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. 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 = 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 0.95, indicating that it a good model of the data.

Explanation:

Step1: Calculate slope and intercept

Use a statistical software or calculator with linear - regression function. Let the linear function be $y = mx + b$. Using the given data points $(x_i,y_i)$ for $i = 1,\cdots,10$, most calculators or software will directly give the values of $m$ and $b$. Assume we use a calculator with linear - regression capabilities. After inputting the data, we get the linear regression equation $y=1.73x + 55.97$.

Step2: Predict height for $x = 22$

Substitute $x = 22$ into the equation $y=1.73x + 55.97$. Then $y=1.73\times22+55.97=38.06 + 55.97=94.03\approx94$.

Step3: Analyze correlation coefficient

A correlation coefficient $r = 0.95$ is close to 1. Since the absolute value of the correlation coefficient $|r|$ is close to 1, it indicates a strong positive linear relationship, so it is a good model of the data.

Answer:

The data is best modeled by the function $y = 1.73x+55.97$.
Based on the linear model, the approximate height of a tree with a diameter of 22 inches is 94 feet.
The correlation coefficient for this model is 0.95, indicating that it is a good model of the data.