Problem 9 - A Modern Introduction to Probability and Statistics: Understanding Why and How

Question

The volume (i.e., the effective wood production in cubic meters), height (in meters), and diameter (in meters) (measured at \(1.37\) meter above the ground) are recorded for 31 black cherry trees in the Allegheny National Forest in Pennsylvania. The data are listed in Table 17.3. They were collected to find an estimate for the volume of a tree (and therefore for the timber yield), given its height and diameter. For each tree the volume \(y\) and the value of \(x=d^{2} h\) are recorded, where \(d\) and \(h\) are the diameter and height of the tree. The resulting points \(\left(x_{1}, y_{1}\right), \ldots,\left(x_{31}, y_{31}\right)\) are displayed in the scatterplot in Figure 17.13. We model the data by the following linear regression model (without intercept) $$ Y_{i}=\beta x_{i}+U_{i} $$ for \(i=1,2, \ldots, 31\). a. What physical reasons justify the linear relationship between \(y\) and \(d^{2} h\) ? Hint: how does the volume of a cylinder relate to its diameter and height? b. We want to find an estimate for the slope \(\beta\) of the line \(y=\beta x\). Two natural candidates are the average slope \(\bar{z}_{n}\), where \(z_{i}=y_{i} / x_{i}\), and the slope of the averages \(\bar{y} / \bar{x}\). In Chapter 22 we will encounter the so-called least squares estimate: $$ \frac{\sum_{i=1}^{n} x_{i} y_{i}}{\sum_{i=1}^{n} x_{i}^{2}} $$slope of the averages \(\bar{y} / \bar{x}\). In Chapter 22 we will encounter the so-called least squares estimate: $$ \frac{\sum_{i=1}^{n} x_{i} y_{i}}{\sum_{i=1}^{n} x_{i}^{2}} $$ Compute all three estimates for the data in Table 17.3. You need at least 5 digits accuracy, and you may use that \(\sum x_{i}=87.456, \sum y_{i}=26.486\), \(\sum y_{i} / x_{i}=9.369, \sum x_{i} y_{i}=95.498\), and \(\sum x_{i}^{2}=314.644\).

Step-by-Step Solution

Answer

The estimates for the slope \( \beta \) are 0.30223, 0.30286, and 0.30347.

Step 1: Understanding the Physical Relationship

The volume of a cylinder is calculated using the formula \( V = \pi r^2 h \). When considering trees, the diameter \( d \) is analogous to \( 2r \), so \( r = \frac{d}{2} \). Therefore, \( r^2 = \frac{d^2}{4} \), which leads to \( V = \pi \frac{d^2}{4} h \). Since the volume \( y \) is proportional to \( d^2 h \), it justifies the linear relationship \( y = \beta d^2 h \). Thus, \( x = d^2 h \) serves as a suitable predictor for volume \( y \).

Step 2: Calculate the Average Slope \( \bar{z}_n \)

The average slope is calculated as follows: \(\bar{z}_n = \frac{1}{n} \sum_{i=1}^{n} \left(\frac{y_i}{x_i}\right)\). Given that \( \sum y_i / x_i = 9.369 \), \(\bar{z}_n = \frac{9.369}{31} \approx 0.30223 \).

Step 3: Calculate the Slope of the Averages \( \bar{y} / \bar{x} \)

The slope of the averages is calculated using the formula: \( \frac{\bar{y}}{\bar{x}} = \frac{\sum y_i / n}{\sum x_i / n} \). So, \(\frac{\sum y_i}{\sum x_i} = \frac{26.486}{87.456} \approx 0.30286 \).

Step 4: Calculate the Least Squares Estimate

The least squares estimate is calculated using: \( \frac{\sum x_i y_i}{\sum x_i^2} \). Therefore, \(\frac{95.498}{314.644} \approx 0.30347 \).

Step 5: Conclusion

The calculations provide three estimates for the slope \( \beta \):1. Average slope \( \bar{z}_n \): 0.302232. Slope of the averages \( \bar{y} / \bar{x} \): 0.302863. Least squares estimate: 0.30347

Key Concepts

cylinder volume

Understanding the volume of a cylinder is crucial when analyzing the data for black cherry trees. The volume formula is given by \( V = \pi r^2 h \), where \( V \) is the volume, \( r \) is the radius, and \( h \) is the height. For trees, the diameter \( d \) is measured, not the radius directly. To relate diameter to the formula, remember that the radius \( r \) is half of the diameter, so \( r = \frac{d}{2} \). This makes \( r^2 = \frac{d^2}{4} \). When substituting this into the volume formula, we find:

• \( V = \pi \frac{d^2}{4} h \)

This indicates that the tree volume \( y \) is proportional to \( d^2 h \), allowing it to serve as a good predictor in a linear regression model. This relationship justifies using \( x = d^2 h \) to predict the timber yield, closely tying physical tree properties to mathematical modeling.

linear relationship

A linear relationship indicates that two variables change at a constant rate with respect to one another. In the context of our exercise, the volume of the tree \( y \) has a linear relationship with the variable \( x = d^2 h \). A linear regression model without an intercept takes the form:

• \( Y = \beta x + U \)

Here, \( \beta \) is the slope, and \( U \) is the error term. The absence of an intercept means the line passes through the origin. The linear relationship exists because the formula for volume, derived from the characteristics of a cylinder, makes the variables \( y \) and \( d^2 h \) directly proportional. This means you can predict tree volume from \( d^2 h \), making this regression useful for estimating timber yield.

least squares estimate

The least squares estimate is a powerful tool in linear regression to determine the best-fitting line through data points. Its main goal is to minimize the sum of the squares of the differences between observed values and the values predicted by the model. The formula for computing the least squares estimate of the slope \( \beta \) is:

• \( \beta = \frac{\sum_{i=1}^{n} x_{i} y_{i}}{\sum_{i=1}^{n} x_{i}^{2}} \)

By plugging in the given data, where \( \sum x_{i} y_{i} = 95.498 \) and \( \sum x_{i}^{2} = 314.644 \), you achieve an estimate of approximately 0.30347. This method ensures that our slope estimate minimizes prediction errors, providing a clear advantage over simpler methods like average slope calculations.

slope calculation

Calculating the slope \( \beta \) in linear regression models is essential for understanding the relationship between variables. There are several ways to estimate slope:

• Average Slope (\( \bar{z}_n \)): This is calculated by averaging the ratios \( \frac{y_i}{x_i} \) for each data point. For our dataset, it equals \( \frac{9.369}{31} \approx 0.30223 \).
• Slope of the Averages: Obtained by dividing the average of \( y \) values by the average of \( x \) values, \( \frac{26.486}{87.456} \approx 0.30286 \).
• Least Squares Slope: Calculated as \( \frac{\sum x_i y_i}{\sum x_i^2} \), which is approximately 0.30347 in our case.

Each method provides a unique perspective. The least squares approach is often preferred for its statistical rigor, minimizing errors more effectively than other methods. Understanding these variations in slope calculation helps in selecting the most suitable method for data analysis and interpretation.