Plant Emissions. Plants emit gases that trigger the ripening of fruit, attract pollinators, and cue other physiological responses. N. Agelopolous et al. examined factors that affect the emission of volatile compounds by the potato plant Solanum numerous and published their findings in the paper "Factors Affecting Volatile Emissions of Intact Potato Plants. Solanum tuberosum: Variability of Quantities and Stability of Ratios" (Journal of Chemical Ecology, Vol. 26. No. 2, pp. 497-511). The volatile compounds analyzed were hydrocarbons used by other plants and animals. Following are data on plant weight (x), in grams, and quantity of volatile compounds emitted (y), in hundreds of nanograms, for 11 potato plants. For part (g), predict the number of volatile compounds emitted by a potato plant that weighs 75 grams.

X	57	85	57	65	52	67	62	80	77	53	68
Y	8.0	22.0	10.5	22.5	12.0	11.5	7.5	13.0	16.5	21..0	12.0

• a. fond the regression equation for the data points.
• b. graph the regression equation and the data points.
• c. describe the apparent relationship between the two variables under consideration.
• d. interpret the slope of the regression line.
• e. identify the predictor and response variables.
• f. identify outliers and potential influential observations.
• g. predict the values of the response variable for the specified values of the predictor variable, and interpret your results.

Given Data points:

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

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

The slope of the regression line is,

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

The y-intercept is,

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

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

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

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