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. (table content: diameter, x: 8.3, 10.5, 11, 12, 12.9, 14, 16.3, 17.3, 17.9, 18; height, y: 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, indicating that it drop - down a good model of the data.

Explanation:

Step1: Calculate mean of x

$\bar{x} = \frac{8.3+10.5+11+12+12.9+14+16.3+17.3+17.9+18}{10} = 13.82$

Step2: Calculate mean of y

$\bar{y} = \frac{70+72+75+75+74+78+77+81+80+81}{10} = 76.3$

Step3: Calculate slope (m)

First compute numerator:
$\sum_{i=1}^{10}(x_i-\bar{x})(y_i-\bar{y}) = (8.3-13.82)(70-76.3)+(10.5-13.82)(72-76.3)+...+(18-13.82)(81-76.3) = 132.676$
Denominator:
$\sum_{i=1}^{10}(x_i-\bar{x})^2 = (8.3-13.82)^2+(10.5-13.82)^2+...+(18-13.82)^2 = 116.296$
$m = \frac{132.676}{116.296} \approx 1.14$

Step4: Calculate y-intercept (b)

$b = \bar{y} - m\bar{x} = 76.3 - 1.14\times13.82 \approx 60.5$

Step5: Predict height at x=22

$y = 1.14\times22 + 60.5 = 25.08 + 60.5 = 85.58 \approx 86$

Step6: Interpret correlation coefficient

A coefficient of 0.95 is close to 1, so it is a good model.

Answer:

The data is best modeled by the function $y = 1.14x + 60.5$
Based on the linear model, the approximate height of a tree with a diameter of 22 inches is 86 feet.
The correlation coefficient for this model is 0.95, indicating that it is a good model of the data.