Problem 16 - Algebra 2 and Trigonometry

Question

Steve kept a record of the height of a tree that he planted. The heights are shown in the table. $$ \begin{array}{|l|l|l|l|l|l|l|}\hline \text { Age of Tree in Years } & {1} & {3} & {5} & {7} & {9} & {11} & {13} \\ \hline \text { Height in lnches } & {7} & {12} & {15} & {16.5} & {17.8} & {19} & {20} \\ \hline\end{array} $$ a. Write an equation that best fits the data. b. What was the height of the tree after 2 years? c. If the height of the tree continues in this same pattern, how tall will the tree be after 20 years?

Step-by-Step Solution

Answer

a. Equation: \( y = 1.37x + 6.57 \). b. 9.31 inches after 2 years. c. 34.97 inches after 20 years.

Step 1: Interpret the Data

We are given the age of the tree in years as independent variable values \( x = [1, 3, 5, 7, 9, 11, 13] \) and the corresponding heights in inches as dependent variable values \( y = [7, 12, 15, 16.5, 17.8, 19, 20] \). We need to find an equation that best fits this data.

Step 2: Choose the Type of Regression

Given the data characteristics, a linear regression seems appropriate as the height appears to increase steadily over time. We assume a linear relationship of the form \( y = mx + b \). We'll calculate the slope \( m \) and the intercept \( b \) using the least squares method.

Step 3: Calculate the Slope (m)

Using the formula for slope in regression \( m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2} \), we calculate:- \( \sum xy = 1*7 + 3*12 + 5*15 + 7*16.5 + 9*17.8 + 11*19 + 13*20 = 880.7 \)- \( \sum x = 1 + 3 + 5 + 7 + 9 + 11 + 13 = 49 \)- \( \sum y = 7 + 12 + 15 + 16.5 + 17.8 + 19 + 20 = 107.3 \)- \( \sum x^2 = 1^2 + 3^2 + 5^2 + 7^2 + 9^2 + 11^2 + 13^2 = 369 \)- \( n = 7 \)So, \( m = \frac{7(880.7) - (49)(107.3)}{7(369) - (49)^2} \approx 1.37 \).

Step 4: Calculate the Intercept (b)

We use the intercept formula \( b = \frac{(\sum y) - m(\sum x)}{n} \).Plugging in the values, \( b = \frac{107.3 - 1.37 * 49}{7} \approx 6.57 \).

Step 5: Write the Best Fit Equation

The best fit equation for the data is \[ y = 1.37x + 6.57 \]

Step 6: Calculate Height After 2 Years

Using the equation \( y = 1.37x + 6.57 \), substitute \( x = 2 \) to find the height after 2 years.\[ y = 1.37(2) + 6.57 = 9.31 \]So, the height after 2 years is approximately 9.31 inches.

Step 7: Calculate Height After 20 Years

Using the equation \( y = 1.37x + 6.57 \), substitute \( x = 20 \) to find the expected height after 20 years.\[ y = 1.37(20) + 6.57 = 34.97 \]So, the height after 20 years is approximately 34.97 inches.

Key Concepts

Slope Calculation

When dealing with linear regression, calculating the slope ( m) of the line is crucial because it indicates the rate at which the dependent variable changes with respect to the independent variable. In this context, the independent variable is the age of the tree, and the dependent variable is its height.
To find the slope, we use the formula:\[ m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2} \]Here:

• n is the number of data pairs.
• \(\sum xy\) is the sum of the product of the independent and dependent values.
• \(\sum x\) and \(\sum y\) are the sums of the independent and dependent values, respectively.
• \(\sum x^2\) is the sum of the squares of the independent values.

Using these values, we achieve the slope as approximately 1.37, suggesting that for each year the tree's height increases by approximately 1.37 inches.
This slope indicates a fairly steady growth pattern of the tree.

Intercept Determination

Once we have the slope, the next step is to determine the y-intercept, denoted as \(b\). The intercept is the value of the dependent variable when the independent variable equals zero. In simpler terms, it is the initial height of the tree when it was planted.
We calculate the intercept using the following formula:\[ b = \frac{\sum y - m(\sum x)}{n} \]Here:

• \(\sum y\) is the total sum of the heights (dependent variable).
• m is the slope we calculated earlier.
• \(\sum x\) is the total sum of the ages (independent variable).
• n remains the number of observations.

By inserting our calculated slope and data sums, we find the intercept \(b\) to be approximately 6.57.
This means, theoretically, the tree started at a height of 6.57 inches.

Least Squares Method

The least squares method is a standard approach in statistics for finding the best-fit line through a set of data points. It minimizes the sum of the squares of the vertical distances of the points from the line, ensuring the most accurate representation of the data.
The idea is simple but powerful. By reducing these squared differences, we find the most optically pleasing and mathematically correct line that generalizes the model beyond the given data points.
Using linear regression, this method involves:

• Calculating the slope and intercept using specific standard formulas, which we applied earlier.
• Aligning these values to form the equation \(y = mx + b\).

For Steve's tree data, after applying the least squares method, we found the equation of best fit to be \(y = 1.37x + 6.57\). This showcases the efficiency of the least squares method in yielding reliable predictions based on clear past trends. It's why this technique is so widely used in predictive analysis.