Question

1. We have the data as follows. There are three independent variables and three dependent variables

(You may use the following table to solve this problem)

	x	y
	3	11
	5	6
	7	4
Total	15	21

a) Calculate b1 and b0, and write the equation of the least squares line.

b) Determine the values of SSE and SST.

c) Calculate the standard error.

d) Find the rejection point for the t statistic at α = .05 and test H₀: β₁ = 0 vs. H_a: β₁ ≠ 0.

e) What is the correlation coefficient?

2. Consider the following partial computer output from a simple linear regression analysis.

Regression Statistics

R Square              (          )

Standard Error           0.4449

ANOVA            df            SS            MS           F                P

Regression          1         (         )                  (         )        0.000

Residual            13        (         )    (          )  

Total               14          12.3240

                Coefficient         Standard Error       t Stat             P-Value

Intercept          4.8615            (        )        (        )           (       )

Ind. Variable      -0.3475             0.0487          (        )

a) Write the equation of the least squares line.

b) Calculate the t statistic used to test H0: β1 = 0 versus Ha: β1 ≠ 0 at α = 0.01.

What do you conclude about the relationship between y and x?

c) What is the predicted value of y when x = 9.00?

d) Calculate the MSE and SSE.

e) What are the unexplained variance and the explained variance?

f) What is the coefficient of determination and the correlation coefficient?

3. Consider the following partial computer output from a multiple regression analysis.

ANOVA            df            SS            MS           F                P

Regression          4         12.5268         4.0089      (       )

Residual            25          1.4318         0.0551  

Total               29         13.9586

a) Calculate the F(model) statistic by using the explained and unexplained variations and other relevant quantities. Find F(model) on the output to check your answer.

b) Use the F(model) statistic and the appropriate critical value to test the significance of the linear regression model under consideration by setting α equal to 0.05.

Answer 1

1)

a)

	ΣX	ΣY	Σ(x-x̅)²	Σ(y-ȳ)²	Σ(x-x̅)(y-ȳ)
total sum	15.00	21.00	8.00	26.00	-14.00
mean	5.00	7.00	SSxx	SSyy	SSxy

sample size ,   n =   3          
here, x̅ = Σx / n=   5.00   ,     ȳ = Σy/n =   7.000  
                  
SSxx =    Σ(x-x̅)² =    8.0000          
SSxy=   Σ(x-x̅)(y-ȳ) =   -14.0          
                  
estimated slope , ß1 = SSxy/SSxx =   -14.0   /   8.000   =   -1.7500
                  
intercept,   ß0 = y̅-ß1* x̄ =   15.75000          
so, regression line is   Ŷ =   15.750   +   -1.750   *x
..

b)

SSE=   (SSxx * SSyy - SS²xy)/SSxx =    1.50
SST =   26.000
................

c)

std error ,Se =    √(SSE/(n-2)) =    1.2247
...

d)

Ho:   ß1=   0          
H1:   ß1╪   0          
n=   3              
alpha =   0.05              
estimated std error of slope =Se(ß1) = Se/√Sxx =    1.225   /√   8.00   =   0.4330
                  
t stat = estimated slope/std error =ß1 /Se(ß1) =    -1.7500   /   0.4330   =   -4.0415
                  
t-critical value=    12.7062   [excel function: =T.INV.2T(α,df) ]          

rejection poitn:

|test stat| > |critical |

.......

e)

correlation coefficient ,    r = SSxy/√(SSx.SSy) =   -0.9707

.........................................

.....................

2)

a)

Y =4.8615 -0.3175*x

..........

b)

t-test statistic =    t = estimated slope / std error =   -0.3475   /   0.0487   =   -7.14

Df =    n - 2 =   13  
p-value =    0.0000   [excel function: =t.dist.2t(t-stat,df) ]   
decision:   p value < α , so, reject the null hypothesis

linear relation exists

.

c)

Y =4.8615 -(0.3175*9)

=2.004

...

d)

SSE =2.5732

MSE =2.572/13

=0.1978

............

THANKS