Stats 110B, Spring 99 Quiz 6
NAME:
Below is output from a regression studying the relation between the volume of lumber (cubic feet) provided by trees based on the height (feet) and diameter (inches).
Data set = Trees, Name of Fit = L1
Normal Regression
Kernel mean function = Identity
Response = Vol
Terms = (D Ht)
Coefficient Estimates
Label Estimate Std. Error t-value
Constant -57.9877 8.63823 -6.713
D 4.70816 0.264265 17.816
Ht 0.339251 0.130151 2.607
R Squared: 0.94795
Sigma hat: 3.88183
Number of cases: 31
Degrees of freedom: 28
Summary Analysis of Variance Table
Source df SS MS F p-value
Regression 2 7684.16 3842.08 254.97 0.0000
Residual 28 421.921 15.0686
Lack of fit 26 421.716 16.2199 158.24 0.0063
Pure Error 2 0.205 0.1025
1. Interpret the parameters of the regression line for a lumberjack who wants to know which trees are the best for providing as much lumber as possible.
The regression line says that if there are two trees with the same height, the tree with the larger diameter will, on average, provide more lumber. One extra inch in diameter leads to an average increase of 4.71 cubic feet of lumber. If two trees have the same diameter, then the tallest tree will generally produce more lumber. Each additional foot in height is worth .34 cubic feet of lumber.
2. Let beta1 be the coefficient of height in the multiple regression model. The null hypothesis "beta1 = 0" has as a test statistic 2.607, and a calculation with ARC shows that the p-value is 0.0145. Interpret this for the lumberjack.
If we were to have examined a different sample of the same type of trees (and the same number of trees), we would have had slightly different values for our coefficients because of the variation in the population. It is possible that there really is NO relation, and the true value of beta1 is 0, but we just got unlucky and got a non-zero estimate (even though the true value is 0.) However, the p-value says that the probability of getting an estimate as far from zero or further as the one we got, if the true value is 0, is only 0.0145. So this happens only 1.45% of the time. For this reason, we reject the null hypothesis that beta1 is zero, and conclude that volume really does vary with height.