91Ó°ÊÓ

Given Data points:

$\begin{array}{|llll|}\hline x& y& xy& x^2\\ 57& 8& 456& 3249\\ 85& 22& 1870& 7225\\ 57& 10.5& 598.5& 3249\\ 65& 22.5& 1462.5& 4225\\ 52& 12& 624& 2704\\ 67& 11.5& 770.5& 4489\\ 62& 7.5& 465& 3844\\ 80& 13& 1040& 6400\\ 77& 16.5& 1270.5& 5929\\ 53& 21& 1113& 2809\\ 68& 12& 816& 4624\\ ∑x=723& ∑y=156.5& ∑xy=10486& ∑x{x}^2=48747\\ \hline\end{array}$

We first need to compute the $b_1$and $b_0$to find the regression equation.

The slope of the regression line is,

$$\begin{gathered} b_1=\frac{S_{xy}}{S_{xx}} \\ =\frac{∑x_i{y}_i-\left(∑x_i\right)\left(∑y_i\right)/n}{∑x_i^2-{\left(∑x_i\right)}^2/n} \\ =\frac{10486-\left(723\right)(156.5)/11}{48747-(723{)}^2/11} \\ =\frac{199.6818}{1226.182} \\ =0.162848 \\ ≈0.16 \end{gathered}$$

The y-intercept is,

$$\begin{gathered} b_0=\frac{1}{n}\left(∑y_i-b_1∑x_i\right) \\ =\frac{1}{11}(156.5-(0.162848\left)\right(723\left)\right) \\ =3.523688 \\ ≈3.52 \end{gathered}$$

So the regression equation isrole="math" localid="1652583005548" $y=3.52+0.16x$

The graph representation for the equation:

From the above graph we can see that the number of volatile compounds emitted increases as the plant weight increases.

The slope of the regression line is 0.16, which means that a unit change in the potato plant weight causes a change of 16 nanograms increases in volatile compounds emitted.

From the regression equation we have the predictor variable as potato plant weight and the response variable is the quality of volatile compounds emitted.

From the obtained regression equation we can see that all the points are near the straight line so there are no outliers in the data given.

The predicted quantity of volatile compounds emitted by the potato plant that weighs 75 grams,

We have the regression line as $\hat{y}=3.523688+0.162848x$

Substituting the value $x=75$, we get

$$\begin{gathered} \hat{y}=3.523688+0.162848x \\ =3.523688+0.162848\left(75\right) \\ ≈15.74\text{(in hundreds)} \end{gathered}$$

Therefore, the predicted quantity of volatile compounds emitted by a potato plant that weighs 75 grams is 1574 nanograms.