Question

The following data are the monthly salaries y and the grade point averages x for students who obtained a bachelor's degree in business administration. GPA Monthly Salary ($) 2.7 3,600 3.5 3,800 3.6 4,200 3.1 3,700 3.5 4,100 2.8 2,500 The estimated regression equation for these data is ŷ = -55.3 + 1,157.9x and MSE =209,013. a. Develop a point estimate of the starting salary for a student with a GPA of 3.0 (to 1 decimal). $ b. Develop a 95% confidence interval for the mean starting salary for all students with a 3.0 GPA (to 2 decimals). $ ( , ) c. Develop a 95% prediction interval for Ryan Dailey, a student with a GPA of 3.0 (to 2 decimals). $ ( , )

Answer 1

a)

Ŷ =   -55.3   +   1157.9   *x

Predicted Y at X=   3   is
Ŷ =   -55.3   +   1157.9   *3=3418.4

b)

std error = √MSE = 457.17957

X	Y	(x-x̅)²
2.7	3600	0.25
3.5	3800	0.09
3.6	4200	0.16
3.1	3700	0.01
3.5	4100	0.09
2.8	2500	0.16

here, x̅ = Σx / n=   3.20
SSxx =    Σ(x-x̅)² =    0.7600

X Value=   3                      
Confidence Level=   95%                      
                          
                          
Sample Size , n=   6                      
Degrees of Freedom,df=n-2 =   4                      
critical t Value=tα/2 =   2.776   [excel function: =t.inv.2t(α/2,df) ]                  
                          
X̅ =    3.20                      
Σ(x-x̅)² =Sxx   0.8                      
Standard Error of the Estimate,Se=   457.18                      
                          
Predicted Y at X=   3   is                  
Ŷ =   -55.263   +   1157.895   *   3   =   3418.421
                          
standard error, S(ŷ)=Se*√(1/n+(X-X̅)²/Sxx) =    214.094                      
margin of error,E=t*Std error=t* S(ŷ) =   2.7764   *   214.0939   =   594.4201      
                          
Confidence Lower Limit=Ŷ +E =    3418.421   -   594.4201   =   2824.00   
Confidence Upper Limit=Ŷ +E =   3418.421   +   594.4201   =   4012.84   

c)

For Individual Response Y                  
standard error, S(ŷ)=Se*√(1+1/n+(X-X̅)²/Sxx) =   504.8261              
margin of error,E=t*std error=t*S(ŷ)=    2.7764   *   504.83   =   1401.6219
                  
Prediction Interval Lower Limit=Ŷ -E =   3418.421   -   1401.622   =   2016.80
Prediction Interval Upper Limit=Ŷ +E =   3418.421   +   1401.622   =   4820.04