Question

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 . 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.

Answer 1

Using technology, the linear function that best models this set of data is .

To find the approximate height of a tree with a diameter of 22 inches, substitute 22 for x in the linear model and solve for y.

So based on the model, the approximate height of a tree with a diameter of 22 inches is 84 feet.

The correlation coefficient, r, is a value between 0 and 1 that shows how well a linear function models a data set. The r value of 0.95 is very close to 1, indicating that it is a good model of the data.

Explanation:

Using technology, the linear function that best models this set of data is .

To find the approximate height of a tree with a diameter of 22 inches, substitute 22 for x in the linear model and solve for y.

So based on the model, the approximate height of a tree with a diameter of 22 inches is 84 feet.

The correlation coefficient, r, is a value between 0 and 1 that shows how well a linear function models a data set. The r value of 0.95 is very close to 1, indicating that it is a good model of the data.

Answer 2

The linear function that best models the data is y = 2.36x + 46.82. The approximate height of a tree with a diameter of 22 inches is 97.98 feet. The correlation coefficient for this model is 0.95.

To find the linear function that best models the data in the table, we can calculate the slope and y-intercept using the least-squares regression method.

Step 1: Calculate the mean of the x-values and the mean of the y-values.

Mean of x: (8.3 + 10.5 + 11 + 12 + 12.9 + 14 + 16.3 + 17.3 + 17.9 + 18) / 10 = 13.24

Mean of y: (70 + 72 + 75 + 75 + 74 + 78 + 77 + 81 + 80 + 81) / 10 = 76.3

Step 2: Calculate the differences between each x-value and the mean of x (x - mean of x) and the differences between each y-value and the mean of y (y - mean of y).

Differences in x: (-4.94, -2.74, -2.24, -1.24, -0.34, 0.76, 3.06, 4.06, 4.64, 4.76)

Differences in y: (-6.3, -4.3, -1.3, -1.3, -2.3, 1.7, 0.7, 4.7, 3.7, 4.7)

Step 3: Calculate the sum of the products of the differences in x and the differences in y and the sum of the squares of the differences in x.

Sum of products of differences: (-4.94 * -6.3) + (-2.74 * -4.3) + (-2.24 * -1.3) + (-1.24 * -1.3) + (-0.34 * -2.3) + (0.76 * 1.7) + (3.06 * 0.7) + (4.06 * 4.7) + (4.64 * 3.7) + (4.76 * 4.7) = 217.88

Sum of squares of differences in x: (-4.94)^2 + (-2.74)^2 + (-2.24)^2 + (-1.24)^2 + (-0.34)^2 + (0.76)^2 + (3.06)^2 + (4.06)^2 + (4.64)^2 + (4.76)^2 = 92.32

Step 4: Calculate the slope of the line (b) by dividing the sum of the products of the differences by the sum of the squares of the differences in x.

Slope (b) = 217.88 / 92.32 = 2.36

Step 5: Calculate the y-intercept (a) by subtracting the product of the slope and the mean of x from the mean of y.

Y-intercept (a) = 76.3 - (2.36 * 13.24) = 46.82

Therefore, the linear function that best models the data is y = 2.36x + 46.82.

To find the approximate height of a tree with a diameter of 22 inches, we can substitute x = 22 into the linear function and solve for y.

Height (y) = 2.36(22) + 46.82 = 97.98 feet.

The correlation coefficient for this model is 0.95, indicating that it is a good model of the data.