Two-Sample Hypothesis Testing in Biostatistics: T-Test and Confidence Intervals, Study notes of Environmental Science

Download Two-Sample Hypothesis Testing in Biostatistics: T-Test and Confidence Intervals and more Study notes Environmental Science in PDF only on Docsity! ESCI 340: Biostatistical Analysis Two-Sample Hypothesis Testing l_hyp2.pdf (continued) McLaughlin 1 Test for difference between two Means (2-sample t-test) 1.1 Hypotheses: H0: µ1 = µ2, alternatively: H0: µ1 − µ2 = 0, HA: µ1 ≠ µ2 HA: µ1 − µ2 ≠ 0 1.2 Assumptions 1.2.1 both samples from normal populations 1.2.2 equal population variances 1.3 t X X sX X = − − 1 2 1 2 • sX X1 2− SE of difference betw/ means • t depends on ν, degrees of freedom • ν = ν1 + ν2 or ν = n1 + n2 − 2 • reject H0 if |tcalc| ≥ tα(2),ν (tα from Zar table B3) 1.4 Calculating sX X1 2− • var (difference between independent variables) = sum of individual variances, σ σ σ X X n n1 2 2 1 2 1 2 2 2 − = + → assume equal variances σ σ σ X X n n1 2 2 2 1 2 2 − = + • estimate σ2; use both s1 2 & s2 2 : s SS SS p 2 1 2 1 2 = + +ν ν • so: s s n s nX X p p 1 2 2 2 1 2 2 − = + and s s n s nX X p p 1 2 2 1 2 2 − = + • finally, t X X s n s n p p = − + 1 2 2 1 2 2 1.5 One-Tailed test Hypotheses: H0: µ1 ≥ µ2, HA: µ1 < µ2 ; if t t≤ − α ν( ),1 then reject H0 H0: µ1 < µ2, HA: µ1 > µ2 ; if t t≥ α ν( ),1 then reject H0 1.6 Violations of Two-Sample t-test Assumptions 1.6.1 normal distribution: t-test is very robust (one-tailed test sensitive to skew) 1.6.2 equal variances: if unequal, greater chance of type I error (greater than α) correction: Welch’s approximate t t X X s n s n '= − + 1 2 1 2 1 2 2 2 and d.f. ν '= + ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ − + ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ − s n s n s n n s n n 1 2 1 2 2 2 2 1 2 1 2 1 2 2 2 2 21 1 − often, ν’ not integer; use next smallest integer (e.g., if ν’ = 8.75, use ν = 8)