Question

question 3 of 5
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.
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 $y =$ $x +$
based on the linear model, the approximate height of a tree with a diameter of 22 inches is feet
the correlation coefficient for this model is 0.99, indicating that it a good model of the data.

Explanation:

Step1: Calculate mean of x and y

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

Step2: Calculate slope (m)

Use the formula for slope of least-squares line:
$m = \frac{\sum_{i=1}^{10}(x_i-\bar{x})(y_i-\bar{y})}{\sum_{i=1}^{10}(x_i-\bar{x})^2}$
Calculate numerator:
$\sum(x_i-\bar{x})(y_i-\bar{y}) = (8.3-13.82)(70-76.3)+(10.5-13.82)(72-76.3)+...+(18-13.82)(81-76.3) = 157.24$
Calculate denominator:
$\sum(x_i-\bar{x})^2 = (8.3-13.82)^2+(10.5-13.82)^2+...+(18-13.82)^2 = 112.156$
$m = \frac{157.24}{112.156} \approx 1.40$

Step3: Calculate y-intercept (b)

Use $\bar{y} = m\bar{x} + b$ to solve for $b$:
$b = \bar{y} - m\bar{x} = 76.3 - 1.40\times13.82 \approx 76.3 - 19.35 = 56.95 \approx 57.0$

Step4: Predict height for x=22

Substitute $x=22$ into $y=1.40x+57.0$:
$y = 1.40\times22 + 57.0 = 30.8 + 57.0 = 87.8$

Step5: Interpret correlation coefficient

A correlation coefficient of 0.99 is very close to 1, so it is a good model.

Answer:

The data is best modeled by the function $y = 1.40x + 57.0$
Based on the linear model, the approximate height of a tree with a diameter of 22 inches is 87.8 feet
The correlation coefficient for this model is 0.99, indicating that it is a good model of the data