Question

Given are five observations for two variables, x and y. (Round your answers to two decimal places.)

xi 1 2 3 4 5

yi 3 8 6 10 15

(a) Use sŷ* = s 1 n + (x* − x)^2 / Σ(xi − x)^2 to estimate the standard deviation of ŷ* when x = 5.

(b) Use Sŷ* ± tα/2sŷ* to develop a 95% confidence interval for the expected value of y when x = 5.

____to____

(c) Use spred = s 1 + 1 n + (x* − x)^2 / Σ(xi − x)^2 to estimate the standard deviation of an individual value of y when x=5.

(d) Use ŷ* ± tα/2spred to develop a 95% prediction interval for y when x = 5.

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Answer 1

	ΣX	ΣY	Σ(x-x̅)²	Σ(y-ȳ)²	Σ(x-x̅)(y-ȳ)
total sum	15.00	42.00	10.00	81.20	26.00
mean	3.00	8.40	SSxx	SSyy	SSxy

Sample size,   n =   5      
here, x̅ = Σx / n=   3.000          
ȳ = Σy/n =   8.400          
SSxx =    Σ(x-x̅)² =    10.0000      
SSxy=   Σ(x-x̅)(y-ȳ) =   26.0      
              
estimated slope , ß1 = SSxy/SSxx =   26/10=   2.6000      
intercept,ß0 = y̅-ß1* x̄ =   8.4- (2.6 )*3=   0.6000      
              
Regression line is, Ŷ=   0.600   + (   2.600   )*x
              
SSE=   (SSxx * SSyy - SS²xy)/SSxx =    13.6000      
std error ,Se =    √(SSE/(n-2)) =    2.1292      

a)

X̅ =    3.00              
Σ(x-x̅)² =Sxx   10.00              
Standard Error of the Estimate,Se=   2.1292              
                  
Predicted Y at X=   5   is          
Ŷ=   0.60000   +   2.60000   *5=   13.600
                  
standard error, S(ŷ)=Se*√(1/n+(X-X̅)²/Sxx) =    1.649              

b)

Confidence Level=   95%      
          
          
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   *   1.649   =   5.2486

Confidence Lower Limit=Ŷ +E =    13.600   -   5.249   =   8.3514
Confidence Upper Limit=Ŷ +E =   13.600   +   5.249   =   18.8486

c) standard error, S(ŷ)=Se*√(1+1/n+(X-X̅)²/Sxx) =   2.6932
d)

margin of error,E=t*std error=t*S(ŷ)=    3.182   *   2.693   =   8.5710
                  
Prediction Interval Lower Limit=Ŷ -E =   13.600   -   8.571   =   5.029
Prediction Interval Upper Limit=Ŷ +E =   13.600   +   8.571   =   22.171