Example: Boats and Manatees

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1 Figure 9-6 Example: Boats and Manatees Slide 1 Given the sample data in Table 9-1, find the value of the linear correlation coefficient r, then refer to Table A-6 to determine whether there is a significant linear correlation between the number of registered boats and the number of manatees killed by boats. Use method 2. Using the same procedure previously illustrated, we find that r = Method 2: Referring to Table A-6, we conclude that there is a significant linear correlation between number of registered boats and number of manatee deaths from boats.

2 Use method 1 Slide 2 t = r 1 r 2 n t = =

3 Using either of the two methods, we find Slide 3 Method 1: > Method 2: > 0.632; That is, the test statistic falls in the critical region. Conclusion: We therefore reject the null hypothesis. There is sufficient evidence at significance level 0.05 to support the claim of a linear correlation between the number of registered boats and the number of manatee deaths from boats.

4 FIGURE 9-4 Testing for a Linear Correlation Slide 4

5 Interpreting r: Explained Variation Slide 5 The value of r 2 is the proportion of the variation in y that is explained by the linear relationship between x and y. Manatee example: With r = 0.922, we get r 2 = We conclude that (or about 85%) of the variation in manatee deaths can be explained by the linear relationship between the number of boat registrations and the number of manatee deaths from boats. This implies that 15% of the variation of manatee deaths cannot be explained by the number of boat registrations.

6 9.3 Regression Slide 6 Regression Equation Variables: X =independent variable, predictor variable or explanatory variable Y= dependent variable or response variable. A straight line (linear relationship): Y = a X + b, where a is the y-intercept and b is the slope.

7 For each x-value, Assumptions Slide 7 Normality: Y is a random variable having a normal (bell-shaped) distribution. Homogeneity: All of these y distributions have the same variance. Linearity: 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.)

8 Regression Equation Slide 8 Given paired sample data (x,y) of size n satisfying population parameter equation below. Population Parameter Sample Statistic y-intercept of regression equation β 0 b 0 Slope of regression equation β 1 b 1 Equation of the regression line y = β 0 + β 1 x ^ y = b 0 + b 1 x where b 0 estimates β 0 and b 1 estimates β 1. Regression line Q: How to get b 0 and b 1?

9 Formula for b 0 and b 1 Slide 9 Formula 9-2 b 1 = n(σxy) (Σx) (Σy) n(σx 2 ) (Σx) 2 (slope) Formula 9-3 b 0 = y b 1 x (y-intercept) calculators or computers can compute these values

10 Slide 10 The regression line fits the sample points best.

11 Data x y n = 4 Σx = 10 Σy = 20 Σx 2 = 36 Σy 2 = 120 Σxy = 48 b 1 = b 1 = Calculating the Regression Equation n(σxy) (Σx) (Σy) n(σx 2 ) (Σx) 2 4(48) (10) (20) 4(36) (10) 2 b 1 = = The estimated equation of the regression line is: Slide 11 In Section 9-2, we used these values to find that the linear correlation coefficient of r = Use this sample to find the regression equation. b 0 = y b x 1 = 5- ( )(2.5) = 5.45 yˆ = x

12 Example: Boats and Manatees Slide 12

13 Example: Boats and Manatees Slide 13 Given the sample data in Table 9-1, find the regression equation. Using the same procedure as in the previous example, we find that b 1 = 2.27 and b 0 = 113 or computer, the estimated regression equation is: ^ y = x

14 Predictions Slide 14 In predicting a value of y based on some given value of x 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.

15 Slide 15 Figure 9-8 Predicting the Value of a Variable

16 Revisit Boats and Manatees Example Slide 16 The best regression equation is y = x. ^ 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. Q: Given that x = 85, find the best predicted value of y, the number of manatee deaths from boats. We do have a significant linear correlation (with r = 0.922).

17 Example: Boats and Manatees Slide 17 y ^ = x (85) = 80.0 The predicted number of manatee deaths is The actual number of manatee deaths in 2001 was 82, so the predicted value of 80.0 is quite close.

18 Guidelines for Using The Regression Equation Slide 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.

19 Definitions Slide 19 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.

20 Residuals and the Slide 20 Residual Least-Squares Property Definitions ^ 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.

21 Residuals and the Slide 21 Least-Squares Property x y ^y = 5 + 4x Figure 9-9

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