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Unformatted text preview: , and Hunter’s Statistics for Experimenters. X 140 148 175 195 245 250 250 Pop (1000’s) 55.5 55.5 64.9 67.5 69 72 75.5 Regression and Correlation Page 12 Minitab Output
Regression Analysis: Population versus X DoStat Output The regression equation is
Population = 35.5 + 0.151 X Predictor
0.15073 S = 2.86360 SE Coef
0.02422 R-Sq = 88.6% T
0.002 R-Sq(adj) = 86.3% Analysis of Variance
0.002 Regression and Correlation Page 13 Correlation The linear regression model assumes the X’s are measured with negligible error. Think about the snake data here…the researcher measured length to predict weight! Why not the other way around? I mean, if we go out to collect snake measurements, I am not volunteering to get the length – I’m volunteering to hook the snake and
throw it in a bag to weigh it. But, if we tried to use the weight to predict length, the variability in weight due to eating, pregnancy, etc. could lead to bad predictions of length. For instance, a snake in our data set that just ate a mouse, would have a shorter length that what would be predicted for a snake that actually weighed little snake + food = big snake pounds. In other words, Y|X is the mean of Y given X. We use this type of model to make predictions of Y, based on our model for a given value of X. For the situation where we’d like to make statements about the joint relationship of X and Y, we’ll need for X and Y to both be random. When we’re interested in examining the joint relationship of two random variables, we are interested in their joint distribution (the joint distribution of two random variables is called a bivariate distribution). Definition The bivariate random sampling model views the pairs (Xi, Yi) as joint random variables, with population means, X, Y, population standard deviations, X, Y, and a correlation parameter, . In this model, measures the level of dependence between two random variables, X and Y. ‐1 ≤ ≤ 1 1 X & Y become more correlated 0 X & Y become uncorrelated We’ll measure the sample correlation coefficient, called r. r ∑n
i 1 1 xi ‐x yi ‐y xi ‐x 2 ∑n
i1 yi ‐y 2 Notice w...
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- Fall '09