AP Statistics Unit 5 Sampling Distributions

Topic 5.3: The Central Limit Theorem

Sampling Distributions via Simulation & Randomization Distributions
Video Follow-Along Worksheet
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Learning Objective: Essential Knowledge:

Key Vocabulary

Sampling Distribution The distribution of values for a statistic from all possible samples of a given size from a population
Central Limit Theorem When sample size is sufficiently large, the sampling distribution of the mean will be approximately normal, regardless of population shape
Randomization Distribution A collection of statistics generated by simulation, reassigning response values to treatment groups to model chance alone
Simulation Using repeated random sampling to approximate a sampling distribution
Sample Mean (x) The average of values in a sample; varies from sample to sample

Video 1: Central Limit Theorem & Simulation

[Video: ~6 min]
Sampling Distribution Definition: [0:24] A sampling distribution of a statistic is the distribution of values for the statistic for all possible samples of the same size from a given population. For most populations, getting all possible samples is nearly impossible, but we can simulate a sampling distribution.

Rice University Applet Simulation

[1:32] The applet takes 10,000 random samples, finds the mean of each, and creates a sampling distribution. Key requirement: individual observations must be independent.
1. [2:27] Normal Population: With a larger sample size (n = 25), the sampling distribution has a standard deviation than when n = 2.
2. [2:53] Skewed Population, Small Sample (n = 2): If the population is skewed right, the sampling distribution is also skewed right — it is approximately normal.
3. [3:20] Skewed Population, Larger Sample (n = 25): With a larger sample size, even from a skewed population, the sampling distribution becomes approximately .
4. [4:40] The "Amazing" Result: Even with an extreme U-shaped population, when sample size is large enough, the sampling distribution is approximately — no matter what the population looks like!
Central Limit Theorem (CLT): [5:17] When the sample size is sufficiently , the sampling distribution of the mean will be approximately normal, regardless of the population's shape.

Video 2: Randomization Distributions for Experiments

[Video: ~5.5 min]
Melatonin Experiment: [0:18] In a double-blind experiment, 16 nursing mothers were randomly assigned to take either Melatonin or Placebo. Response variable: total hours each newborn slept during the night.
6. [0:48] Sample Statistics: Melatonin mean = 11.07 hours, Placebo mean = 9.38 hours.
Difference in sample means = hours (Pretty significant when you're sleep-deprived!)
7. [1:09] To investigate whether the difference could reasonably be due to , we simulate a randomization distribution.

Randomization Distribution

A collection of statistics generated by repeatedly reassigning the response values to treatment groups, with total disregard for where they originally came from.
8. [1:28] Simulation Process: Take all 16 values, randomly assign 8 to each group, calculate means, find the difference. Repeat 1,000 times to build a distribution of differences that could occur by alone.
9. [3:22] The actual experiment's difference of 1.69 hours is located in the tail of the randomization distribution.
10. [4:09] Probability Calculation: Only out of 1,000 simulations produced a difference ≥ 1.69 hours.
Probability =
11. [4:50] Conclusion: Since 0.012 is a probability, the observed difference is unlikely to have happened by chance. This provides evidence that melatonin increases newborn sleep.
Key Takeaway: [5:07] A randomization distribution assesses the likelihood of an observed outcome happening by chance alone. Small probability = evidence the treatment works.

Reflection Questions

[AI-Graded]
R1. Explain in your own words what the Central Limit Theorem says and why it is "amazing" (as the video describes). Use examples from the simulations in Video 1 to support your explanation. Why does sample size matter?
R2. In the melatonin experiment (Video 2), the probability of observing a difference of at least 1.69 hours by chance alone was 0.012. Explain what this probability means in context, and why it provides evidence that melatonin is effective. What would a larger probability (like 0.30) have suggested instead?

Exit Ticket

A researcher wants to test whether a new fertilizer increases plant growth. She randomly assigns 20 plants to receive either the new fertilizer (10 plants) or no fertilizer (10 plants). After 4 weeks, she measures the height increase for each plant.

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