Question

select the correct answer from each drop - down menu. this table shows the relationship between the diameter, x, in inches, and the height, y, in feet, of trees in a national park. (the table has columns: diameter, x with values 8.3, 10.5, 11, 12, 12.9, 14, 16.3, 17.3, 17.9, 18 and height, y with values 70, 72, 75, 75, 74, 78, 77, 81, 80, 81) what linear function best models the data in this table? based on the model, what is the approximate height of a tree with a diameter of 22 inches? the data is best modeled by the function y = drop - down x + drop - down. based on the linear model, the approximate height of a tree with a diameter of 22 inches is drop - down feet. the correlation coefficient for this model is 0.95, so it drop - down a good model of the data.

Explanation:

Step1: Calculate mean of $x$ and $y$

First, find the average of diameter ($x$) values:
$$\bar{x} = \frac{8.3+10.5+11+12+12.9+14+16.3+17.3+17.9+18}{10} = \frac{138.2}{10} = 13.82$$
Find the average of height ($y$) values:
$$\bar{y} = \frac{70+72+75+75+74+78+77+81+80+81}{10} = \frac{763}{10} = 76.3$$

Step2: Calculate slope $m$

Use the formula for the slope of the least-squares line:
$$m = \frac{\sum_{i=1}^{10}(x_i-\bar{x})(y_i-\bar{y})}{\sum_{i=1}^{10}(x_i-\bar{x})^2}$$
First compute numerator:

$$\begin{align*} \sum(x_i-\bar{x})(y_i-\bar{y}) &= (8.3-13.82)(70-76.3)+(10.5-13.82)(72-76.3)+(11-13.82)(75-76.3)\\ &+(12-13.82)(75-76.3)+(12.9-13.82)(74-76.3)+(14-13.82)(78-76.3)\\ &+(16.3-13.82)(77-76.3)+(17.3-13.82)(81-76.3)+(17.9-13.82)(80-76.3)\\ &+(18-13.82)(81-76.3)\\ &= (-5.52)(-6.3)+(-3.32)(-4.3)+(-2.82)(-1.3)+(-1.82)(-1.3)\\ &+(-0.92)(-2.3)+(0.18)(1.7)+(2.48)(0.7)+(3.48)(4.7)+(4.08)(3.7)+(4.18)(4.7)\\ &=34.776+14.276+3.666+2.366+2.116+0.306+1.736+16.356+15.096+19.646\\ &=110.34 \end{align*}$$

Now compute denominator:

$$\begin{align*} \sum(x_i-\bar{x})^2 &= (8.3-13.82)^2+(10.5-13.82)^2+(11-13.82)^2+(12-13.82)^2+(12.9-13.82)^2\\ &+(14-13.82)^2+(16.3-13.82)^2+(17.3-13.82)^2+(17.9-13.82)^2+(18-13.82)^2\\ &=30.4704+11.0224+7.9524+3.3124+0.8464+0.0324+6.1504+12.1104+16.6464+17.4724\\ &=105.014 \end{align*}$$

Calculate slope:
$$m = \frac{110.34}{105.014} \approx 1$$

Step3: Calculate y-intercept $b$

Use $\bar{y}=m\bar{x}+b$, solve for $b$:
$$b = \bar{y} - m\bar{x} = 76.3 - 1\times13.82 = 62.48 \approx 62$$
So the linear function is $y = 1x + 62$.

Step4: Predict height for $x=22$

Substitute $x=22$ into the linear model:
$$y = 1\times22 + 62 = 84$$

Step5: Evaluate model fit

A correlation coefficient of $0.95$ is close to 1, so the model is a good fit.

Answer:

The data is best modeled by the function $y = 1x + 62$.
Based on the linear model, the approximate height of a tree with a diameter of 22 inches is 84 feet.
The correlation coefficient for this model is 0.95, so it is a good model of the data.