91Ó°ÊÓ

$\begin{array}{|lll|}\hline \text{Mammal}& \text{Heart weight(grams)}& \text{Length of left ventricle cavity}\\ \text{Mouse}& 0.13& 0.55\\ \text{Rat}& 0.64& 1\\ \text{Rabbit}& 5.8& 2.2\\ \text{Dog}& 102& 4\\ \text{Sheep}& 210& 6.5\\ \text{Ox}& 2030& 12\\ \text{Horse}& 3900& 16\\ \hline\end{array}$

The horizontal dimension is length, and the vertical axis represents the weight of the heart.

The scatterplot is swinging higher, hence the direction is positive.

Because there is no straight line connecting the ends, the form is curved.

Because all spots in the same pattern are so close together, the strength is solid.

$\begin{array}{|lll|}\hline \text{Mammal}& \text{Heart weight(grams)}& \text{Length of left ventricle cavity}\\ \text{Mouse}& 0.13& 0.55\\ \text{Rat}& 0.64& 1\\ \text{Rabbit}& 5.8& 2.2\\ \text{Dog}& 102& 4\\ \text{Sheep}& 210& 6.5\\ \text{Ox}& 2030& 12\\ \text{Horse}& 3900& 16\\ \hline\end{array}$

Exponential Model

Power Model

The length logarithm is in the horizontal axis in the Exponential Model, and the heart weight logarithm is in the y axis in the Power Model, and the length logarithm is in the horizontal axis in the Power Model.

The Power Model is a model for the interaction of variables, and the related scatterplot has a more linear structure.

$\begin{array}{|lll|}\hline \text{Mammal}& \text{Heart weight(grams)}& \text{Length of left ventricle cavity}\\ \text{Mouse}& 0.13& 0.55\\ \text{Rat}& 0.64& 1\\ \text{Rabbit}& 5.8& 2.2\\ \text{Dog}& 102& 4\\ \text{Sheep}& 210& 6.5\\ \text{Ox}& 2030& 12\\ \text{Horse}& 3900& 16\\ \hline\end{array}$

The length of the left ventricular cavity (X) and the weight of the heart (grams) are taken as logarithmic values.

$\begin{array}{|rrll|}\hline \text{Length of left ventricle cavity (X)}& \text{Heart weight(grams) (Y)}& \backslash \mathrm{multicolumn}1\left|l\right|\text{X.X}& \backslash \mathrm{multicolumn}1l|\text{XY}\\ -0.597837001& -2.040220829& 0.357409& 1.21972\\ 0.78845736& -0.446287103& 0& 0\\ 1.386294361& 1.757857918& 0.621665& 1.385996\\ 1.871802177& 4.624972813& 1.921812& 6.411574\\ 2.48490665& 5.347107531& 3.503643& 10.00873\\ 2.772588722& 7.615791072& 6.174761& 18.92453\\ X=8.70621227& ∑Y=25.12795323& =20.26654& ∑X^2=60.87634\\ \hline\end{array}$

$a=\frac{∑Y·∑X^2-∑X·∑XY}{n∑X^2-{\left(∑X\right)}^2}=-0.314$

$b=\frac{n∑XY-∑X·∑Y}{n∑X^2-{\left(∑X\right)}^2}=3.139$

role="math" localid="1652862916129" $$\begin{gathered} y=a+bx \\ =-0.314+3.139x \end{gathered}$$

From part(c)

$\ln \hat{y}=-0.314+3.139\ln x$

Using the value of $x=6.8$in the component equation (c)

$$\begin{gathered} \ln \hat{y}=-0.314+3.139\ln x \\ =-0.314+3.139\ln (6.8) \\ =5.70322 \end{gathered}$$

We take the Exponential of above equation

$$\begin{gathered} \hat{y}=e^{5.70322} \\ =299.831 \end{gathered}$$