Confidence Intervals
- estimate ± z* · SE
- 95% CI: z* = 1.96
- Wider interval = more confident
- Larger n → narrower CI
Hypothesis Testing
- H₀: null hypothesis (=)
- Hₐ: alternative (<, >, ≠)
- p-value < α → reject H₀
- Type I error: reject true H₀
- Type II: fail to reject false H₀
Key Formulas
t = (x̄ − μ₀) / (s/√n) [one-sample t-test, df=n−1]
CI for μ: x̄ ± t* · (s/√n)
χ² = Σ(O − E)² / E [df = (r−1)(c−1)]
t = b / SEb [slope test, df=n−2]
Example
x̄=52, s=8, n=16, H₀: μ=50. Compute t.
t = (52−50)/(8/√16) = 2/2 = 1.0. With df=15, p-value ≈ 0.33 (two-tailed). Fail to reject H₀.