Question

In: Statistics and Probability

Listed below are altitudes​ (thousands of​ feet) and outside air temperatures​ (°F) recorded during a flight....

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

Solutions

Expert Solution

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


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