Problem 89 - Algebra and Trigonometry Real Mathematics, Real People

Question

Four test plots were used to explore the relationship between wheat yield \(y\) (in bushels per acre) and amount of fertilizer applied \(x\) (in hundreds of pounds per acre). The results are given by the ordered pairs \((1.0,32),(1.5,41),(2.0,48),\) and (2.5,53) (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 regression feature of a graphing utility to confirm the result in part (a). (c) Use the graphing utility to plot the data and graph the linear model from part (a) in the same viewing window. (d) Use the linear model from part (a) to predict the yield for a fertilizer application of 160 pounds per acre.

Step-by-Step Solution

Answer

The regression line for the given system of equations is \(y = 14x +19\). By using this model, the predicted yield for a fertilizer application of 160 pounds per acre is 42.4 bushels per acre.

Step 1: Solve for a and b

You can solve the system of linear equations using substitution or elimination method. \Let's use elimination. \Multiply the first equation by 7 and the second by 4 to have the coefficients of a being the same: \\[\begin{aligned}&28b + 49a = 1218\&28b + 54a = 1288\end{aligned}\]Subtract the first equation from the second: \\(5a = 70\)So, \(a = 70/5 =14 \)\Then substitute a=14 into the first equation back: \\(4b + 7*14 = 174\)So we got \(b = (174-98)/4 = 19\)

Step 2: Using a graphing utility feature

Plug values into the graphing utility tool for confirmation. Since you should provide your own graphing tool, there is no practical way to demonstrate this. However, the regression line should have the equation \(y = 14x +19\) as a result.

Step 3: Plotting the data and graphing the linear model

Using a graphing tool, plot the original data points and the linear model \(y =14x +19\). The points should closely follow the linear model since the model is made to fit those points.

Step 4: Predicting the yield based on the linear model

For predicting the yield, use x = 160/100 = 1.6 in the equation \(y = 14x +19\), giving \(y = 14 * 1.6 +19 = 42.4\) bushels per acre

Key Concepts

Linear Model

A linear model is a mathematical representation used to describe the relationship between two variables by fitting a straight line. The standard form of a linear equation is represented 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. In the context of the original exercise, the linear model describes how wheat yield varies with changes in the amount of fertilizer applied. By solving the equations for the slope \(a\) and y-intercept \(b\), we determine the best-fit line that minimizes the sum of the squared differences between the observed values and the values predicted by the model. This process is known as least squares regression. Understanding the linear model is essential, as it helps predict outcomes for values of \(x\) that haven't been directly measured.

Graphing Utility

A graphing utility is a powerful tool often used in mathematics to visualize data and verify calculated models. These tools can be physical graphing calculators, online applications, or software packages that provide a graphical representation of equations and data points. For the exercise in question, a graphing utility can display the data points of wheat yield against fertilizer amount and the least squares regression line. This visual confirmation is valuable, as it helps students observe how well the linear model aligns with the actual data. Having access to a graphing utility also allows for experimentation by altering models quickly and visualizing the impact, which aids in deeper understanding and learning.

Data Plotting

Data plotting is the process of graphically representing data sets on a coordinate plane. This involves displaying data points that show the relationship between two variables – in our case, fertilizer amount and wheat yield. Each data point is represented as an ordered pair \((x, y)\), corresponding to the amount of fertilizer applied and the resulting yield. Data plots can help identify trends, patterns, and potential outliers in data. In this exercise, plotting the data helps to initially visualize whether a linear relationship might exist. This step is crucial before applying any statistical model, as it can inform the choice of whether a linear model is suitable and guide further steps in data analysis.

Yield Prediction

Yield prediction involves using a mathematical model to estimate outcomes based on given data inputs. In agricultural contexts, such as this exercise, predicting the yield of crops like wheat based on fertilizer applied is an essential tool for optimizing resources and maximizing production efficiency. By using the linear model derived from the least squares regression, calculations can extend beyond observed data to make predictions, such as estimating the wheat yield for 160 pounds of fertilizer per acre. Here, substituting \(x = 1.6\) (since 160 pounds is 1.6 hundreds) into the equation \(y = 14x + 19\) results in a prediction of 42.4 bushels per acre. This prediction aids farmers and agribusinesses in making informed decisions about resource allocation and expected crop outcomes.