Question

The relationship between the amount of remaining carbon monoxide (CO) in an individual’s lungs and the time since that person last smoked a cigarette can be summarized using a linear regression approach.

Write down a simple linear regression model and the underlying assumptions. The following summary data was collected from 12 different smokers.

x = time since last smoked a cigarette (hours) y = amount of CO in ppm

n=12, x=1.88 sxx =25.8, syy =1805 SSE=877.4 Fitted regression line: ?̂ = ??. ?? − ?. ??? ?

(k) What is the residual at ?0 = 2.25 if the corresponding observed amount CO is

28ppm

(l) Estimate the mean amount of CO in the lungs for an elapsed time of 2.25 hours.

(m) Construct a 95% confidence interval for the true mean amount of CO in the lungs

for an elapsed time of 2.25 hours.

Please show all work.

Answer 1

Answer:

K) residual = 28-31.679 =  -3.6790

L) Ŷ =   45.1700   +   -5.9960   *2.25=   31.679

M)

X Value=   2.25              
Confidence Level=   95%              
                  
                  
Sample Size , n=   12              
Degrees of Freedom,df=n-2 =   10              
critical t Value=tα/2 =   2.228   [excel function: =t.inv.2t(α/2,df) ]          
                  
X̅ =    1.88              
Σ(x-x̅)² =Sxx   25.80              
Standard Error of the Estimate,Se=   9.3670              
                  
Predicted Y at X=   2.25   is          
Ŷ =   45.1700   +   -5.9960   *2.25=   31.679
                  
standard error, S(ŷ)=Se*√(1/n+(X-X̅)²/Sxx) =    2.789              
margin of error,E=t*Std error=t* S(ŷ) =   2.2281   *   2.789   =   6.2138
                  
Confidence Lower Limit=Ŷ +E =    31.679   -   6.214   =   25.465
Confidence Upper Limit=Ŷ +E =   31.679   +   6.214   =   37.893

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