? = a + bx for the least-squares line. In some computer printouts, the least-squ

Question

? = a + bx

for the least-squares line. In some computer printouts, the least-squares equation is not given directly. Instead, the value of the constant a is given, and the coefficient b of the explanatory or predictor variable is displayed. Sometimes a is referred to as the constant, and sometimes as the intercept. Data from a report showed the following relationship between elevation (in thousands of feet) and average number of frost-free days per year in a state.

A Minitab printout provides the following information.

4. -6 points BBUnderStat12 9.2.005.M My Notes Ask Your Teacher We use the form ý-a bx for the least-squares line. In some computer printouts, the least-squares equation is not given directly. Instead, the value of the constant a is given, and the coefficient b of the explanatory or predictor variable is displayed. Sometimes a is referred to as the constant, and sometimes as the intercept. Data from a report showed the following relationship between elevation (in thousands of feet) and average number of frost-free days per year in a state A Minitab printout provides the following information. Predictor Constant Elevation s 11.8603 Cocf 316.89 -28.842 R-Sq- 95.2% SE Cocf 28.31 3.511 11.24 0.002 -8.79 0.003 Notice that "Elevation" is listed under "Predictor." This means that elevation is the explanatory variable x. Its coefficient is the slope b. "Constant" refers to a in the equation (a) Use the printout to write the least-squares equation (b) For each 1000-foot increase in elevation, how many fewer frost-free days are predicted? (Round your answer to three decimal places.) (c) The printout gives the value of the coefficient of determination r2. What is the value of r? Be sure to give the correct sign for r based on the sign of b. (Round your answer to four decimal places.) (d) what percentage of the variation in y can be explained by the corresponding variation in x and the least-squares line? What percentage is unexplained?

Explanation / Answer

a) yhat = 316.89 - 28.842x is the least square equation. (a = constant = 316.89 and b = elevation = -28.842)

b) Now if elevation is 1000 ft, then expected frost free days are (yhat) = 316.89 - 28.842 * 1 = 288.048

We have taken 1 because in the regression equation, the unit of explanatory variable is in thousands of feet, so 1000 feet is equivalent to 1.

c) R2 is 95.2%. Thus r is square root of 0.952, which is 0.976 with a negative sign, ie, r = - 0.9757. This is because r and slope of regression line have the same sign.

d) 95.2% of the variation in frost free days can be explained by elevation. (Since it is the coefficient of determination)

4.8% of the variation is unexplained. ((100 - 95.2)%,ie, Total variation(100%) - explained variation(95.2%) = unexplained variation)

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