Question

Calculate the correlation coefficient of the following data:

x y
1 16
4 20
8 15
6 22
2 9

.14

-.14

-0.37

0.37

Answer 1

To calculate the correlation coefficient for the given data, we can use the Pearson correlation coefficient formula:

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

where:
- \( n \) = number of data points
- \( \Sigma xy \) = sum of the product of paired scores
- \( \Sigma x \) = sum of x scores
- \( \Sigma y \) = sum of y scores
- \( \Sigma x^2 \) = sum of squared x scores
- \( \Sigma y^2 \) = sum of squared y scores

Let's denote the data points:

| x | y |
|---|----|
| 1 | 16 |
| 4 | 20 |
| 8 | 15 |
| 6 | 22 |
| 2 | 9 |

Now we can calculate the necessary sums.

1. **Calculate \( n \)**:
- \( n = 5 \)

2. **Calculate sums:**
- \( \Sigma x = 1 + 4 + 8 + 6 + 2 = 21 \)
- \( \Sigma y = 16 + 20 + 15 + 22 + 9 = 82 \)
- \( \Sigma xy = (1 \cdot 16) + (4 \cdot 20) + (8 \cdot 15) + (6 \cdot 22) + (2 \cdot 9) = 16 + 80 + 120 + 132 + 18 = 366 \)
- \( \Sigma x^2 = 1^2 + 4^2 + 8^2 + 6^2 + 2^2 = 1 + 16 + 64 + 36 + 4 = 121 \)
- \( \Sigma y^2 = 16^2 + 20^2 + 15^2 + 22^2 + 9^2 = 256 + 400 + 225 + 484 + 81 = 1446 \)

3. **Substituting into the formula:**

\[
r = \frac{5(366) - (21)(82)}{\sqrt{[5(121) - (21)^2][5(1446) - (82)^2]}}
\]

Calculating each component step-by-step:

- Numerator:
\[
5(366) - (21)(82) = 1830 - 1722 = 108
\]

- Denominator:
1. First part:
\[
5(121) - (21)^2 = 605 - 441 = 164
\]
2. Second part:
\[
5(1446) - (82)^2 = 7230 - 6724 = 506
\]
3. Now combine for the denominator:
\[
\sqrt{(164)(506)} = \sqrt{82984} \approx 288.18
\]

Now calculating \( r \):

\[
r \approx \frac{108}{288.18} \approx 0.375
\]

Rounding to two decimal places gives \( 0.37 \).

Thus, the correlation coefficient for the data is approximately **0.37**.