Question

Question 47 (20 points) A local tire dealer wants to predict the number of tires sold each month. He believes that the number of tires sold is a linear function of the amount of money invested in advertising. He randomly selects 6 months of data consisting of tire sales (in thousands of tires) and advertising expenditures (in thousands of dollars). Based on the data set with 6 observations, the simple linear regression model yielded the following results. X = 24 Ex2=124 Y = 42 EY2 =338 EXY = 196 Calculate the coefficient of determination and the coefficient of correlation between X and Y. Interpret the coefficient of Determination. Also find the slope and intercept and interpret the estimated Regression equation. What would the predicted sales of tires be if he spends five thousand dollars in advertising? Perform the test of significance for the slope coefficient. Use 5% level of significance. (Ch13)

Answer 1

	X	Y	XY	X²	Y²
total sum	24	42	196	124	338

sample size ,   n =   6          
here, x̅ =Σx/n =   4.0000   ,   ȳ = Σy/n =   7.00  
                  
SSxx =    Σx² - (Σx)²/n =   28.00          
SSxy=   Σxy - (Σx*Σy)/n =   28.00          
SSyy =    Σy²-(Σy)²/n =   44.00          
estimated slope , ß1 = SSxy/SSxx =   28.000   /   28.000   =   1.00000
                  
intercept,   ß0 = y̅-ß1* x̄ =   3.00000          
                  
so, regression line is   Ŷ =   3.0000   +   1.0000   *x
interpretation: for every $1 increase in adv expenditure, tire sale get increase by 1

correlation coefficient ,    r = Sxy/√(Sx.Sy) =   0.7977
      
coefficient of determination,R² =    (Sxy)²/(Sx.Sy) =    0.6364
about 63.64% of variation in observation of variable y is explained by linear regression line.

----------

Predicted Y at X=   5   is                  
Ŷ =   3.000   +   1.000   *   5   =   8.000

predicted sale = 8000 tires

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

Ho:   ß1=   0          
H1:   ß1╪   0          
n=   6              
alpha=   0.05              
estimated std error of slope =Se(ß1) = Se/√Sxx =    2.000   /√   28   =   0.3780
                  
t stat = estimated slope/std error =ß1 /Se(ß1) =    1.0000   /   0.3780   =   2.646
                  
Degree of freedom ,df = n-2=   4              
p-value =    0.0572              
decision :    p-value>α , do not reject Ho              
Conclusion:   do not Reject Ho and conclude that slope is not significanty different from zero