11+Regression

11 Regression

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Unformatted text preview: ample 2. Refer to example 1. n 1 SSyy = 26 − 102 = 6, 5 2 yi = 26, i=1 SSE = 6 − 0.7 · 7 = 1.1, s2 = 1.1 = 0.3667, 5−2 s = 0.605. The test for slope 1 . In view of four assumptions about ε2 the sampling distribution of the ˆ least squares estimator β1 will be normal with mean β1 and standard deviation σ s σβ1 = √ ≈ sβ1 = √ . ˆ ˆ SSxx SSxx ˆ s ˆ is called the estimated standard error of the least squares slope β1 . β1 Usually σ is unknown and appropriate test statistic is a t-statistic t= ˆ β √1 . s/ SSxx Let we’d like to test H0 = {β1 = 0}. Alternative hypothesis is One-tailed test Ha = {β1 < 0} (or Ha = {β1 > 0}) and reject region is RR = {t < −tα } (or Ha = {t > tα }), where tα are based on (n − 2) degree of freedom and we can find it using Table VI (McClave and Benson). Two-tailed test Ha = {β1 = 0} and reject region is RR = {|t| > tα/2 }. 5 A. Zhensykbaev Regression Example 3. Refer to example 1. Apply two-tailed test with α = 0.05 for H0 = {β1 = 0}. √ ˆ S Sxx = 10 = 3.162 β1 = 0.7, s = 0.605, ˆ β 0.7 · 3.162 √1 = = 3.66, tα/2 = t0.025 = 3.182. 0.605 s/ SSxx Since t = 3.66 > 3.182 = t0.025 , we reject H0 . Hence, our data don’t contradict the linear dependence of y on x. t= A 100(1 − α)% confidence interval. Another way to make inferences about the slope β1 is a confidence interval: s ˆ β1 ± tα/2 · sβ1 , , sβ1 = ˆ ˆ SSxx where tα/2 is based on (n − 2) degree of freedom. In considered example 95% confidence interval is 0.605 ˆ β1 ± t0.025 · sβ1 = 0.7 ± 3.182 √ = 0.7 ± 0.61 ˆ 10 β1 ∈ (0.09, 1.31). Measures of strength of a relationship. A bivariate relationship describes a relationship between two variables x and y . A numerical measure of it is provided by the coefficient of correlation. Coefficient of correlation r is a measure of the strength of the linear relationship between x and y . r= SSxy . S Sxx SSyy r=√ 7 = 0.904. 10 · 6 For considered case 6 A. Zhensykbaev Regression Coefficient of determination r2 is another measure of the streng...
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This note was uploaded on 02/11/2014 for the course MATH 1390 taught by Professor Christopherstocker during the Spring '13 term at Marquette.

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