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# Notes Ch8

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Linear Regression Notes

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### Notes Ch8

1. 1. Chapter 8 Linear Regression
2. 2. Regression Line <ul><li>Straight line that describes how a response variable y changes as an explanatory variable x changes. </li></ul><ul><li>LSRL - Least Squares Regression Line </li></ul><ul><li>LSRL is a predictor of y given x. It serves as a mathematical model for the data. </li></ul>
3. 3. LSRL cont. <ul><li>Error = observed – predicted </li></ul><ul><li>The LSRL minimizes the vertical distance of the data points from the line (error). </li></ul>
4. 4. LSRL cont. <ul><li>LSRL of y on x is the line that makes the sum of the squares of the vertical distances of the data points from the line as small as possible. </li></ul>
5. 5. Formulas to know
6. 6. Slope and Intercept <ul><li>Slope is the rate of change of y when x increases by 1 unit. </li></ul><ul><li>The intercept is the value of the predicted response when x is equal to 0. The intercept is not always statistically meaningful. </li></ul>
7. 7. Coefficient of Determination <ul><li>The coefficient of determination, r^2, gives the proportion that is not error. </li></ul><ul><li>If error is large, r^2 will be close to 0. </li></ul><ul><li>If error is small, r^2 will be close to 1. </li></ul><ul><li>Coefficient of Determination = (Correlation Coefficient)^2 </li></ul>
8. 8. Interpretation of r^2 <ul><li>Convert r^2 to a per cent and state that: </li></ul><ul><li>About _______% of the variation in y is explained by the LSRL on x . </li></ul>
9. 9. Calculate LSRL using formulas <ul><li>Data was taken from all 78 7 th grade students in a rural midwestern town. Find the equation of the LSRL for predicting GPA from IQ given the following summary statistics. </li></ul>
10. 10. What percent of the observed variation in these student’s GPAs can be explained by the linear relationship between GPA and IQ?
11. 11. Do Part I of the Archaeopteryx worksheet
12. 12. Residuals <ul><li>Residuals (errors) = observed – predicted </li></ul><ul><li>Residual Plot (x, residuals) </li></ul><ul><ul><li>There should be no patterns in the plot only random scatter. </li></ul></ul>
13. 13. Outliers <ul><li>Points that are outside overall pattern of the other observations. They show up clearly in the residual plot. </li></ul>
14. 14. Influential points <ul><li>A point is influential if by removing it, it would markedly change the position of the regression line. Points in the x direction are often influential. </li></ul>