Question

A study was conducted to determine whether certain features could be used to explain the variability in the prices of air conditioners. For a sample of nineteen (19) air conditioners, the following regression was estimated:

Yhat = -68.236 +.0023X1 +19.729X2 +7.653X3

(.005) (8.992) (3.082)

where Y = price (in dollars)

X1 = rating of the air conditioner, in BTU per hour

X2 = energy efficiency ratio

X3 = number of settings

The coefficient of multiple determination is 0.84. The figures in parentheses beneath the coefficient estimates are the corresponding estimated standard errors.

a. Find a 95% confidence interval for the expected increase in price resulting from an additional setting, other things constant.

b. Test the null hypothesis that, all else being equal, the energy efficiency ratio of air conditioners does not affect their price against the alternative that the higher the energy efficiency ratio, the higher the price. Use alpha = .05.

Answer 1

A)

n =   19                  
alpha,α =    0.05                  
estimated slope=   7.653                  
std error =    3.082                  
                      
Df = n-4 =   15                  
t critical value =    2.1314   [excel function: =t.inv.2t(α,df) ]              
                      
margin of error ,E = t*std error =    2.1314   *   3.082   =   6.5691  
                      
95%   confidence interval is ß1 ± E                   
lower bound = estimated slope - margin of error =    7.653   -   6.5691   =   1.0839  
upper bound = estimated slope + margin of error =    7.653   +   6.5691   =   14.2221  

B)

Ho:   ß = 0                  
Ha:   ß > 0                  
                      
n =   19                  
alpha,α =    0.05                  
estimated slope=   19.729                  
std error =    8.992                  
                      
t-test statistic =    t = estimated slope / std error =   19.729   /   8.992   =   2.194
                      
Df =    n - 2 =   15              
p-value =    0.0212   [excel function: t-dist.rt(t-stat,df) ]               
decision:   p value < α , so, reject the null hypothesis                  
                      

THANKS

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