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Unformatted text preview: Notation: Y|X Defiinition: A conditional standard deviation is the standard deviation of a variable conditional on another variable. Notation: Y|X The linear regression model of Y on X assumes Y = Y|X + where the conditional mean is linear with Y|X = βO + β1X and = 0 and = Y|X We use ______ to estimate βO, ______ to estimate β1, and __________ to estimate Y|X. Then, we can estimate (or predict) Y|X=x at any X so that μY|X x = bO + b1x Assuming the linear model is appropriate here, find estimates of the mean and standard deviation of female Vipera bertis weight at a length of 65 cm. Should we estimate female Vipera bertis weight at a length of 75 cm? Why or why not? Regression and Correlation Page 5 Inference on β1 Normal Error Model In our discussion on the linear statistical model, we stated that the linear regression model of Y on X assumes a linear conditional mean, with the errors having mean 0 and standard deviation Y|X. To make inference on β1, we need to update the conditions on this model to include a normal distribution on the errors.
Y = Y|X + Y|X = βO + β1X ~ N(0, σ2 X ) Y| Assumptions to check εi must be independent εi must be normally distributed εi must have equal variance εi must have mean zero Regression and Correlation Page 6 How to Check Assumptions εi independent εi normally distributed εi must have equal variance εi must have mean zero Looking at Residuals vs. Predicted (or X) Plot Regression and Correlation Page 7 Check the assumptions (that can be checked) for the female Vipera bertis regression using the plots below. Regression and Correlation Page 8 Confidence Interval for β1 Under the normal error model, b1 is unbiased for β1 with SE b1 sY|X
i 1 xi ‐ x 2 This confidence interval uses a t critical point: t/2,df=n‐2 Then the CI is b1 ± tα/2,df=n‐2SE(b1) These intervals will be computed for you this semester. Interpret the 95% confidence interval for β1 in the female Vipera bertis regression. (4.938, 9.446) Regression and Correlation Page 9 Hypothesis Test for β1 Similar to the development of the confidence interval for β1, we can use the t distribution to conduct a hypothesis test for β1 with ts b1 SE b1 Under HO, ts ~ tdf=n‐2 We’ll test HO: _________________________________________________________________ HA: ____________...
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This note was uploaded on 02/27/2013 for the course STAT 205 taught by Professor Hendrix during the Fall '09 term at South Carolina.
- Fall '09