AP Statistics Unit 5 Sampling Distributions

Topic 5.5: Sampling Distributions for Sample Proportions

How Do Sample Proportions Behave?
Video Follow-Along Worksheet
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Learning Objectives: Essential Knowledge:

Key Vocabulary

Sample Proportion (p̂) The proportion of successes in a random sample: p̂ = (number of successes) / n
Population Proportion (p) The true proportion of successes in the entire population
Sampling Distribution of p̂ The distribution of all possible p̂ values from all samples of a given size from a population
Large Counts Condition np ≥ 10 and n(1−p) ≥ 10 — required for the sampling distribution to be approximately normal
10% Condition Sample size must be less than 10% of the population size for the standard deviation formula to apply

Part 1: Building a Sampling Distribution of p̂

[Video 1: 0:00 – 2:50]
Setup — Sampling From a High School: [0:39] In a high school with 2,000 students, 30% have a driver's license. We select a random sample of n = 50 students and calculate the sample proportion p̂ with a driver's license.
1. [1:05] In the population, % have a driver's license and % do not.
2. [1:21] The first sample of 50 students has 12 with a driver's license. So p̂ = 12/50 = .
3. [1:37] A second sample gives 19 out of 50: p̂ = . The fact that different samples give different values of p̂ is called sampling .
4. [1:53] A third sample gives p̂ = 16/50 = .
5. [2:10] After taking 500 random samples of size 50, the distribution of sample proportions is:
Shape: approximately
Center:
Spread: varies from about to about

Part 2: Mean and Standard Deviation of the Sampling Distribution

[Video 1: 2:50 – 4:40]

Formulas for the Sampling Distribution of p̂

μp̂ = p
σp̂ = √(p(1 − p) / n)
*Requires: sample size < 10% of the population size (or sampling with replacement)
6. [2:50] The mean of the sampling distribution of p̂ equals , the .
7. [3:20] The standard deviation formula requires that the sample size is less than % of the .
AP Exam Note: [3:48] These formulas are provided on the AP Statistics formula sheet! You can find them under "Parameters of Sampling Distribution" for the random variable p̂.
8. [4:04] For the driver's license example (n = 50, p = 0.30):
μp̂ =
σp̂ = √(0.3 × 0.7 / 50) =
10% condition: Is 50 < 10% of 2,000? (50 < 200)

Part 3: Shape — The Large Counts Condition

[Video 1: 4:40 – 6:09]
Large Counts Condition: [4:39] The sampling distribution of p̂ will be approximately normal when both:
np ≥ 10     AND     n(1 − p) ≥ 10
9. [5:03] Check the Large Counts condition for the driver's license example (n = 50, p = 0.3):
np = 50(0.3) = ≥ 10?
n(1 − p) = 50(0.7) = ≥ 10?
10. [5:03] Since both conditions are met, the sampling distribution of p̂ will be approximately .

Part 4: Interpreting the Parameters

[Video 2: 0:00 – 2:50]
11. [1:26] Interpret μp̂ = 0.30: For all random samples of size n = from this population, the sample proportions of students who have a driver's license will have a of 0.30.
12. [1:58] Interpret σp̂ = 0.065: For all random samples of size n = 50 from this population, the sample proportions of students who have a driver's license vary by about from the population proportion of 0.30.
Why This Matters: [2:38] Knowing how much a sample statistic typically varies from the truth is one of the most important reasons to study sampling distributions. This concept connects to (Units 6–9).

Part 5: Calculating and Interpreting Probabilities

[Video 2: 2:50 – 5:51]
Question: [2:52] Would it be unusual to get a sample proportion of p̂ = 0.12 or less in a random sample of size n = 50 from this population?
13. [3:18] Step 1 — Draw a picture: Sketch a normal curve centered at with standard deviation . Shade to the of 0.12.
14. [3:40] Step 2 — Calculate the z-score:
z = ( − ) / =
15. [4:02] Step 3 — Find the probability: Using Table A or technology,
P(p̂ ≤ 0.12) =
16. [4:14] Step 4 — Interpret: Getting a sample proportion of 0.12 or less happens in only about % of all possible samples of size 50 from this population. This is very !
Second Key Reason: [4:28] Knowing what values of a statistic count as unusual is one of the most important reasons we study sampling distributions. This concept connects to (Units 6–9).

Part 6: Key Takeaways

[Video 2: 4:44 – 5:51]
[4:47] Interpreting parameters:
  1. Always interpret in .
  2. Reference "all possible of this size from the population."
  3. For standard deviation, include "" when describing how much values vary.
Calculating & interpreting probabilities:
  1. Use procedures from Topic 5.2 with sampling distribution parameters.
  2. Interpretations should be in context and reference all possible samples.

Reflection Questions

[AI-Graded]
R1. In the driver's license example, the standard deviation of the sampling distribution is 0.065. Explain what this value tells us in context. Why is understanding this variability important for statistics? (Hint: Josh Tabor says this is "one of the most important reasons to study sampling distributions.")
R2. A polling company surveys n = 100 people from a large city where 40% support a new policy (p = 0.40). (a) Calculate the mean and standard deviation of the sampling distribution of p̂. (b) Verify that the Large Counts condition is met. (c) Would it be surprising to get a sample proportion of 0.52 or higher? Calculate the z-score and explain.

Exit Ticket

A factory produces light bulbs and historically 5% are defective (p = 0.05). A quality inspector takes a random sample of n = 200 bulbs.

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