AP Statistics Unit 5 Sampling Distributions

Topic 5.6: Sampling Distributions for Differences in Sample Proportions

Comparing Two Populations — Parameters, Shape, & Probability
Video Follow-Along Worksheet
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Learning Objectives: Essential Knowledge:

Key Vocabulary

Difference in Sample Proportions The statistic p̂1 − p̂2, computed from independent random samples drawn from two different populations
Sampling Distribution The distribution of a statistic (here, the difference in proportions) across all possible pairs of random samples of given sizes
Large Counts Condition For approximate normality, all four products n1p1, n1(1−p1), n2p2, and n2(1−p2) must be ≥ 10
10% Condition Each sample size must be less than 10% of its population size for the standard deviation formula to be valid
Unbiased Estimator A statistic whose sampling distribution is centered at the true parameter value — on average, it hits the target
z-score The number of standard deviations a value lies from the mean: z = (value − mean) / SD. Used to calculate probabilities from the normal distribution

Part 1: Building a Sampling Distribution for p̂A − p̂B

[0:00 – 3:57]
Setup: [0:55] High School A: 30% have a driver's license (pA = 0.30). High School B: 22% have a driver's license (pB = 0.22). We take random samples of n = 50 from each school.
1. [1:36] Each dot plotted on the sampling distribution represents a calculated from one pair of random samples — not an individual student's response.
2. [2:15] The second pair of samples produced p̂A − p̂B = −0.02. A negative difference is possible because the sample from School happened to have a higher proportion than School A in that particular pair of samples.
3. [3:16] After simulating 500 pairs of samples, describe the sampling distribution using shape, center, and spread:
Shape: approximately
Center: approximately
Spread: ranges from about −0.15 to about
4. [3:16] The center of the simulated distribution (0.08) matches the true difference in population proportions: 0.30 − 0.22 = 0.08. This suggests that p̂1 − p̂2 is an estimator of p1 − p2.

Part 2: Mean & Standard Deviation of the Sampling Distribution

[3:57 – 6:37]

Formulas (on your AP formula sheet)

Mean: μp̂1−p̂2 = p1 − p2
SD: σp̂1−p̂2 = √[ p1(1−p1)/n1 + p2(1−p2)/n2 ]
5. [4:41] The mean formula tells us: the sampling distribution is centered at the . In other words, the statistic targets the parameter on average.
6. [4:41] Notice the standard deviation formula adds the two variance terms (one from each population). We can add variances because the two samples are of each other.
7. [5:08] Two conditions must be met for the SD formula to be valid:
(a) Both sample sizes are less than of their population sizes.
(b) The two samples are independent of each other.
8. [5:25] Calculate for the example (pA = 0.30, pB = 0.22, nA = nB = 50):
Mean = 0.30 − 0.22 = 0.08
SD = √[ 0.30(0.70)/50 + 0.22(0.78)/50 ] =

Part 3: Shape — When Is the Distribution Approximately Normal?

[7:04 – 9:48]
Large Counts Condition (for two proportions): [7:35]
The sampling distribution of p̂1 − p̂2 is approximately normal when all four of these are ≥ 10:
   n1p1 ≥ 10     n1(1 − p1) ≥ 10     n2p2 ≥ 10     n2(1 − p2) ≥ 10
9. [7:35] For a single proportion (Topic 5.5), we checked two products: np ≥ 10 and n(1 − p) ≥ 10. For the difference in two proportions, we must check products because we need both individual sampling distributions to be approximately normal.
10. [8:06] Check for the example:
Population n × p n × (1 − p) ≥ 10?
School A (p = 0.30, n = 50) Yes
School B (p = 0.22, n = 50) Yes
Since all four values are ≥ 10, the sampling distribution of p̂A − p̂B is approximately .

Part 4: Interpreting Parameters & Calculating Probabilities

[Video 2: 0:00 – 7:42]
From Video 1: [1:34] The sampling distribution of p̂A − p̂B is approximately normal with mean = 0.08 and SD = 0.087.
11. [2:05] Interpreting the mean: A proper interpretation must reference " random samples" of these sizes from these populations — not just one pair of samples.
12. [2:24] The order of subtraction matters. If we had subtracted B − A instead of A − B, the mean would be instead of 0.08. Always be clear about which direction you are subtracting.
13. [3:08] Interpreting the SD: The differences in sample proportions vary by about 0.087 from the true difference of 0.08. (Use the word "typically" or "on average" when interpreting SD.)
14. [3:51] Knowing how much a statistic typically varies from the truth is one of the most important reasons to study . This idea connects directly to confidence intervals in later units.
Probability Question: [4:09] Would it be unusual to get a greater proportion of students with a license in the School B sample, even though School A has the higher population proportion?
15. [4:51] For School B's sample to have a higher proportion, the difference p̂A − p̂B must be . So we need P(p̂A − p̂B < 0).
16. [5:38] Calculate the z-score: z = (0 − 0.08) / 0.087 = . This means a difference of 0 is less than one standard deviation below the mean — not very far out.
17. [5:55] Using Table A or technology: P(p̂A − p̂B < 0) ≈
18. [6:11] Since about 18% of all sample pairs would produce School B with the higher proportion, this result is unusual. It could reasonably happen by chance.
Key Takeaway: [6:25] Determining whether a value of a statistic is unusual is one of the most important reasons we study sampling distributions. This connects directly to significance tests in later units.

Reflection Questions

[AI-Graded]
R1. The video emphasizes that interpreting the standard deviation of the sampling distribution is "one of the most important reasons to study sampling distributions." Explain what the SD tells us in practical terms. Why is it more useful to know how much a statistic typically varies than to know just the mean of the sampling distribution?
R2. In the probability calculation, we found P(p̂A − p̂B < 0) ≈ 0.18. Explain in your own words what this probability means in context. If the probability had been 0.001 instead, how would your interpretation change? Connect this to the idea of whether a result is "unusual."

Exit Ticket

A polling company surveys voters in two cities about support for a ballot measure. In City 1, the true proportion in favor is p1 = 0.58 and they sample n1 = 100 voters. In City 2, the true proportion is p2 = 0.45 and they sample n2 = 80 voters.

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