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Chapter 14: Problem 14

If we knew the width and height of cylindrical tin cans of food, could we predict the volume of these cans with precision and accuracy? a. Give the equation that would allow us to make such predictions. b. Is the relationship between volume and its predictors, height and width, a linear one? c. Should we use an additive multiple regression model to predict a volume of a can from its height and width? Explain. d. If you were to take logarithms of each side of the equation in Part (a), would the relationship be linear?

Short Answer

Expert verified
a. The equation to predict the volume of tins is \(V = \pi r^2 h\). b. This relationship is non-linear. c. No, we should not use a multiple regression model due to the non-linear relationship. d. By taking logarithms, the equation becomes a linear relationship.

Step by step solution

01

Equation for cylindrical volume

Firstly, the equation for the volume of a cylinder is given by \[V = \pi r^2 h\] where \(V\) is the volume of the cylinder, \(r\) the radius (which is width divided by 2), and \(h\) is the height of the cylinder.
02

Checking if the relationship is linear

In the equation \[V = \pi \left(\frac{w}{2}\right)^2 h\], we see that the volume is not a linear function of the width and height. This is because width appears as a squared term, not a first-order term. Consequently, the relationship is non-linear.
03

Opining on the usage of additive multiple regression model

An additive multiple regression model might not be the best because such models are used to predict the outcome (volume, in this case) based on a linear relationship. But as we determined in the previous step, the relationship between volume and its predictors (height and width) is non-linear.
04

Effect of taking logarithm on each side of equation

If we take the logarithm of each side, the equation becomes \[\ln(V) = \ln(\pi) + 2\ln(w) + \ln(h)\]. This new transformation of the equation is a linear relationship. The dependent variable and independent variables are connected with '+', which denotes a linear relationship.

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