# Lec21 - Two Sample Mean Tests Summary Sample Mean Tests...

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IOE/Stat 265, Fall 2009 Lecture #21: Hypothesis Tests for Two Means 9-1 Differences in Means w/Variances Known / 9-2 Differences in Means w/Variances Unknown 9-3 Paired t -Test 9-4 Differences Between Proportions 9-5 Ratios of Variances from Normal Distributions 1 wo Sample Mean Tests Summary Two Sample Mean Tests Summary ests of Two Population Means Case USE Test Statistic Formula Tests of Two Population Means Δ 1 known σ ’s , and normal populations z o 0 22 12 xy z nn σ −− Δ = + −− Δ 2 nknown ’ ,andnorma l t o 0 2 t Δ = unknown σ ’s , and normal populations (equal variances) 0 2 11 p s = ⎛⎞ + ⎜⎟ ⎝⎠ unknown σ s , and normal populations (unequal variances) 3 t o ss + 4t Paired t-test 0 D T −Δ = 2 o 0 sn ase 1: Two ample Z Test Case 1: Two-Sample Z Test ± Z Tests and Confidence Intervals for a Difference between the means of two ifferent population distributions: - different population distributions: μ 1 μ 2 ± Assumptions: ± X 1i ’s are a random sample from a population with mean μ 1 and variance σ 1 2 ± X 2i ’s are a random sample from a population with mean μ 2 and variance σ 2 2 ± The X 1i ’s and X 2i ’s are independent from one another ± The variances of both populations are known 3 ± Since both populations are normal, then sample means are also normal and we can btain the standard normal variable: obtain the standard normal variable: () ( ) ( ) XX μ μμ = 2 Z σσ == + 1 Number of samples in first population umber of samples in second population n = = 4 2 Number of samples in second population

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ase 1: Z test Case 1: Z test we set the difference between the ± If we set the difference between the population means as: ull Hypothesis: 2 1 0 μ = Δ Null Hypothesis: est statistic: 01 2 0 : H μμ 120 22 xx z −− Δ = Alternative Hypothesis: 12 nn σσ + 11 2 0 1 2 0 : : Hz α −> Δ −< Δ 5 2 0 2 2 : o r −≠ Δ Example 1: Drying Time nown Variances) (Known Variances) he drying times for two formulations of paint are tested: ± The drying times for two formulations of paint are tested: standard vs. express. It is known that the population standard deviation of drying time is 8 minutes for both formulations. ± Standard Formulation: 10 samples, xpress Formulation: 10 samples, 112 min. x = 1 121 min. = ± Express Formulation: 10 samples, ± What can you say about the differences between the formulations? (use α =0.05) 2 6 tep y tep Approach Step-by-Step Approach ) entify the parameter of interest 1) Identify the parameter of interest.
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Lec21 - Two Sample Mean Tests Summary Sample Mean Tests...

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