Question

Given are five observations for two variables, x and y. xi 1 2 3 4 5 yi 4 7 6 12 14 a. Using the following equation: 12.32a.jpg Estimate the standard deviation of ŷ* when x = 3. b. Using the following expression: 12.32b.jpg Develop a 95% confidence interval for the expected value of y when x = 3. c. Using the following equation: 12.32c.jpg Estimate the standard deviation of an individual value of y when x = 3. d. Using the following expression: 12.32d.jpg Develop a 95% prediction interval for y when x = 3.

Answer 1

X	Y	(x-x̅)²	(y-ȳ)²	(x-x̅)(y-ȳ)
1	4	4.000	21.160	9.200
2	7	1.000	2.560	1.600
3	6	0.000	6.760	0.000
4	12	1.000	11.560	3.400
5	14	4.00	29.160	10.80

	ΣX	ΣY	Σ(x-x̅)²	Σ(y-ȳ)²	Σ(x-x̅)(y-ȳ)
total sum	15	43	10.000	71.20	25.000
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 =   1.1000   +   2.5000   *x
                  
SSE=   (SSxx * SSyy - SS²xy)/SSxx =    8.700          
                  
std error ,Se =    √(SSE/(n-2)) =    1.70294      

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

a)

x=3

X̅ =    3.00
Σ(x-x̅)² =Sxx   10.0

standard dev of y^ , S(ŷ)=Se*√(1/n+(X-X̅)²/Sxx) =    0.762

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   =   6.176
Confidence Upper Limit=Ŷ +E =   8.600   +   2.4237   =   11.024

c)

  standard dev, S(ŷ)=Se*√(1+1/n+(X-X̅)²/Sxx) =   1.8655  

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   =   2.6632
Prediction Interval Upper Limit=Ŷ +E =   8.600   +   5.937   =   14.5368