Data Analysis An agricultural scientist used four test plots...

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.

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.

Key Concepts

Unit Conversion and Scaling

In data analysis, careful attention to units and scaling is crucial, especially when variables are recorded in non-standard units or scaled values. Converting and appropriately handling these units ensures that predictions and interpretations are accurate and consistent across different measurements.

Prediction Using Linear Models

A linear model established through regression allows for the estimation of the response variable based on new values of the predictor variable. This predictive capability is central in data analysis for making informed decisions and projections based on observed trends.

System of Linear Equations

When applying the least squares method, a system of linear equations—often called the normal equations—is derived by setting the partial derivatives of the sum of squared errors with respect to each parameter equal to zero. Solving these equations provides the regression coefficients needed to define the best-fit line.

Least Squares Regression

Least squares regression is a statistical technique used to determine the line that best fits a set of data points by minimizing the sum of the squared differences between the observed values and those predicted by the linear model. This method is fundamental in identifying trends and making predictions based on the relationship between two variables.