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# Stat_Chap10 - Inference for two samples Chapter 10 Not...

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Inference for two samples Chapter 10 Not covered: 10-5 1

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Comparing two populations Does a diet and exercise program reduce blood cholesterol level? Compare means of blood cholesterol level before/after Does Aspirin reduce heart attack risk? Compare heart attack rates for those who take/do not take Aspirin 2
Tools to be developed Difference of two population means/proportions Hypothesis testing Test statistics and sampling distributions Rejection criteria Power and sample size determination Confidence intervals Connection to hypothesis testing 3

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Road octane numbers Two different formulations of a motor fuel are tested to study their road octane numbers. The standard deviations are known to be σ 1 = 2 =1.2 for both formulations from historical data. Two random samples of size n 1 =15 and n 2 =20 are tested • Compare mean octane numbers # 1 # 2 87.45 91.83 88.38 93.16 88.27 92.12 90.55 94.17 90.15 89.89 87.98 92.70 89.94 93.64 87.42 93.26 87.84 93.68 89.68 94.02 91.13 93.61 87.88 94.42 87.69 91.45 89.04 91.78 89.33 92.07 92.91 91.87 92.75 90.00 91.48 4
Visualize data using box plot 5

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Hypothesis testing Is there evidence that mean RON of #1 is smaller than that of #2 by at least 3 H 0 : μ 1 - 2 = 3= Δ 0 H 1 : 1 2 < 3 A plausible statistic to consider: Difference b/w sample means 88.849 - 92.541= - 3.692 6 2 1 X X
Sampling distribution Sampling distribution if are normal and independent Verify normality (sample sizes not large enough for CLT): normal plot + 2 2 2 1 2 1 2 1 2 1 , ~ n n N X X σσ μμ 7 12 and XX

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Check normality 8
Test statistic Under H 0 : μ 1 - 2 = Δ 0 • Strong evidence against H 0 significantly different from Δ 0 Z 0 significantly different from 0 ) 1 , 0 ( ~ 2 2 2 1 2 1 0 2 1 0 N n n X X Z σσ + Δ = 9 + Δ 2 2 2 1 2 1 0 2 1 , ~ n n N X X 2 1 X X

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Two sample z-test Null hypothesis H 0 : μ 1 −μ 2 = Δ 0 H 1 : 1 2 ≠Δ 0 , reject H 0 if |z 0 |>z α /2 H 1 : 1 2 > 0 , reject H 0 if z 0 >z H 1 : 1 2 < 0 , reject H 0 if z 0 < - z Type I error probability = α = significance level p-value: same way as in one sample z-test 10