Chapter 7.2, Problem 85E

a.

To determine

To find: the least regression line.

Answer to Problem 85E

The least squares regression line is $y=14x+19$ .

Explanation of Solution

Given:

Since, $y$ represents wheat yield in bushels per acre and $x$ represents amount of fertilizer in hundreds of pounds per acre.

The result can be arranged in table format as shown below:

$x$	$y$
$1.0$	$32$
$1.5$	$41$
$2.0$	$48$
$2.5$	$53$

Calculation:

The system of linear equations for getting values of $a$ and $b$ :

  $4b+7a=174$  ...(1)

  $7b+13.5a=322$  ...(2)

Multiply $7$ and $4$ in equation (1) and (2) respectively:

  $28b+49a=1218$  ...(3)

  $28b+54a=1288$  ...(4)

Subtracting equation (3) from equation (4):

  $\begin{array}{l}28b+54a-28b-49a=1288-1218\\ 5a=70\\ a=14\end{array}$

Put the value of $a$ in equation (1) to get the value of $b$ :

  $\begin{array}{l}4b+\left(7\times 14\right)=174\\ 4b=174-98\\ 4b=76\\ b=19\end{array}$

Therefore, the value of $a$ and $b$ by elimination method are $14$ and $19$ respectively.

Calculate the least squares regression line.

  $\begin{array}{l}y=ax+b\\ y=14x+19\end{array}$

b.

To determine

To confirm:the result by using a regression feature of graphing utility.

Answer to Problem 85E

The least squares regression line by graphing utility is $y=14x+19$ .

Explanation of Solution

Given:

Since, $y$ represents wheat yield in bushels per acre and $x$ represents amount of fertilizer in hundreds of pounds per acre.

The result can be arranged in table format as shown below:

$x$	$y$
$1.0$	$32$
$1.5$	$41$
$2.0$	$48$
$2.5$	$53$

Calculation:

Use regression feature of a graphing utility to confirm the result.

  

Therefore,the least squares regression line by graphing utility is $y=14x+19$ .

c.

To determine

To plot :the given data and graph the linear model .

Answer to Problem 85E

Plotted the given data and graphed the linear model.

Explanation of Solution

Given:

Since, $y$ represents wheat yield in bushels per acre and $x$ represents amount of fertilizer in hundreds of pounds per acre.

The result can be arranged in table format as shown below:

$x$	$y$
$1.0$	$32$
$1.5$	$41$
$2.0$	$48$
$2.5$	$53$

Plotting data on graph:

  

d.

To determine

To predict:the yield for a fertilizer application of $160\text{ pounds per acre}$ .

Answer to Problem 85E

The wheat yield for a for a fertilizer of $1.6\text{ hundreds of pounds per acre}$

is $41.4\text{ bushels per acre}$ .

Explanation of Solution

Given:

Since, $y$ represents wheat yield in bushels per acre and $x$ represents amount of fertilizer in hundreds of pounds per acre.

The result can be arranged in table format as shown below:

$x$	$y$
$1.0$	$32$
$1.5$	$41$
$2.0$	$48$
$2.5$	$53$

Amount of fertilizer

  $\begin{array}{l}=\frac{160}{100}\\ =1.6\text{ hundreds of pounds per acre}\end{array}$

Calculation:

To predict the yield for a fertilizer of $1.6\text{ hundreds of pounds per acre}$ , use the least square regression line that is

  $y=14x+19$  ...(1)

Here, $x=1.6$

Putting the value of $x$ in equation (1):

  $\begin{array}{l}y=\left(14\times 1.6\right)+19\\ y=22.4+19\\ y=41.4\end{array}$

Therefore, the wheat yield for a for a fertilizer of $1.6\text{ hundreds of pounds per acre}$

is $41.4\text{ bushels per acre}$ .