Find the correlation coefficient, r, of the data described below. Ariel owns a car wash and has noticed that her business fluctuates throughout the year. She is curious to know whether these fluctuations are related to changes in the local pigeon population. Over several mornings, Ariel counted the number of pigeons that were sitting on power lines in her neighborhood each day, x, and the number of car washes that were purchased during that day, y.

Question

Pigeons counted Car Washes
6 3
22 9
26 31
34 27
38 0

1. 0.7
2. 0.2
3. -0.1
4. -0.9

Answer 1

2. 0.2

To calculate the correlation coefficient:
1. Calculate the mean of x and y:
mean of x = (6 + 22 + 26 + 34 + 38) / 5 = 25.2
mean of y = (3 + 9 + 31 + 27 + 0) / 5 = 14

2. Calculate the deviations from the mean for x and y:
For x: 6-25.2 = -19.2, 22-25.2 = -3.2, 26-25.2 = 0.8, 34-25.2 = 8.8, 38-25.2 = 12.8
For y: 3-14 = -11, 9-14 = -5, 31-14 = 17, 27-14 = 13, 0-14 = -14

3. Calculate the sum of the product of the deviations:
(-19.20 * -11) + (-3.20 * -5) + (0.80 * 17) + (8.80*13) + (12.80 * -14) = 211.6

4. Calculate the sum of the squares of the deviations for x and y:
For x: (-19.20^2) + (-3.20^2) + (0.80^2) + (8.80^2) + (12.80^2) = 808.48
For y: (-11^2) + (-5^2) + (17^2) + (13^2) + (-14^2) = 630

5. Calculate the square root of the sum of the squares of the deviations for x multiplied by the sum of the squares of the deviations for y:
√(808.48 * 630) = √509942.4 ≈ 713.3

6. Calculate the correlation coefficient:
r = 211.6 / 713.3 = 0.2967 ≈ 0.2

Therefore, the correlation coefficient, r, is approximately 0.2.