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4 Pages

### regression_145

Course: MATH 145, Fall 2009
School: Wisc La Crosse
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Word Count: 661

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145 Math - Elementary Statistics Simple Linear Regression December 6, 2007 A statistical relation, unlike a functional relation, is not a perfect one. If X is the independent variable and Y the dependent variable, a statistical relation is of the form: Y = f (X) + . In such cases, we call X an explanatory variable and Y a response variable. The response variable Y is linearly related to one explanatory...

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145 Math - Elementary Statistics Simple Linear Regression December 6, 2007 A statistical relation, unlike a functional relation, is not a perfect one. If X is the independent variable and Y the dependent variable, a statistical relation is of the form: Y = f (X) + . In such cases, we call X an explanatory variable and Y a response variable. The response variable Y is linearly related to one explanatory variable X. That is, Yi = (a + bXi ) + i . Assumptions: 1. The mean of i i = 1, 2, . . . , n. is 0 and the variance of i i is 2 . 2. The random errors are uncorrelated. 3. a and b are parameters. 4. Xi is a known constant. For example, let x denote the distance of a marathon and y the time that it will take a certain runner to finish it. Consider the following 22 practice finish times of our runner. 1 2 25 12 4 47 2 2 22 13 4 48 3 3 35 14 4 48 4 3 36 15 4.5 56 5 2 23 16 4.5 53 6 2.5 30 17 5 62 7 2.5 31 18 5 60 8 3 35 19 5 61 9 3.5 41 20 3.5 42 10 3.5 40 21 3.5 41 11 4 49 22 4 47 Distance (x) Time (y) Distance (x) Time (y) Determine his expected finish time for a 6-mile marathon. Finish Time 30 2.0 40 50 60 2.5 3.0 3.5 Distance 4.0 4.5 5.0 Equation of the Least-Squares Regression Line . Suppose we have data on an explanatory variable x and a response variable y for n individuals. The means and standard deviations of the sample data are x and sx for x and y and sy for y, and the correlation between x and y is r. The equation of the least-squares regression line of y on x is y = a+^ ^ ^ bx with slope 1 ^ = SSxy = (xy) - n (x)(y) = r sy b 1 SSxx sx (x2 ) - n (x)2 (1) (2) and intercept Practice. a = y - ^x ^ b 1. The table below displays data on age (in years) and price (in \$100)for a sample of 11 cars. Age (x) 5 4 6 6 5 5 Price (y) 85 102 70 80 89 98 a. Determine the values of SSxx , SSyy , and SSxy . 6 66 6 90 2 169 7 68 7 50 b. Determine the correlation coefficient r. c. What can you say about the linear relationship of x and y? Is it a strong linear relationship. d. Determine the regression line. e. Estimate the expected value of a car that is 3 years old. 2. Tree Circumference and Height. Listed below are the circumferences (in feet) and the heights (in feet) of trees in Marshall, Minnesota on (based data from "Tree Measurements" by Stanley Rice, American Biology Teacher. x (circ) y (height) x (circ) y (height) 1.8 21.0 5.1 45.3 1.9 33.5 8.3 53.5 1.8 24.6 13.7 93.8 2.4 40.7 5.3 64.0 5.1 73.2 4.9 62.7 3.1 24.9 3.7 47.2 5.5 40.4 3.8 44.3 a. Determine the correlation coefficient r. b. What can you say about the linear relationship of x and y? Is it a strong linear relationship. c. Determine the regression line. d. Estimate the expected height of a tree that has a circumference of 10 feet. 3. A criminologist studying the relationship between population density and robbery rate in medium-sized U.S. cities collected the following data for a random sample of 16 cities; X is the population density of the city (number of people per unit area), and Y is the robbery rate last year (number of robberies per 100,000 people). Assume that the simple linear regression model is appropriate. i Xi Yi i Xi Yi 1 59 209 9 69 213 2 49 180 10 83 201 3 75 195 11 88 214 4 54 192 12 94 212 5 78 215 13 47 205 6 56 197 14 65 186 7 60 208 15 89 200 8 82 189 16 70 204 a. Determine the correlation coefficient r. b. What can you say about the linear relationship of x and y? Is it a strong linear relationship. c. Determine the regression line. d. Estimate the expected value of a car that is 3 years old. 4. To study the relationship b...

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