Problem 57 - Precalculus with Limits

Question

Data Analysis An agricultural scientist used four test plots to determine the relationship between wheat yield \(y\) (in bushels per acre) and the amount of fertilizer \(x\) (in hundreds of pounds per acre). The table shows the results. $$ \begin{array}{|c|c|}\hline \text { Fertilizer, } x & {\text { Yield, } y} \\\ \hline 1.0 & {32} \\ \hline 1.5 & {41} \\ \hline 2.0 & {48} \\ \hline 2.5 & {53} \\ \hline\end{array} $$ (a) Find the least squares regression line \(y=a x+b\) for the data by solving the system for \(a\) and \(b .\) $$ \left\\{\begin{array}{l}{4 b+7.0 a=174} \\ {7 b+13.5 a=322}\end{array}\right. $$ (b) Use the linear model from part (a) to estimate the yield for a fertilizer application of 160 pounds per acre.

Step-by-Step Solution

Answer

The least squares regression line is \(y = 14x + 19\) and the yield for a fertilizer application of 160 pounds per acre is 42.4 bushels.

Step 1: Solve the system of equations

Let's start by solving the given system of equations to find 'a' and 'b' in the equation \(y = ax + b\). The system of equation are: \n 1) \(4b + 7.0a = 174\) \n 2) \(7b + 13.5a = 322\). By substituting the value of 'b' from equation 1 to equation 2, we can solve for 'a'.

Step 2: Find the coefficient a in the regression line

Let's multiply the first equation by 7 and the second equation by 4 and subtract the resulting second equation from the first:\n \[28b + 49a = 1218;\]\[28b + 54a = 1288.\]\n Subtracting these equations yields:\[0b - 5a = -70,\] thus, \(a = 14\).

Step 3: Find the intercept b in the regression line

We can solve for 'b' by substituting 'a = 14' in equation 1. Meaning,\[4b + 7.0*14 = 174.\] That simplifies to \[4b + 98 = 174,\] which results in \(b = 19\).

Step 4: Apply the linear model to determine the yield

Now we can apply the linear regression model \(y = 14x + 19\) to predict the yield for a fertilizer application of 160 pounds per acre. Keep in mind that 'x' was in hundreds of pounds per acre, so 'x' should be \(160/100 = 1.6\). Substituting 'x' into the equation, we get \(y = 14*1.6 + 19 = 42.4\) bushels.

Key Concepts

linear model

A linear model is a fundamental concept used in statistics to describe and predict relationships between two variables in a straight-line graph. This model is represented by a linear equation, typically written as \( y = ax + b \), where \( y \) is the dependent variable, \( x \) is the independent variable, \( a \) is the slope of the line, and \( b \) is the y-intercept.

Understanding the slope \( a \) and intercept \( b \) is crucial. The slope \( a \) indicates how much \( y \) changes for a unit change in \( x \). If \( a \) is positive, \( y \) increases as \( x \) increases. If negative, the opposite occurs. The intercept \( b \) is the value of \( y \) when \( x = 0 \).
This model assumes a linear relationship between the variables and is used in various fields to make predictions.

In this context, we use the linear model to predict wheat yield based on fertilizer applied, illustrating the utility of linear relationships in real-world applications.

agricultural data analysis

Agricultural data analysis involves leveraging statistical methods to interpret data specific to agriculture, improving decision-making processes like crop yield predictions. By analyzing patterns and relationships within these datasets, scientists can enhance agricultural efficiency.

For example, in our case, the problem presents data about the relationship between fertilizer applied and wheat yield. This is typical of agricultural data analysis, identifying key variables affecting production.

The role of linear regression in this context is to establish a model that explains how yield varies with changes in fertilizer amount. Such insights enable prediction of yields for specific fertilizer amounts that were not directly observed, helping farmers to optimize input usage and maximize crop output.

This model not only aids in understanding the current data but also paves the way for future agricultural advancements through more informed data-driven strategies.

system of equations

A system of equations consists of multiple equations that share the same set of variables. Solving these equations allows us to find values of variables that satisfy all the equations simultaneously.

In the context of linear regression, as shown in our problem, we use a system of equations to determine the coefficients \( a \) and \( b \) for the regression line \( y = ax + b \).

For example, the equations \( 4b + 7.0a = 174 \) and \( 7b + 13.5a = 322 \) come from manipulating the data points to ensure the best fit line minimizes error. We solve these using substitution or elimination methods, which test varying combinations to satisfy both equations.

Here, solving this system reveals \( a = 14 \) and \( b = 19 \), giving us the equation to predict yields based on fertilizer used.
Understanding systems of equations is pivotal in extracting meaningful insights from data, enabling the creation of predictive models essential in fields like agricultural science.