Question

data were gathered on how high dolphins jump from the surface of the water based on body length. the following data was obtained from 10 randomly selected dolphins in miami, florida
body length (in feet) height of jump (in feet)
5.5 15
6 15
6.7 17.5
7.2 17
7.5 18.1
8.2 18.7
8.2 17.6
8.5 19.2
9 22.3
9.5 19.3
what is the value of r and its interpretation?
-0.87, there is a strong, negative association
-0.75, there is a moderate, negative association
0.75, there is a moderate, positive association
0.87, there is a strong, positive association

Explanation:

Step1: Define variables and list data

Let $x$ = Body Length, $y$ = Height of Jump.
Data pairs:
$(5.5,15), (6,15), (6.7,17.5), (7.2,17), (7.5,18.1), (8.2,18.7), (8.2,17.6), (8.5,19.2), (9,22.3), (9.5,19.3)$
$n=10$

Step2: Calculate sums for correlation formula

Compute required sums:
$\sum x = 5.5+6+6.7+7.2+7.5+8.2+8.2+8.5+9+9.5 = 76.3$
$\sum y = 15+15+17.5+17+18.1+18.7+17.6+19.2+22.3+19.3 = 179.7$
$\sum xy = (5.5×15)+(6×15)+(6.7×17.5)+(7.2×17)+(7.5×18.1)+(8.2×18.7)+(8.2×17.6)+(8.5×19.2)+(9×22.3)+(9.5×19.3) = 1438.18$
$\sum x^2 = 5.5^2+6^2+6.7^2+7.2^2+7.5^2+8.2^2+8.2^2+8.5^2+9^2+9.5^2 = 594.33$
$\sum y^2 = 15^2+15^2+17.5^2+17^2+18.1^2+18.7^2+17.6^2+19.2^2+22.3^2+19.3^2 = 3266.69$

Step3: Compute correlation coefficient $r$

Use Pearson's correlation formula:

$$ r = \frac{n\sum xy - \sum x \sum y}{\sqrt{[n\sum x^2 - (\sum x)^2][n\sum y^2 - (\sum y)^2]}} $$

Substitute values:
Numerator: $10×1438.18 - 76.3×179.7 = 14381.8 - 13711.11 = 670.69$
Denominator first term: $10×594.33 - 76.3^2 = 5943.3 - 5821.69 = 121.61$
Denominator second term: $10×3266.69 - 179.7^2 = 32666.9 - 32292.09 = 374.81$
Denominator: $\sqrt{121.61×374.81} = \sqrt{45580.6441} ≈ 213.496$
$r = \frac{670.69}{213.496} ≈ 0.87$

Step4: Interpret the correlation

$r≈0.87$, which is close to 1, indicating a strong positive linear association between body length and jump height.

Answer:

0.87, there is a strong, positive association