Key formulas
From the Sampling & CLT formula sheet
- Standard Error of xbar:
SE_xbar = sigma / sqrt(n)— Standard deviation of the sampling distribution of the sample mean.
Module 4 · Sampling & CLT · Topic 5
CLT Applications
Using the CLT. The CLT allows us to compute probabilities about sample means using the normal distribution, even when the population itself is not normal.
Proportions and the CLT. The CLT also applies to sample proportions. If np ≥ 10 and n(1 − p) ≥ 10, then p̂ is approximately normal with mean p and SE = √(p(1 − p)/n).
Worked example
Probability About x̄
Population: μ = 500, σ = 100. Sample of n = 25. P(x̄ > 520) = P(Z > (520 − 500)/(100/√25)) = P(Z > 1.0) ≈ 0.1587.
Proportion Example
60% of voters favor a policy (p = 0.6). In a sample of 200, SE = √(0.6 × 0.4 / 200) = √(0.0012) ≈ 0.0346. P(p̂ > 0.65) = P(Z > (0.65 − 0.60)/0.0346) ≈ P(Z > 1.45) ≈ 0.0735.
Related glossary terms
- Parameter: A number that describes an entire population. Example: The true mean height of all students at a university.
- Statistic: A number computed from a sample. Example: The mean height of 80 sampled students.
- Standard Error: The standard deviation of a statistic across repeated samples. Example: The standard error of xbar is sigma divided by sqrt(n).
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