Question

Given are five observations for two variables, x and y.

xi	1	2	3	4	5
yi	4	7	6	12	14

Round your answers to two decimal places. Use Table 1 of Appendix B 

a. Using the following equation: 

 b. Using the following expression: 

c. Using the following equation: 

d. Using the following expression:

Answer 1

X	Y	(x-x̅)²	(y-ȳ)²	(x-x̅)(y-ȳ)
1	4	4.00	21.16	9.20
2	7	1.00	2.56	1.60
3	6	0.00	6.76	0.00
4	12	1.00	11.56	3.40
5	14	4.00	29.16	10.80

	ΣX	ΣY	Σ(x-x̅)²	Σ(y-ȳ)²	Σ(x-x̅)(y-ȳ)
total sum	15	43	10	71.2	25.00
mean	3.00	8.60	SSxx	SSyy	SSxy

sample size ,   n =   5          
here, x̅ = Σx / n=   3.00   ,     ȳ = Σy/n =   8.60  
                  
SSxx =    Σ(x-x̅)² =    10.0000          
SSxy=   Σ(x-x̅)(y-ȳ) =   25.0          
                  
estimated slope , ß1 = SSxy/SSxx =   25.0   /   10.000   =   2.5000
                  
intercept,   ß0 = y̅-ß1* x̄ =   1.1000          
                  
so, regression line is   Ŷ =   1.100   +   2.500   *x

a)

SSE=   (SSxx * SSyy - SS²xy)/SSxx =    8.700
      
std error ,Se =    √(SSE/(n-2)) =    1.70294

X̅ =    3.00
Σ(x-x̅)² =Sxx = 10.0
Predicted Y at X=   3   is                  
Ŷ =   1.100   +   2.500   *   3   =   8.600
                          
standard error, S(ŷ)=Se*√(1/n+(X-X̅)²/Sxx) = 1.703*√(1/5+(3-3)²/10) =   0.76   

b)

Sample Size , n=   5      
Degrees of Freedom,df=n-2 =   3      
critical t Value=tα/2 =   3.182 [excel function: =t.inv.2t(α/2,df) ]  

margin of error,E=t*Std error=t* S(ŷ) =   3.1824   *   0.7616   =   2.4237
                  
Confidence Lower Limit=Ŷ +E =    8.600   -   2.4237   =   6.18
Confidence Upper Limit=Ŷ +E =   8.600   +   2.4237   =   11.02

c)

standard error, S(ŷ)=Se*√(1+1/n+(X-X̅)²/Sxx) =  1.703*√(1+1/5+(3-3)²/10) = 1.87

d)

  margin of error,E=t*std error=t*S(ŷ)=    3.1824   *   1.87   =   5.9368
                  
Prediction Interval Lower Limit=Ŷ -E =   8.600   -   5.937   =   2.66
Prediction Interval Upper Limit=Ŷ +E =   8.600   +   5.937   =   14.54