Problem 1. The cost of the maintenance of shipping tractors seems to increase wi

Question

Problem 1. The cost of the maintenance of shipping tractors seems to increase with the age of the tractor. The following data are collected Age(yr) 6 Months Cost ($) 4.5 4.5 4.5 4.0 4.0 4.0 5.0 5.0 5.5 5.0 0.5 0.5 6.0 6.0 1.0 1.0 1.0 619 1049 1033 495 723 681 890 1522 987 1194 163 182 764 1373 978 466 549 a. Drive the ANOVA table for the above data. Please show the split of sum square of residuals b. Test whether the linear regression model is significant at 1% c. Conduct a test to check whether the regression model is adequate

Explanation / Answer

Lets first address the part (c).

<(c)>

Before proceeding to do an ANOVA, we must understand if the linear model assumption holds or not.

For this, the simplest measure is to find the Pearson product moment correlation coeffecient between X and Y. For our sample the value of correlation coeffecient = Cov(x,y)/S.D(X)*S.D(Y) = 0.69 which is quite high. So a regression assumption can hold and we can proceed to do an ANOVA.

<(a) and (b) >

The total sum of Squares can be split into the sum of squares due to regression + sum of squares due to error (residuals).

SS(Total) = Sum [(y(ij)-y(bar))^2]

SS(Regression) = Sum [(y(i bar) - y(bar))^2]

SS(Residual) = Sum [(y(ij) - y(i bar))^2]

where y(ij) is the jth observation under the ith category of (X), here the categories are different age types - 5.0, 1.0, etc.

y(i bar) is the mean of the j observations in the ith category of age

and y(bar) is the overall mean of all the observations. The above split is also shown in the regression table below.

The MS are derived as SS/df where df is determined according to the following rule:

df (Regression) = no. of regressor variables - 1

df(total) = total number of observations - 1

df(Residual) = df(total) - df(Regression)

and lastly F(statistic) is calculated as MS(Regression) / MS(Residual) which ollows F-distribution with

( df(regression),df(residual) ) degrees of freedom. F-significant is the value of the same F-distribution at 1% level of signifiacnce derived from Biometrika table.

Since the F_statistic value > F-significant value so we reject the underlyinh null hypotheses of Regression / ANOVA that the Age has no effect on the cost of maintanence. Hence from the data we conclude that the age of tractor has linear effect on the cost of the maintanence.

ANOVA df SS MS F Significance F Regression 1 1099634.977 1099634.977 13.68378013 0.002142775 Residual 15 1205407.023 80360.46817 Total 16 2305042

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