1\ge 10$, $n_1(1-{\hat{p}}_1)\ge 10$, $n_2{\hat{p}}_2\ge 10$, and $n_2(1-{\hat{p}}_2)\ge 10$.

2. Describing the Sampling Distribution of ${\hat{p}}_1-{\hat{p}}_2$

Once the conditions are met, we can describe the distribution's characteristics:

• Shape: The sampling distribution of ${\hat{p}}_1-{\hat{p}}_2$ is approximately Normal. This is justified by the Large Counts Condition.

• Center: The mean of the sampling distribution is the true difference between the population proportions. This means ${\hat{p}}_1-{\hat{p}}_2$ is an unbiased estimator of $p_1-p_2$.

  • Mean Formula:${\mu }_{{\hat{p}}_1-{\hat{p}}_2}=p_1-p_2$

• Spread: The standard deviation of the sampling distribution measures the typical distance of the statistic ${\hat{p}}_1-{\hat{p}}_2$ from the true difference $p_1-p_2$. To calculate it, we add the variances of the individual sampling distributions.

  • Standard Deviation Formula:${\sigma }_{{\hat{p}}_1-{\hat{p}}_2}=\surd [(p_1(1-p_1)/n_1)+(p_2(1-p_2)/n_2)]$

  • This formula is only valid if the Random and 10% conditions are met for both samples.

Key Vocabulary

• Sampling Distribution for the Difference in Sample Proportions: The theoretical distribution of all possible values of the statistic ${\hat{p}}_1-{\hat{p}}_2$ that would be obtained from all possible pairs of independent random samples of size $n_1$ and $n_2$ from their respective populations.

• Independent Samples: Two samples are independent if the selection of individuals or objects for one sample does not affect or influence the selection of individuals or objects for the other sample.

• Unbiased Estimator: A statistic whose sampling distribution has a mean that is exactly equal to the population parameter it is intended to estimate. ${\hat{p}}_1-{\hat{p}}_2$ is an unbiased estimator of $p_1-p_2$.

• Large Counts Condition: The guideline ($np\ge 10$ and $n(1-p)\ge 10$ for both samples) used to check if the shape of a sampling distribution for a proportion or difference in proportions is approximately Normal.

• 10% Condition: The rule used when sampling without replacement to ensure observations within a sample are reasonably independent. It requires the sample size to be no more than 10% of the population size ($n\le 0.10N$).

Calculator Tech (TI-84)

No major calculator functions are required for describing the sampling distribution itself. The calculations for the mean and standard deviation are straightforward arithmetic. Later, in Unit 6, you will use $2-PropZInt$ and $2-PropZTest$ for inference, which builds upon the concepts learned here.

How to Show Work on the FRQ

On the AP exam, you will be asked to describe the sampling distribution of ${\hat{p}}_1-{\hat{p}}_2$ or to calculate a probability based on it. Your response must be a clear, written description that addresses shape, center, and spread, explicitly checking all necessary conditions.

Template for Describing the Sampling Distribution of ${\hat{p}}_1-{\hat{p}}_2$:

1. State the Distribution: Identify the sampling distribution of ${\hat{p}}_1-{\hat{p}}_2$, where ${\hat{p}}_1$ is the sample proportion of [context for group 1] and ${\hat{p}}_2$ is the sample proportion of [context for group 2].

2. Check Conditions:

   • Random: "The problem states we have two independent random samples: a random sample of $n_1$ [subjects from population 1] and a random sample of $n_2$ [subjects from population 2]."

   • 10% Condition: "Assuming the population of [population 1] is at least $10\ast n_1$ and the population of [population 2] is at least $10\ast n_2$, the 10% condition is met. This allows us to calculate the standard deviation."

   • Large Counts Condition: "We check the Large Counts Condition to ensure a normal approximation is appropriate:

     • $n_1{p}_1$ = [calculation] \ge 10

     • $n_1(1-p_1)$ = [calculation] \ge 10

     • $n_2{p}_2$ = [calculation] \ge 10

     • $n_2(1-p_2)$ = [calculation] \ge 10

     Since all expected counts are at least 10, the sampling distribution is approximately normal."

3. Describe Shape, Center, and Spread:

   • Shape: "The sampling distribution of ${\hat{p}}_1-{\hat{p}}_2$ is approximately Normal."

   • Center: "The mean of the sampling distribution is ${\mu }_{{\hat{p}}_1-{\hat{p}}_2}=p_1-p_2=[value]-[value]=[result]$."

   • Spread: "The standard deviation of the sampling distribution is ${\sigma }_{{\hat{p}}_1-{\hat{p}}_2}=\surd [(p_1(1-p_1)/n_1)+(p_2(1-p_2)/n_2)]=\surd [([calculation])+([calculation])]=[result]$."

