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Unformatted text preview: STAT 350 Lecture 23 Chapter 11 Inferential Methods in Regression and Correlation linear regression • Simple Linear Regression: – Fit a line in the data when you see a linear trend – Minimizing the errors using LS method – Get estimates of slope and intercept accordingly linear regression • Sample Least Square Regression Line: – Minimizes – Random residuals • Residuals: • What is the sum of residuals? • Population Least Square Regression Line – Linear Regression model for population – Statistical Inference based on random samples SS Re sid SSE y i ˆ y i 2 e i y i ˆ y i Simple Linear Regression Model The true regression line as a model: y i = α + β x i + e i • In this model: – e i is assumed to follow a normal distribution with mean 0 and standard deviation σ – all e i ’s are assumed independent of each other Example Suppose the simple linear regression equation is given as y=5.000.01x+e with σ =0.075 1) What is the mean value of y when x=100? x=200? x=500? 2) What is the standard deviation of y when x=100? x=200? x=500? Example Suppose the simple linear regression equation is given as y=5.000.01x+e with σ =0.075 1) What is the mean value of y when x=100? x=200? x=500? Solution: when x=100, y=4+e. Since e has a normal distribution with mean 0, the mean value for y is 4. Similarly, when x=200, the mean value of y is 3; And when x=500, the mean value of y is 0. Example Suppose the simple linear regression equation is given as y=5.000.01x+e with σ =0.075 2) What is the standard deviation of y when x=100? x=200? x=500? Solution: When x=100, y=4+e. Since e has a normal distribution with σ =0.075, the standard deviation of y is also σ=0.075. Adding a constant term “4” does not change the standard deviation. In general, the standard deviation of y given any x value is σ =0.075. Estimating the slope β and intercept α • A point estimate of β is b • A point estimate of α is a ˆ b S xy S xx x i y i 1 n ( x i )( y i ) x i 2 1 n ( x i ) 2 ˆ a y bx Example 11.2 on page 494Example 11....
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 Spring '08
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 Statistics, Correlation, Linear Regression, Normal Distribution, Regression Analysis, Errors and residuals in statistics

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