Independent Two-sample z-test: Hypothesis Testing for Difference of Means, Study notes of Experimental Design

Download Independent Two-sample z-test: Hypothesis Testing for Difference of Means and more Study notes Experimental Design in PDF only on Docsity! Independent Two-sample z-test 1. Assumptions • Experimental Design: The sample forms two independent treatment groups. • Null Hypothesis: The population means of the two treatment groups are not significantly different from each other. • Population Distribution: Arbitrary distribution within each treat- ment group. • Sample Size: Size of each treatment group is equal to or greater than 30. 2. Inputs for independent two-sample z-test • Sample sizes of treatment groups: n1 and n2 • Sample means of treatment groups: x̄1 and x̄2 • Standard deviations of treatment groups: s1 and s2 • Standard errors of treatment group means: SE1 = s1√ n1 and SE2 = s2√ n2 • Standard error of the differences: sdiff = √ SE2 1 + SE2 2 • Null hypothesis: µ1 = µ2 • The level of the test: α 3. The Five Steps for Performing the Test of Hypothesis 1. State null and alternative hypotheses: H0 : µ1 = µ2, H1 : µ1 6= µ2 1 2. Compute test statistic: z = (x̄2 − x̄1)− (µ2 − µ1) SEdiff , assuming the null hypothesis. 3. Compute 100(1− α)% confidence interval I for z. 4. If z ∈ I, accept H0; if z /∈ I, reject H1 and accept H0. 5. Compute p-value. 4. Discussion Since we are assuming that n ≥ 30, the sample standard deviations s1 and s1 are close approximations to the population standard deviations σ1 and σ2, so we will assume that the population standard deviations are known and equal to the respective sample standard deviations. Furthermore E(x̄2 − x̄1) = E(x̄2)− E(x̄1) = µ2 − µ1. Also, if the two treatment groups are independent, Var(x̄2 − x̄1) = 12 · Var(x̄2) + (−1)2 Var(x̄1) = σ2 2 n1 + σ2 1 n2 , the standard deviation of x̄2 − x̄1 is SEdiff = √ Var(x̄2 − x̄1) = √ σ2 2 n1 + σ2 1 n2 . Finally, because the expected value and variance of z = (x̄2 − x̄1)− (µ2 − µ1) SE2 diff are µ2 − µ1 and SE2 diff, respectively, E(z) = 0 and σz = 1. By the central limit theorem, z ∼ N(0, 1), so we can use the standard normal table to find confidence intervals and p-values for z. 2