Practice Problems

Problem 1:

A large university wants to compare the proportion of in-state and out-of-state students who are satisfied with the campus dining services. The true proportion of satisfied in-state students is $p_1=0.75$, and the true proportion of satisfied out-of-state students is $p_2=0.68$. The university plans to take a simple random sample of 200 in-state students and a separate simple random sample of 150 out-of-state students. Let ${\hat{p}}_1-{\hat{p}}_2$ be the difference in the sample proportions of satisfied students. Describe the sampling distribution of ${\hat{p}}_1-{\hat{p}}_2$.

Solution:

The sampling distribution of ${\hat{p}}_1-{\hat{p}}_2$, the difference in sample proportions of satisfied in-state and out-of-state students, can be described by its shape, center, and spread after checking the necessary conditions.

• Conditions: The problem states we have two independent simple random samples. We must assume the university has at least 10(200) = 2000 in-state students and at least 10(150) = 1500 out-of-state students to satisfy the 10% condition. For the Large Counts Condition: $n_1{p}_1=200(0.75)=150\ge 10$, $n_1(1-p_1)=200(0.25)=50\ge 10$, $n_2{p}_2=150(0.68)=102\ge 10$, and $n_2(1-p_2)=150(0.32)=48\ge 10$. All conditions are met.

• Shape, Center, and Spread: Because the Large Counts Condition is met, the sampling distribution of ${\hat{p}}_1-{\hat{p}}_2$ is approximately Normal. The mean of the distribution is ${\mu }_{{\hat{p}}_1-{\hat{p}}_2}=p_1-p_2=0.75-0.68=0.07$. The standard deviation is ${\sigma }_{{\hat{p}}_1-{\hat{p}}_2}=\surd [(0.75(0.25)/200)+(0.68(0.32)/150)]=\surd [0.0009375+0.0014507]\approx 0.0489$.

Problem 2:

A national polling agency reports that 44% of adults in the Eastern U.S. approve of the current president, while 52% of adults in the Western U.S. approve. Assume these are the true population proportions. The agency takes a simple random sample of 100 adults from the East and an independent simple random sample of 120 adults from the West. What is the probability that the sample proportion of approvers from the West is at least 10 percentage points higher than the sample proportion from the East?

Solution:

We want to find $P({\hat{p}}_W-{\hat{p}}_E\ge 0.10)$. First, we must describe the sampling distribution of the difference ${\hat{p}}_W-{\hat{p}}_E$.

• Conditions: We have two independent random samples. We assume there are at least 10(120) = 1200 adults in the West and 10(100) = 1000 adults in the East (10% condition). For Large Counts: $n_W\ast p_W=120(0.52)=62.4\ge 10$, $n_W(1-p_W)=120(0.48)=57.6\ge 10$, $n_E\ast p_E=100(0.44)=44\ge 10$, and $n_E(1-p_E)=100(0.56)=56\ge 10$. All conditions are met.

• Shape, Center, and Spread: The sampling distribution of ${\hat{p}}_W-{\hat{p}}_E$ is approximately Normal with a mean of $\mu =p_W-p_E=0.52-0.44=0.08$ and a standard deviation of $\sigma =\surd [(0.52(0.48)/120)+(0.44(0.56)/100)]=\surd [0.00208+0.002464]\approx 0.0674$.

• Probability Calculation: We standardize the value 0.10 to find the z-score: $z=(0.10-0.08)/0.0674\approx 0.297$. The probability is $P(Z\ge 0.297)$, which is approximately 0.383. There is about a 38.3% chance that the sample proportion of approvers from the West will be at least 10 percentage points higher than that from the East.

Common Mistakes to Avoid

• Adding Standard Deviations: You cannot add standard deviations. You must add the variances and then take the square root. Notice the formula $\surd [(Va{r}_1)+(Va{r}_2)]$. A common mistake is to calculate ${\sigma }_1$ and ${\sigma }_2$ separately and add them together.

• Forgetting to Check Conditions for Both Samples: The Large Counts Condition and the 10% Condition must be explicitly checked and verified for $n_1$ and $n_2$ independently. It is not sufficient to check them for just one group.

• Confusing $p$ and $\hat{p}$: When describing the sampling distribution, use the true population proportions ($p_1$ and $p_2$) for the mean and standard deviation formulas. Only use sample proportions (${\hat{p}}_1$ and ${\hat{p}}_2$) to check the Large Counts Condition if the true parameters are unknown.

• Saying the "Sample" is Normal: The data in the samples are categorical (e.g., yes/no, satisfied/not satisfied). The samples themselves are not Normal. It is the sampling distribution of the statistic${\hat{p}}_1-{\hat{p}}_2$ that is approximately Normal. Be precise with your language.

• Ignoring Sample Independence: The formulas for the center and spread of the difference rely on the two samples being independent of each other. If the samples are paired or dependent (e.g., a pre-test and post-test on the same group), these methods are not appropriate.