Average height of a species of tree (in feet) after a certain number of years
Years----------Height
1 ------------- 2.1
2 ------------- 3.2
3 ------------- 6.8
4 ------------- 7.3
5 ------------- 11.2
6 ------------- 12.6
7 ------------- 13.4
8 ------------- 15.9
What is the correlation coefficient for the set of data? Round your answer to the nearest thousandth.
a. 0.014
b. 0.989
c. 0.075
d. -0.977***
About how tall do you expect one of these trees to be after 22 years?
a. 22.31 ft
b. 35.2 ft
c. 44.25 ft
d. 46.2 ft
should be b then c
To find the correlation coefficient for the given set of data, we can use the formula:
Correlation coefficient (r) = (nΣxy - ΣxΣy) / sqrt((nΣx^2 - (Σx)^2) * (nΣy^2 - (Σy)^2))
Where:
n = number of data points
Σxy = sum of the product of x and y values
Σx = sum of x values
Σy = sum of y values
Σx^2 = sum of squared x values
Σy^2 = sum of squared y values
Using the given data, we can calculate the values needed for the above formula:
n = 8
Σxy = (1 * 2.1) + (2 * 3.2) + (3 * 6.8) + (4 * 7.3) + (5 * 11.2) + (6 * 12.6) + (7 * 13.4) + (8 * 15.9) = 345.6
Σx = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = 36
Σy = 2.1 + 3.2 + 6.8 + 7.3 + 11.2 + 12.6 + 13.4 + 15.9 = 72.5
Σx^2 = (1^2) + (2^2) + (3^2) + (4^2) + (5^2) + (6^2) + (7^2) + (8^2) = 204
Σy^2 = (2.1^2) + (3.2^2) + (6.8^2) + (7.3^2) + (11.2^2) + (12.6^2) + (13.4^2) + (15.9^2) = 722.39
Plugging these values into the formula, we get:
r = (8 * 345.6 - 36 * 72.5) / sqrt((8 * 204 - 36^2) * (8 * 722.39 - 72.5^2))
Simplifying, we have:
r = (2764.8 - 2616) / sqrt((1632 - 1296) * (5779.12 - 5280.25))
r = 148.8 / sqrt(336 * 490.87)
r ≈ 148.8 / sqrt(164991.12)
r ≈ 148.8 / 406.18
r ≈ 0.366
Rounding this to the nearest thousandth, the correlation coefficient is 0.366.
For the second part of the question, we don't have a direct equation to calculate the height for a given number of years. However, we can observe the relationship between years and height to estimate the height after 22 years based on the given data.
From the given data, we can see that as the number of years increases, the height of the tree also tends to increase. Therefore, we can assume there is a positive correlation between years and height.
Based on this assumption, we can estimate the height for 22 years by extrapolating from the existing data points. Based on the trend observed in the data, it is reasonable to expect that the height of a tree after 22 years would be greater than the height at year 8 (which is 15.9 ft).
Among the given options, the closest estimate to a height greater than 15.9 ft is 46.2 ft (option d).