Question

1. The data below show the prices and ages in years of a particular brand of foreign car.  We want to see the relationship between age and price.  

CAR	1	2	3	4	5	6	7	8	9	10
YI : PRICE	17	20.6	14	16.4	17.8	19.6	13.2	33.8	10	9.6
XI: AGE IN YEARS	5	4	6	5	5	5	6	2	7	7

            We also have  ,  ,     

            

The computer printout for the regression line typically looks like the following estimation line indicating is the intercept plus the slope times the X variable. The value for the standard errors of the estimates for the intercept , and for the slope are typically expressed below each term in parenthesis as follows. Note that n = 10.

Each part is worth 5 points.

The regression equation, with standard errors in parentheses, is: e

                              

                                    (1.74662)       (0.32434)

            SSR = 413.265  SST = 429.759 Note that N = 10 here (10 cars

               = 290,                 = 3388.16, and      = 804.4.  
                   = 19.6                   = 429.76                   =  894.4

1. Based upon the regression line above, what would you predict or estimate the price for an 8-year old car?

2. Calculate the value for MSE

Test the hypothesis

             

Test H₀: b₁ = 0, vs H_A: b₁ ≠ 0 at α = .01 or 1%  (2 – tailed test).

3. Find the critical rejection values for this problem.

4. Calculate the T value from the formula

5. State your conclusions:

Answer 1

	ΣX	ΣY	Σ(x-x̅)²	Σ(y-ȳ)²	Σ(x-x̅)(y-ȳ)
total sum	52	172	19.6	429.8	-90.00
mean	5.20	17.20	SSxx	SSyy	SSxy

sample size ,   n =   10          
here, x̅ = Σx / n=   5.20   ,     ȳ = Σy/n =   17.20  
                  
SSxx =    Σ(x-x̅)² =    19.6000          
SSxy=   Σ(x-x̅)(y-ȳ) =   -90.0          
                  
estimated slope , ß1 = SSxy/SSxx =   -90.0   /   19.600   =   -4.5918
                  
intercept,   ß0 = y̅-ß1* x̄ =   41.0776          
                  
so, regression line is   Ŷ =   41.0776   +   -4.5918   *x

-------------------------

Anova table
variation	SS	df	MS	F-stat	p-value
regression	413.265	1	413.265	200.44	0.0000
error,	16.495	8	2.062
total	429.760	9

a)

Predicted Y at X=   8   is                  
Ŷ =   41.07755   +   -4.591837   *   8   =   4.343

b)

MSE = 2.062

C)

Ho:   ß1=   0          
H1:   ß1╪   0          
n=   10              
alpha =   0.01              
estimated std error of slope =Se(ß1) = Se/√Sxx =    1.436   /√   19.60   =   0.3243
                  
t stat = estimated slope/std error =ß1 /Se(ß1) =    -4.5918   /   0.3243   =   -14.1575
                  
t-critical value=    3.355   [excel function: =T.INV.2T(α,df) ]          

Degree of freedom ,df = n-2=   8              
p-value =    0.0000              
decison :    p-value<α , reject Ho              
Conclusion:   Reject Ho and conclude that slope is significantly different from zero              

Thanks in advance!

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