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MVS 250: V. Katch

S TATISTICS. Chapter 5 Regression. MVS 250: V. Katch. Definition Regression Equation. ^. ( y = m x + b ). Regression. Given a collection of paired data, the regression equation. algebraically describes the relationship between the two variables. Regression Line

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MVS 250: V. Katch

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  1. STATISTICS Chapter 5 Regression MVS 250: V. Katch

  2. Definition Regression Equation ^ (y= mx + b) Regression Given a collection of paired data, the regression equation algebraically describes the relationship between the two variables • Regression Line • (line of best fit or least-squares line) the graph of the regression equation

  3. Regression Line Plotted on Scatter Plot

  4. Regression Line

  5. Two different lines, one to predict X and one to predict Y.

  6. x is the independent variable (predictor variable) The Regression Equation ^ y is the dependent variable (response variable) b= slope y = mx +b

  7. 1. We are investigating only linearrelationships. 2. For each x value, y is a random variable having a normal (bell-shaped) distribution. All of these y distributions have the same variance. Also, for a given value of x, the distribution of y-values has a mean that lies on the regression line. (Results are not seriously affected if departures from normal distributions and equal variances are not too extreme.) Assumptions

  8. SD2x SD2x (xy/n) - (x/n) (y/n) m = (x2/n) - (x/n)2 Formula for y-intercept and slope (y-intercept) (y/n) (x2/n) - (x/n) (xy/n) Formula 1 b = (x2/n) - (x/n)2 (slope) Formula 2

  9. Formula 3 b = y - mx m = r sy/sx slope = Intercept = If you find r, then where y is the mean of the y-values and x is the mean of the x values Formula 4 where y is the mean of the y-values, x is the mean of the x-values and m is the slope

  10. Round to three significant digits If you use the formulas 1 and 2, and 3 try not to round intermediate values. Rounding the y-intercept and the slope

  11. The regression line fits the sample points best.

  12. Definitions Residual for a sample of paired (x,y) data, the difference (y - y) between an observed sample y-value and the value of y, which is the value of y that is predicted by using the regression equation. Least-Squares Property A straight line satisfies this property if the sum of the squares of the residuals is the smallest sum possible. Residuals and the Least-Squares Property ^ ^

  13. 32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 0 1 2 3 4 5 Residuals and the Least-Squares Property x 1 2 4 5 ^ y= 5 + 4x y 4 24 8 32 y • Residual = 7 • Residual = 11 • Residual = -13 • Residual = -5 x

  14. In predicting a value of y based on some given value of x ... 1. If there is not a significant linear correlation, the best predicted y-value is y. 2. If there is a significant linear correlation, the best predicted y-value is found by substituting the x-value into the regression equation. Predictions

  15. Calculate the value of r and test the hypothesis that  = 0 Use the regression equation to make predictions. Substitute the given value in the regression equation. Is there a significant linear correlation ? Given any value of one variable, the best predicted value of the other variable is its sample mean. Start Yes Predicting the Value of a Variable No

  16. 1. If there is no significant linear correlation, don’t use the regression equation to make predictions. 2. When using the regression equation for predictions, stay within the scope of the available sample data. 3. A regression equation based on old data is not necessarily valid now. 4. Don’t make predictions about a population that is different from the population from which the sample data was drawn. • Guidelines for Using The • Regression Equation

  17. Example X Y Compute r, slope, intercept, regression What is this equation used for?

  18. Data from the Garbage Project x Plastic (lb) 0.27 2 1.41 3 2.19 3 2.83 6 2.19 4 1.81 2 0.85 1 3.05 5 y Household What is the best predicted size of a household that discard 0.50 lb of plastic?

  19. Data from the Garbage Project x Plastic (lb) 0.27 2 1.41 3 2.19 3 2.83 6 2.19 4 1.81 2 0.85 1 3.05 5 y Household What is the best predicted size of a household that discard 0.50 lb of plastic? Using a calculator: b= 0.549 m = 1.48 y = 0.549 + 1.48 (0.50) y = 1.3 A household that discards 0.50 lb of plastic has approximately one person.

  20. Marginal Change the amount a variable changes when the other variable changes by exactly one unit Outlier a point lying far away from the other data points Influential Points points which strongly affect the graph of the regression line Definitions

  21. Regression equation uncorrected data: 15.4 + 0.13 height corrected data: -3.2 + 0.42 height Correlation uncorrected data: r = 0.28 corrected data: r = 0.69 Example 5.4 Height and Foot Length (cont) Three outliers were data entry errors.

  22. Correlation all data: r = 0.73 w/o SF: r = –0.96 Example 5.10 Earthquakes in US San Francisco earthquake of 1906.

  23. Example: Predict the quiz score of a student who spends 30 hours a week watching television. One more step…….

  24. Compute the Standard Error of the Estimate SY*X = SDY√1-r2 SY*X = 13.83√1-(-8.17)2 SY*X = ±7.978 7.978 points The predicted score is 56.56 points +

  25. Definition Multiple Regression Equation A linear relationship between a dependent variable y and two or more independent variables (x1, x2, x3 . . . , xk) Multiple Regression ^ y = m0 + m1x1 + m2x2 + . . . + mkxk

  26. Linear: y = a + bx Quadratic: y = ax2 + bx + c Logarithmic: y = a + b lnx Exponential: y = abx Power: y = axb Logistic: y = Generic Models c 1 + ae -bx

  27. Look for a Pattern in the Graph: Examine the graph of the plotted points and compare the basic pattern to the known generic graphs. Find and Compare Values of R2: Select functions that result in larger values of R2, because such larger values correspond to functions that better fit the observed points. Think: Use common sense. Don’t use a model that lead to predicted values known to be totally unrealistic. Development of a Good Mathematics Model

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