Question

Regression Analysis. Calculate when y hat = 9,500 – 250(x) when age = 10, 15, 20, and 22 with an asking price of $8,000 6,000, 5,000 and 4,200 respectively.

1. Sum of Squared Errors (SSE)

2. Total Sum of Squares (TSS)

3. Sum of Squares of Regression (SSR)

4. The calculated value that represents the explained variation.

5. What is the calculated value that represents the unexplained variation

6. What is the value of the coefficient of determination.

Answer 1

Regression Analysis. Calculate when y hat = 9,500 – 250(x) when

age = 10, 15, 20, and 22 with an asking price of $8,000 6,000, 5,000 and 4,200 respectively.

Thus  

x ~ Age

y ~ Price

Hence x = 10, 15, 20, 22

            y = 8000 , 6000 , 5000 , 4200

Now estimate of slope is   b1 = 250

Also   b1 = Sxy / Sxx

Now Sum of Squared Errors (SSE) =

here are predicted values

And Total Sum of Squares (TSS) =

here is mean of y

And Sum of Squares of Regression (SSR) = TSS- SSE

Note that we will use R-software for calculation purpose only

Now

Mean of x = = = ( 10+15+ 20+22 ) / 4 = 16.75

Mean of y = = = ( 8000 + 6000 +5000 + 4200) / 4 = 5800

Now

Sxx= = { (10-16.75)2 +(15-16.75)2 + (20-16.75)2 + (22 -16.75)2 } = 86.75

Similarly

TSS = Syy = = 8080000

And

Sxy = = -26200

1. Sum of Squared Errors (SSE)

Given = 9,500 – 250(x)

and   x = 10, 15, 20, 22

Thus

= 7000           at   x = 10

= 5750            at   x = 15

= 4500            at   x = 20

= 4000           at   x = 22

Now y = 8000 , 6000 , 5000 , 4200

Sum of Squared Errors (SSE) =

                                         = { (8000 -7000)2 + (6000 -7000)2 + (5000 -7000)2 + 4200 -7000)2   }

                                         = 1352500

Thus Sum of Squared Errors (SSE) = 1352500

2.Total Sum of Squares (TSS)

TSS =

Now = 5800

= { (8000 -5800)2 + (6000 -5800)2 + (5000 -5800)2 + 4200 -5800)2   } = 8080000

Thus Total Sum of Squares (TSS)   = 8080000

3. Sum of Squares of Regression (SSR)

SSR = TSS - SSE = 8080000 - 1352500 = 6727500

Sum of Squares of Regression (SSR) = 6727500

4. The calculated value that represents the explained variation.

Value that represents the explained variation is nothing but SSR

SSR = 6727500

5. What is the calculated value that represents the unexplained variation

Value that represents the unexplained variation is SSE

which is SSE = 1352500

6. What is the value of the coefficient of determination

Formula coefficient of determination r2 :   1 - SSE / TSS

   1 - SSE / TSS = 1 - 1352500 / 8080000 = 0.8326114

coefficient of determination r2 = 0.8326   or 83.26 %