During winter months in Northern Illinois, the Illinois State Police Department

Question

During winter months in Northern Illinois, the Illinois State Police Department notices a strong positive relationship between X-the amount of snow (inches) and Y the number of accidents. Upon examination of the scatterplot, the relationship was determined to be linear. Use this information to answer the next 5 questions. = 4.1, y = 23.2, s. 2.1, ,, = 8.8, r = 0.86 . What is the equation of the regression line? a. 23.34 + 0.21 X b) = 8.44 + 3.6X = 23.2-41x d) = 10.38 + 2.82x 2. The regression line,,is the "BEST" line because it a. Minimizes the square root of the residuals b. Minimizes the sum of the squared residuals c. Minimizes th d. Minimizes the number of residuals e square root of the squared residuals 3. Suppose the equation of the regression line is y = 12.6 + 2.3x. If on a particular winter day, there was a bad storm that drops 7 inches of snow, what would be the predicted number of accidents? a. 14.9 b. 28.7 e. 16.1 d. 19.6 4. Suppose the equation of the regression line is12.6+2.3X. If on a particular winter day, there was a storm that dropped 3 inches of snow and resulted in 11 accidents, what is the residual (error) for this storm? a. 19.5 b.-19.5 c. 8.5 s. What percent of the variation in the data (accidents) is explained by the model? In other words, how good is this model in predicting the number of accidents? a, 93% b. 172% c.86% d. 74%

Explanation / Answer

Part (1)

Regression equation is: ycap = a + bx, where

b = r.sy/sx = (0.86 x 8.8)/2.1 = 3.6 and

a = ybar – b.xbar

= 23.2 – (3.6 x 4.1)

= 8.44.

So, the regression equation is: ycap = 8.44 + 3.6x – Option (b) ANSWER

Part (2)

The regression coefficients are obtained by minimizing the sum of squares of difference between the actual and the predicted y values (i.e., residuals). So, Option (b) ANSWER

Part (3)

Substituting x = 7, predicted number of accidents = ycap = 12.6 + (2.3 x 7) = 12.6 + 16.1

= 28.7 accidents - option (b) ANSWER

Part (4)

Substituting x = 3, predicted number of accidents = ycap = 12.6 + (2.3 x 3) = 12.6 + 6.9

= 19.5.

Given the actual y-value is 11, the residual = y – ycap = 11 – 19.5

= - 8.5 – option (d) ANSWER

Part (5)

Percent of variation in the number of accidents explained by the model is given by r2. So, the answer = 0.862 = 0.7396 = 74% option (d) ANSWER

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