# You are helping a timber harvest company to predict the volume of t...

## Question

You are helping a timber harvest company to predict the volume of timber (which is directly related to profit) that can be harvested from different areas. Volume per hectare is a difficult quantity to measure and can only be determined after the area has been harvested (destructively). It is easy to get information on some others variables, so your goal is to develop a model to predict volume per hectare from these “easy to obtain” variables. You are able to use data from areas that have previously been harvested, and for which you have the volume per hectare measurement (n = 26).

The variables you have information about are the following:

Volume of timber per hectare (response variable) in m3 / ha – volha

Average height of trees (m) - topht
Average diameter at breast height (cm) – dbh
Stems per hectare (number of trees per hectare) – stemsha
Basal area per hectare (obtained from dbh and stems per hectare) m2 / ha – baha
Average age of trees – age
1. Create a linear model that includes baha, topht and the interaction between the two (Interaction Model).
Using this equation, calculate the predicted volume per hectare for Area A with baha = 30 and topht = 24. Calculate the predicted volume per hectare for Area B with baha = 37 and topht = 20. Use R to obtain the 95% prediction intervals for these point estimates.

Give all responses to two decimal places in the following format (do not include any units, spaces, or commas as separators):

predicted volume (lower limit of prediction interval, upper limit of prediction interval)
2. Create a linear model that includes log baha and log topht, and uses log volha as the response variable (Log model). Why does this model make sense based on the type of data?
Using R, convert the predicted values into their original units. Calculate the SSY, and SSE in original units. Calculate the pseudo-R2. Answer as a proportion to four decimal places.
Calculate the residual standard error based on the SSE and SSY in original units. Answer to two decimal places.
In log units, calculate the 95% confidence intervals for the co-efficients. Give answers to three decimal places.
What value would these co-efficients have in the model if tree volume could be modeled perfectly with the equation for volume of a cone? Do the confidence intervals include these values? Explain in 3-6 sentences
Using this equation, calculate the predicted volume per hectare for Area A with baha = 30 and topht = 24. Calculate the predicted volume per hectare for Area B with baha = 37 and topht = 20. Use R to obtain the prediction intervals for these point estimates. Remember to backtransform these values into the original units.

Give all responses to two decimal places in the following format (do not include any units, spaces, or commas as separators):

predicted volume (lower limit of prediction interval, upper limit of prediction interval)
Test the significance of log baha using a partial F-test and alpha = 0.05
Test the significance of log topht using a partial F-test and alpha = 0.05.

## Solution Preview

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model1 <- lm (volha~baha+topht+baha:topht, data = data)
new.dat<-data.frame(baha = 30, topht = 24)
new.dat
predict(model1, newdata = new.dat, interval = 'prediction')
new.dat1<-data.frame(baha = 37, topht = 20)
predict(model1, newdata = new.dat1, interval = 'prediction')
R Output is given below:
> summary (model1)
Call:
lm(formula = volha ~ baha + topht + baha:topht, data = data)

Residuals:
Min, 1Q, Median, 3Q, Max
-34.24 -13.00   1.74 12.30 29.74...

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