Created by Erin Hodgess, Houston, Texas Section 9-3 Regression Created by Erin Hodgess, Houston, Texas

Regression Definition Regression Equation ^ The regression equation expresses a relationship between x (called the independent variable, predictor variable or explanatory variable, and y (called the dependent variable or response variable. The typical equation of a straight line y = mx + b is expressed in the form y = b0 + b1x, where b0 is the y-intercept and b1 is the slope. ^

Assumptions 1. We are investigating only linear relationships. 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.)

Regression Definition Regression Equation y = b0 + b1x Regression Line Given a collection of paired data, the regression equation y = b0 + b1x ^ algebraically describes the relationship between the two variables Regression Line The graph of the regression equation is called the regression line (or line of best fit, or least squares line).

Notation for Regression Equation Population Parameter Sample Statistic ^ y-intercept of regression equation 0 b0 Slope of regression equation 1 b1 Equation of the regression line y = 0 + 1 x y = b0 + b1 x

calculators or computers can compute these values Formula for b0 and b1 Formula 9-2 n(xy) – (x) (y) b1 = (slope) n(x2) – (x)2 b0 = y – b1 x (y-intercept) Formula 9-3 calculators or computers can compute these values

b0 = y - b1x If you find b1 first, then Formula 9-4 Can be used for Formula 9-2, where y is the mean of the y-values and x is the mean of the x values

The regression line fits the sample points best.

Rounding the y-intercept b0 and the slope b1 Round to three significant digits. If you use the formulas 9-2 and 9-3, try not to round intermediate values. page 527 of text

Calculating the Regression Equation 1 2 8 3 6 5 4 Data x y In Section 9-2, we used these values to find that the linear correlation coefficient of r = –0.135. Use this sample to find the regression equation. This is exercise #7 on page 521.

Calculating the Regression Equation 1 2 8 3 6 5 4 Data x y n(xy) – (x) (y) n(x2) –(x)2 b1 = 4(48) – (10) (20) 4(36) – (10)2 –8 44 = –0.181818 n = 4 x = 10 y = 20 x2 = 36 y2 = 120 xy = 48 This is exercise #7 on page 521.

Calculating the Regression Equation 1 2 8 3 6 5 4 Data x y n = 4 x = 10 y = 20 x2 = 36 y2 = 120 xy = 48 b0 = y – b1 x 5 – (–0.181818)(2.5) = 5.45 This is exercise #7 on page 521.

Calculating the Regression Equation 1 2 8 3 6 5 4 Data x y n = 4 x = 10 y = 20 x2 = 36 y2 = 120 xy = 48 The estimated equation of the regression line is: y = 5.45 – 0.182x ^ This is exercise #7 on page 521.

Example: Boats and Manatees Given the sample data in Table 9-1, find the regression equation. Using the same procedure as in the previous example, we find that b1 = 2.27 and b0 = –113. Hence, the estimated regression equation is: This is exercise #7 on page 521. y = –113 + 2.27x ^

Example: Boats and Manatees Given the sample data in Table 9-1, find the regression equation. This is exercise #7 on page 521.

Predictions 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.

Figure 9-8 Predicting the Value of a Variable page 529 of text Figure 9-8 Predicting the Value of a Variable

Example: Boats and Manatees Given the sample data in Table 9-1, we found that the regression equation is y = –113 + 2.27x. Assume that in 2001 there were 850,000 registered boats. Because Table 9-1 lists the numbers of registered boats in tens of thousands, this means that for 2001 we have x = 85. Given that x = 85, find the best predicted value of y, the number of manatee deaths from boats. ^ This is exercise #7 on page 521.

Example: Boats and Manatees Given the sample data in Table 9-1, we found that the regression equation is y = –113 + 2.27x. Given that x = 85, find the best predicted value of y, the number of manatee deaths from boats. ^ We must consider whether there is a linear correlation that justifies the use of that equation. We do have a significant linear correlation (with r = 0.922). This is exercise #7 on page 521.

Example: Boats and Manatees Given the sample data in Table 9-1, we found that the regression equation is y = –113 + 2.27x. Given that x = 85, find the best predicted value of y, the number of manatee deaths from boats. ^ y = –113 + 2.27x –113 + 2.27(85) = 80.0 ^ The predicted number of manatee deaths is 80.0. The actual number of manatee deaths in 2001 was 82, so the predicted value of 80.0 is quite close. This is exercise #7 on page 521.

Guidelines for Using The Regression Equation 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. page 530 of text

Definitions Marginal Change: The marginal change is the amount that a variable changes when the other variable changes by exactly one unit. Outlier: An outlier is a point lying far away from the other data points. Influential Points: An influential point strongly affects the graph of the regression line. page 531 of text The slope b1 in the regression equation represents the marginal change in y that occurs when x changes by one unit.

Residuals and the Least-Squares Property 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. ^ ^ page 533 of text

Residuals and the Least-Squares Property 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. ^ ^ page 533 of text

Residuals and the Least-Squares Property Figure 9-9 x 1 2 4 5 y 4 24 8 32 y = 5 + 4x ^ Data is found in margin on page 532.