Question

Listed below are altitudes​ (thousands of​ feet) and outside air temperatures​ (°F) recorded during a flight. Find the​ (a) explained​ variation, (b) unexplained​ variation, and​ (c) indicated prediction interval. There is sufficient evidence to support a claim of a linear​ correlation, so it is reasonable to use the regression equation when making predictions. For the prediction​ interval, use a​ 95% confidence level with the altitude of 6327 ft​ (or 6.327 thousand​ feet).

Altitude 4 12 15 20 28 31 33

Temperature 60 36 22 1 -34 -41 -51

A) Find explained variation. round to 3 decimal places

B) Find unexplained variation. round to 3 decimal places

C) Find indicated prediction interval. round to 3 decimal places

Answer 1

X	Y	(x-x̅)²	(y-ȳ)²	(x-x̅)(y-ȳ)
4	60	269.90	3721.00	-1002.14
12	36	71.04	1369.00	-311.86
15	22	29.47	529.00	-124.86
20	1	0.18	4.00	-0.86
28	-34	57.33	1089.00	-249.86
31	-41	111.76	1600.00	-422.86
33	-51	158.04	2500.00	-628.57

	ΣX	ΣY	Σ(x-x̅)²	Σ(y-ȳ)²	Σ(x-x̅)(y-ȳ)
total sum	143	-7	697.7142857	10812.0	-2741.00
mean	20.43	-1.00	SSxx	SSyy	SSxy

a)

explained variation = R² =    (Sxy)²/(Sx.Sy) =    0.996

b)

unexplained variation = 1 -  explained variation = 1 - 0.996 = 0.004

c)

X Value=   6.327                      
Confidence Level=   95%                      
                          
                          
Sample Size , n=   7                      
Degrees of Freedom,df=n-2 =   5                      
critical t Value=tα/2 =   2.571   [excel function: =t.inv.2t(α/2,df) ]                  
                          
X̅ =    20.43                      
Σ(x-x̅)² =Sxx   697.7                      
Standard Error of the Estimate,Se=   2.962                      
                          
Predicted Y at X=   6.327   is                  
Ŷ =   79.255   +   -3.929   *   6.327   =   54.399
For Individual Response Y                  
standard error, S(ŷ)=Se*√(1+1/n+(X-X̅)²/Sxx) =   3.5393              
margin of error,E=t*std error=t*S(ŷ)=    2.5706   *   3.54   =   9.0982
                  
Prediction Interval Lower Limit=Ŷ -E =   54.399   -   9.098   =   45.301
Prediction Interval Upper Limit=Ŷ +E =   54.399   +   9.098   =   63.497