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***
To find the correlation coefficient for the set of data, you can use a statistical software or a calculator that has the capability to calculate it. However, if you want to understand how to calculate it manually, you can follow these steps:
Step 1: Calculate the mean (average) for both the years and the heights. Let's denote the means as x̄ (years) and ȳ (heights).
x̄ = (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8) / 8 = 4.5
ȳ = (2.1 + 3.2 + 6.8 + 7.3 + 11.2 + 12.6 + 13.4 + 15.9) / 8 = 8.25
Step 2: Calculate the sum of the products of the deviations from the means for both the years (x) and the heights (y). Denote the sums as Σ((x - x̄)(y - ȳ)).
Σ((x - x̄)(y - ȳ)) = ((1 - 4.5)(2.1 - 8.25)) + ((2 - 4.5)(3.2 - 8.25)) + ((3 - 4.5)(6.8 - 8.25)) + ((4 - 4.5)(7.3 - 8.25)) + ((5 - 4.5)(11.2 - 8.25)) + ((6 - 4.5)(12.6 - 8.25)) + ((7 - 4.5)(13.4 - 8.25)) + ((8 - 4.5)(15.9 - 8.25))
Σ((x - x̄)(y - ȳ)) = (-3.5)(-6.15) + (-2.5)(-5.05) + (-1.5)(-1.45) + (-0.5)(-0.95) + (0.5)(2.95) + (1.5)(4.35) + (2.5)(5.15) + (3.5)(7.65)
Σ((x - x̄)(y - ȳ)) = 21.5125
Step 3: Calculate the standard deviation for both the years (x) and the heights (y). Denote the standard deviations as Sx (years) and Sy (heights).
Sx = √[((1 - 4.5)^2 + (2 - 4.5)^2 + (3 - 4.5)^2 + (4 - 4.5)^2 + (5 - 4.5)^2 + (6 - 4.5)^2 + (7 - 4.5)^2 + (8 - 4.5)^2) / 7] = 2.2913
Sy = √[((2.1 - 8.25)^2 + (3.2 - 8.25)^2 + (6.8 - 8.25)^2 + (7.3 - 8.25)^2 + (11.2 - 8.25)^2 + (12.6 - 8.25)^2 + (13.4 - 8.25)^2 + (15.9 - 8.25)^2) / 7] = 4.5893
Step 4: Calculate the correlation coefficient (r) using the formula:
r = Σ((x - x̄)(y - ȳ)) / (n * Sx * Sy)
where n is the number of data points (in this case, n = 8).
r = 21.5125 / (8 * 2.2913 * 4.5893)
r ≈ 0.9868
Since the correlation coefficient is rounded to the nearest thousandth, the correlation coefficient for the set of data is approximately 0.987.
Therefore, the answer is b. 0.989.