Problem 36 - Thomas Calculus

Question

Ponderosa pines In the table, \(x\) represents the girth (distance around) of a pine tree measured in inches (in.) at shoulder height; \(y\) represents the board feet (bf) of lumber finally obtained. (GRAPH NOT COPY) Formulate and test the following two models: that usable board feet is proportional to (a) the square of the girth and (b) the cube of the girth. Does one model provide a better "explanation" than the other?

Step-by-Step Solution

Answer

Compare the \(R^2\) values of both models after fitting regression; the model with the higher \(R^2\) offers a better explanation.

Step 1: Define the Models

We need to test two different models. Model (a) assumes usable board feet is proportional to the square of the girth: \( y = kx^2 \), and model (b) assumes it is proportional to the cube of the girth: \( y = mx^3 \), where \(k\) and \(m\) are proportionality constants.

Step 2: Prepare Data

Use the given data of girth \(x\) and board feet \(y\). Ensure all data is correctly listed, as it's necessary to fit both equations \(y = kx^2\) and \(y = mx^3\).

Step 3: Linearize the Models

Transform both models into linear form for easier analysis. For model (a), take logarithms: \( \log(y) = \log(k) + 2\log(x)\). For model (b), \( \log(y) = \log(m) + 3\log(x)\).

Step 4: Fit the Linear Models Using Regression

Using the data, regress \( \log(y) \) on \( \log(x) \) for both models to find the best fit lines. For model (a), find the line with slope 2, and for model (b), find the line with slope 3. Calculate both \(k\) and \(m\) using the intercept from the regression.

Step 5: Calculate Goodness of Fit

For each model, calculate the coefficient of determination \(R^2\) from the regression analysis to determine how well the models fit the data. \(R^2\) measures the proportion of variance explained by the models.

Step 6: Compare Models

Compare the \(R^2\) values obtained for both models to determine which model provides a better fit. The model with the higher \(R^2\) value explains more variation in the data and is considered a better fit.

Key Concepts

Understanding Proportionality

Proportionality is a fundamental concept in mathematics. It describes a relationship where one quantity changes in direct relation to another. In the context of the given exercise, we have two models. One suggests that usable board feet is proportional to the square of the girth, and the other suggests proportionality to the cube of the girth. This means:

• In model (a), for every increase in the girth of the pine tree, the board feet increase at a rate proportional to the square of the girth.
• In model (b), the increase is proportional to the cube of the girth.

Such proportional relationships allow us to predict how one variable will change as another variable changes. Understanding proportionality helps in building mathematical models that accurately represent real-world phenomena.

Diving into Linear Regression

Linear regression is a statistical method used to model the relationship between two variables by fitting a linear equation to the observed data. In our exercise, we aim to test whether the board feet from pine trees fit a model that is linear concerning the variables we are testing:

• For the squared girth, we linearize the model by applying a logarithmic transformation which results in a straight-line equation.
• Similarly, for the cubic girth, we perform the same transformation to fit a linear model.

By plotting the logarithms of these models, we can apply linear regression to find the line of best fit. This process helps in determining the constants that define our proportionality and understanding how well these models explain the observed data.

Evaluating Goodness of Fit

Goodness of fit is a statistical measure that describes how well our model reflects the observed data. In simple terms, it tells us how closely our model's predictions align with the actual data points. In this exercise:

• We measure the goodness of fit using the coefficient of determination, often denoted as \( R^2 \).
• An \( R^2 \) value closer to 1 indicates a perfect fit, meaning the model explains most of the variance in the data.

By calculating the \( R^2 \) for each linear model developed (squared or cubed), we can assess which model better fits the observed data, thereby providing insight into the most realistic representation of the relationship between girth and board feet.

Exploring the Coefficient of Determination

The coefficient of determination, denoted \( R^2 \), is a key statistical tool in assessing the fit of a model. It ranges from 0 to 1:

• An \( R^2 \) value of 1 suggests a perfectly fitting model, where all variations in the data are explained by the model.
• An \( R^2 \) value near 0 implies that the model fails to explain the variation in the data.

In our context, once we calculate \( R^2 \) for both squared and cubed models, we compare them:

• The higher \( R^2 \) value indicates a better fit, meaning it provides a more accurate explanation of how the girth relates to board feet.
• This comparison guides us in determining whether the board feet are better explained as proportional to the square or cube of the girth.

Understanding \( R^2 \) helps in evaluating the strength of different predictive models.