When a random sample of size n is taken from a population N(μ, σ²), the sample mean X̄ follows
X̄ 〜 N(μ, σ²/n).
In other words:
・E(X̄) = μ
・V(X̄) = σ²/n, standard deviation = σ/√n
Example ① If the population variance is σ²=400 and the sample size is n=25, find the standard deviation of X̄.
Solution:σ/√n = 20/5 = 4
Solution:σ/√n = 20/5 = 4
Key Point
- The distribution of the sample mean becomes narrower as n increases
- Central limit theorem: even if the population is not normal, the sample mean approaches a normal distribution when n is large
Practice Problems
- Find the standard deviation of X̄ when σ=10 and n=100.
Solutions and Explanations
- Solution:1