Four test plots were used to explore the relationship...

Four test plots were used to explore the relationship between wheat yield (in bushels per acre) and the amount of fertilizer applied (in hundreds of pounds per acre). The results are shown in the table. (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 vicwing window. (d) Use the linear model from part (a) to predict the yield for a fertilizer application of 160 pounds per acre.

Four test plots were used to explore the relationship between wheat yield (in bushels per acre) and the amount of fertilizer applied (in hundreds of pounds per acre). The results are shown in the table.
(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 vicwing window.
(d) Use the linear model from part (a) to predict the yield for a fertilizer application of 160 pounds per acre.

Key Concepts

Graphing Utility

Graphing utilities are software tools that enable users to create visual representations of data and mathematical models. In regression analysis, these tools are essential for plotting data points, fitting a regression line, and validating the model's performance by visually comparing the model with the actual data.

Prediction

Prediction in the context of a linear regression model involves using the derived linear equation to calculate the expected value of the dependent variable for a given value of the independent variable. This is a key application of regression analysis, allowing for estimation and forecasting based on established trends.

Linear Model

A linear model describes the relationship between two variables using a straight line, typically expressed in the form y = ax + b. This model is fundamental in many areas of science and engineering because of its simplicity and the ease with which it can be interpreted and used for short-term predictions and trend analysis.

Least Squares Regression

Least squares regression is a statistical method used to determine the best-fitting line through a set of data points by minimizing the sum of the squares of the vertical distances (errors) between the observed values and the values predicted by the line. It provides a systematic approach for modeling the relationship between an independent variable and a dependent variable.

System of Linear Equations

A system of linear equations consists of two or more linear equations that share the same variables. Solving such a system involves finding the values of the variables that satisfy all equations simultaneously, which is crucial in regression analysis when determining the parameters of the best-fitting line.

Transcript

00:02 So in this problem, we're given some data that were found on the relationship between fertilizer being put on some acreage and the yield of wheat on the acreage.

00:16 And in the first part, they've done the least squares regression for a line through the data of this form, y equals a -x plus b.

00:27 And on that least squares regression, we have come up with these formulas for a and b equals 174, and 7b plus 13 .5a equals 322.

00:50 Okay.

00:53 So on this data then, call this equation 1 and this equation 2.

01:05 Draw my little line just to separate my data here for a minute.

01:09 So i need to put these together so that one of the variables cancels out, right? well, it might be easiest to go minus 7 times equation 1.

01:21 So that's minus 28b plus.

01:25 It won't be plus, will it? it'll be minus 49a equals 174.

01:42 Times negative 7.

01:45 So that's a negative 1218.

01:50 And then do four times equation 2.

01:56 So that's 28b plus 13 and a half times 4 is 54a equals 322 times 4.