Question

3. A response variable is related to a predictor variable through the quadratic regression model u yıx(x) = -8.5– 3.2x + 0.7x2 (a) Give the rate of change of the regression function at x = 0, 2, and 3. (b) Express the model in terms of the centered vari- ables as jyjx(x) = Bo + B1(x – ux) + B2(x - ux)2. If ux = 2, give the values of Bo, B1, and B2. (Hint. Match the coefficients of the different powers of x.)

Answer 1

Answer:

The given question deals with the study of a quadratic equation, and studying their rate of change of the regression equation at the given points, and also obtaining the different coefficients for the shifted variables. The provided quadratic regression equation representing the relationship between the predictor variable, and the response variable is given as, My\x(s) = -8.5-3.2x +0.7x? The rate of change of the quadratic regression equation is obtained by, d My x(x) dx -3.2x +0.7x) = -3.2 +1.4x Thus, the rate of change of the quadratic regression equation at the different provided point is obtained by replacing the value of x. The rate of change of the quadratic regression equation at the point x = 0 is, rate of change = -3.2 +1.4 x 0 = -3.2 The rate of change of the quadratic regression equation at the point, x = 2 is, rate of change = -3.2 +1.4 x 2 = -0.4 The rate of change of the quadratic regression equation at the point x = 3 is, rate of change = -3.2 +1.4x3 = 1.0 Thus, the required change of the quadratic regression equation at the points, x=0,2,3 are, -3.2,-0.4,1.0 respectively.

Now, the quadratic regression equation is reframed with the help of the changing the variable by a shifted variable, that is changing x by, x-u Hence, the new quadratic regression equation with the shifted variables is, My\x(x) = B. +B. (x-4)+B2(x-4) With given information that, y = 2 Thus, the above equation can be written as, B+B (x-)+B.(x - 1) = B. +B. (x - 2)+B. (x - 2) = B. + xß, -28, + B x2 - 4B,x+482 =(B-28, +4B2)+(B. -482)x+ B,x? Now, equating the value of the coefficients with the original provided quadratic regression equation gives B.-26, +4B2 = -8.5 B. -4B2 = -3.2 B, = 0.7 Solving the above equations, gives, B = 0.7 B = -0.4 B. =-12.1 Hence, the obtained value of the coefficients for the shifted quadratic regression equation is as obtained above.

Now, the quadratic regression equation is reframed with the help of the changing the variable by a shifted variable, that is changing x by, x-u Hence, the new quadratic regression equation with the shifted variables is, My\x(x) = B. +B. (x-4)+B2(x-4) With given information that, y = 2 Thus, the above equation can be written as, B+B (x-)+B.(x - 1) = B. +B. (x - 2)+B. (x - 2) = B. + xß, -28, + B x2 - 4B,x+482 =(B-28, +4B2)+(B. -482)x+ B,x? Now, equating the value of the coefficients with the original provided quadratic regression equation gives B.-26, +4B2 = -8.5 B. -4B2 = -3.2 B, = 0.7 Solving the above equations, gives, B = 0.7 B = -0.4 B. =-12.1 Hence, the obtained value of the coefficients for the shifted quadratic regression equation is as obtained above.

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