Short on time? This page is the recap. For explanations and worked examples, use the full study guide.
Recap
- A statistic (x̄, p̂) varies from sample to sample: sampling variability.
- The sampling distribution of a statistic is its distribution over all possible samples of the same size from the same population.
- For a small population you can list every sample. For a large one, simulate.
- A randomization distribution comes from an experiment: shuffle the observed responses between the treatment groups many times and recompute the statistic.
- Central limit theorem: the sampling distribution of a sample mean is approximately normal, and the approximation improves as n increases, even for a skewed population.
Key relationships
| Idea | What to remember |
|---|---|
| Population distribution | one value = one individual in the population |
| One sample’s distribution | one value = one individual in the sample; looks like the population |
| Sampling distribution | one value = the statistic from one whole sample |
| Simulating a sampling distribution | assume the parameter → random sample of size n → record statistic → repeat many times |
| Randomization distribution | keep the responses and group sizes → randomly reassign labels → record statistic → repeat |
| Larger n | centre stays near the parameter; variability shrinks; shape of x̄ gets closer to normal |
| More repetitions | a clearer picture of the same sampling distribution; spread does not shrink |
Assumptions and conventions
- A simulation assumes a value for the parameter (or a model for the population). Say what you assumed.
- Judge “unusual” by the proportion of simulated values as extreme as or more extreme than the observed one.
- Simulation results vary a little each time you run them.
Mistakes to avoid
- Confusing the sample’s data with the sampling distribution.
- Saying the CLT makes the population or the sample normal.
- Mixing up n with the number of repetitions.
- Claiming an unusual result proves a claim false. It is evidence against it.
- Changing group sizes or responses in a randomization.
- Describing a distribution without context or without all three features.
Quick self-check
- A population has 6 individuals. How many different samples of size 2 (without replacement) are there? (15)
- Simulated sample means for n = 5 have standard deviation 2.0. Would n = 50 give a larger or smaller standard deviation? (Smaller: larger samples vary less.)
- In 400 reallocations, 12 differences were as large as the observed difference or larger. Is the observed difference unusual if the treatment has no effect? (Yes: 12 ÷ 400 = 3% of reallocations.)
Next: practice questions.