To explain what determines the price of air conditioners, B. T. Ratchford24 obta
ID: 3223105 • Letter: T
Question
To explain what determines the price of air conditioners, B. T. Ratchford24 obtained the following regression results based on a sample of 19 air conditioners: gamma^_i = -68.236 + 0.023X_2i + 19.729X_3i + 7.653X_iR^2 = 0.84 se = (0.005) (8.992) (3.082) where gamma = the price, in dollars X2 = the BTU rating of air conditioner X3 = the energy efficiency ratio X4 = the number of settings se = standard errors Interpret the regression results. Do the results make economic sense? At a = 5%, test the hypothesis that the BTU rating has no effect on the price of an air conditioner versus that it has a positive effect. Would you accept the null hypothesis that the three explanatory variables explain a substantial variation in the prices of air conditioners? Show clearly all your calculations.Explanation / Answer
Part-a:
The regression model explained 84% of the variability in the price of air conditioners. Corresponding to unit increase in BTU rating of air conditioner there is on an average an increase of $0.023 in the price, holding other predictors fixed. Corresponding to unit increase in energy efficiency ratio of air conditioner there is on an average an increase of $19.729 in the price, holding other predictors fixed. Corresponding to unit increase in number of settings of air conditioner there is on an average an increase of $7.653 in the price, holding other predictors fixed.
Part-b
Coefficient of BTU rating does not make any sense because BTU rating should lead to higher price.
Part-c
We have to test null hypothesis H0: BTU=0 against alternative Ha: BTU>0
Test statistic t=betahat/SE(betahat)= 0.023/0.005 =4.6
Degree of freedom =n-k-1=19-3-1=15
Right tailed p-value=0.000173491 using excel function =TDIST(4.6,15,1)
As p-value>0.05, we reject null and conclude that BTU rating has positive effect onn price.
Part-d
We have to test the null hypothesis that H0: R2=0 against alternative ha: R2>0.
Test statistic F=R2(n-k-1)/[(1-R2)*k]
=0.84*(19-3-1)/[(1-0.84)*3]
=26.25
Degree of freedom=(k,n-k-1)=(3,15)
p-value=0.0000032 using excel function =FDIST(26.25,3,15)
Asp-value<0.05, we reject the null hypothesis and conclude that three explanatory variables explain a substantial variations in prices of air conditioners.