However its close approximate with k degrees of

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Unformatted text preview: tions 15 / 26 Two-sample t procedure Suppose we don’t know the populations’ standard deviations σ1 and σ2 . Approximate σ1 −→ s1 and σ2 −→ s2 like before: t= (¯1 − x2 ) − (µ1 − µ2 ) x ¯ 2 s1 s2 +2 n1 n2 Remember that this is an approximation. This t statistic does not exactly follow a t distribution. However, it’s close. Approximate with k degrees of freedom where k is calculated by software, OR k = min(n1 − 1, n2 − 1) K310 Statistical Techniques Inference for Distributions 15 / 26 Two-sample t significance test Let’s recap the procedure for a hypothesis test: H0 : µ1 − µ2 = 0 1 State the hypotheses: Ha : µ1 − µ2 = 0 2 Find the test statistic. Let’s break this into a couple steps: 1 2 3 Find the standard error: s(¯1 −x2 ) = x¯ Write the test statistic: t = 2 s1 s2 +2 n1 n2 x1 − x2 ¯ ¯ s(¯1 −x2 ) x¯ Critical region for, e.g., α = 0.05, df = 15: t ∗ = 2.131 −2.131 0 2.131 4 t Find the t statistic relative to the critical region K310 Statistical Techniques Inference for Distributions 16 / 26 Two-sample t significance test Let’s recap the procedure for a hypothesis test: H0 : µ1 − µ2 = 0 1 State the hypotheses: Ha : µ1 − µ2 = 0 2 Find the test statistic. Let’s break this into a couple steps: 1 2 3 Write the test statistic: t = 2 s1 s2 +2 n1 n2 Find the standard error: s(¯1 −x2 ) = x¯ x1 − x2 ¯ ¯ s(¯1 −x2 ) x¯ Critical region for, e.g., α = 0.05, df = 15: t ∗ = 2.131 t = 2.5 −2.131 0 2.131 4 t Find the t statistic relative to the critical region Conclusion: t is in the critical region. Reject H0 . K310 Statistical Techniques Inference for Distributions 16 / 26 Two-sample t significance test Let’s recap the procedure for a hypothesis test: H0 : µ1 − µ2 = 0 1 State the hypotheses: Ha : µ1 − µ2 = 0 2 Find the test statistic. Let’s break this into a couple steps: 1 2 3 Find the standard error: s(¯1 −x2 ) = x¯ Write the test statistic: t = 2 s1 s2 +2 n1 n2 x1 − x2 ¯ ¯ s(¯1 −x2 ) x¯ Critical region for, e.g., α = 0.05, df = 15: t ∗ = 2.131 t = 1.7 −2.131 0 2.131 4 t Find the t statistic relative to the critical region Conclusion: t is NOT in the critical region. Accept H0 . K310 Statistical Techniques Inference for Dis...
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