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b.lect26 - Outline Introduction Comparing Two Means...

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Outline Introduction Comparing Two Means Comparing Two Proportions Power and Sample Size Calculation – NOT COVERED Lecture 26 Chapter 10: Comparing Two Populations Michael Akritas Michael Akritas Lecture 26 Chapter 10: Comparing Two Populations
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Outline Introduction Comparing Two Means Comparing Two Proportions Power and Sample Size Calculation – NOT COVERED Introduction Comparing Two Means T CIs for μ 1 - μ 2 T Tests for μ 1 - μ 2 Comparing Two Proportions Z CIs for p 1 - p 2 Z Tests for p 1 - p 2 Power and Sample Size Calculation – NOT COVERED Comparison of Means Comparison of Proportions Michael Akritas Lecture 26 Chapter 10: Comparing Two Populations
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Outline Introduction Comparing Two Means Comparing Two Proportions Power and Sample Size Calculation – NOT COVERED I Here we will use CIs and hypothesis testing for comparing two population means, and two population proportions. I The mean and variance of population i , i = 1 , 2, are denoted by μ i , σ 2 i . If the two populations are Bernoulli, then μ i = p i , and σ 2 i = p i (1 - p i ). I The comparison will be based on a simple r.s. from each of the two populations: X 11 , . . . , X 1 n 1 from population 1, and X 21 , . . . , X 2 n 2 from population 2. I The sample mean and sample variances are denoted by X i = 1 n i n i X j =1 X in i , S 2 i = 1 n i - 1 n i X j =1 ( X ij - X i ) 2 , i = 1 , 2 . I In the Bernoulli case only b p i (or T i = n i b p i ) is typically given. Michael Akritas Lecture 26 Chapter 10: Comparing Two Populations
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Outline Introduction Comparing Two Means Comparing Two Proportions Power and Sample Size Calculation – NOT COVERED The Two Approaches I One approach to the comparison is based on the contrast X 1 - X 2 or b p 1 - b p 2 . I An alternative approach for comparing means uses the ranks of the observations. I With the exception of the section on paired data, we will make the further assumption that the two samples are independent. I Before computing the T CIs and tests for comparing two means it is a good idea to construct boxplots. This is similar to making an x-y scatterplot before fitting the simple linear regression model. Michael Akritas Lecture 26 Chapter 10: Comparing Two Populations
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Outline Introduction Comparing Two Means Comparing Two Proportions Power and Sample Size Calculation – NOT COVERED Example (Comparative Boxplot) To compare the strengths of cold-rolled steel and two-sided galvanized steel, n 1 = 32 strength measurements of cold-rolled and n 2 = 35 measurements of two-sided galvanized steel are made. The data are given in http://www.stat.psu.edu/ ~ mga/401/ Data/SteelStrengthData.txt . Construct comparative boxplots. Solution. The R commands for doing this are: library(psych); df=read.clipboard() # Reads data in data frame df y=df$Value; x=df$Sample # Measurements in y, sample index in x boxplot(y x) [or boxplot(y x, col=”grey”)] boxplot(y x,notch=TRUE) # Gives approximate CIs for medians.
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