Question

calculate the correlation coefficient for the following ordered pairs.

\\(\

$$\begin{array}{c|ccccc} x & 3 & 8 & 2 & 6 & 6 \\\\ y & 5 & 10 & 4 & 4 & 7 \\end{array}$$

\\)

\\(r = \\) (round to three decimal places as needed.)

Explanation:

⚡ Using what you learned: regression, residuals, and model fit

Step 1: Organize the data and calculate summary statistics

We are given the following pairs of \( (x, y) \):

$$ (3, 5), \quad (8, 10), \quad (2, 4), \quad (6, 4), \quad (6, 7) $$

The number of data points is \( n = 5 \).

Calculate the sum of \( x \), \( y \), \( x^2 \), \( y^2 \), and \( xy \):

• \( \sum x = 3 + 8 + 2 + 6 + 6 = 25 \)
• \( \sum y = 5 + 10 + 4 + 4 + 7 = 30 \)
• \( \sum x^2 = 3^2 + 8^2 + 2^2 + 6^2 + 6^2 = 9 + 64 + 4 + 36 + 36 = 149 \)
• \( \sum y^2 = 5^2 + 10^2 + 4^2 + 4^2 + 7^2 = 25 + 100 + 16 + 16 + 49 = 206 \)
• \( \sum xy = (3 \times 5) + (8 \times 10) + (2 \times 4) + (6 \times 4) + (6 \times 7) = 15 + 80 + 8 + 24 + 42 = 169 \)

Step 2: Apply the correlation coefficient formula

The formula for the Pearson correlation coefficient \( r \) is:

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

Substitute the calculated sums into the formula:

$$ r = \frac{5(169) - (25)(30)}{\sqrt{[5(149) - (25)^2][5(206) - (30)^2]}} $$

Step 3: Simplify the expression

Calculate the numerator:

$$ 5(169) - 750 = 845 - 750 = 95 $$

Calculate the terms inside the square root in the denominator:

$$ 5(149) - 25^2 = 745 - 625 = 120 $$

$$ 5(206) - 30^2 = 1030 - 900 = 130 $$

Combine the terms in the denominator:

$$ \sqrt{120 \times 130} = \sqrt{15600} \approx 124.89996 $$

Calculate \( r \):

$$ r = \frac{95}{124.89996} \approx 0.76061 $$

Rounding to three decimal places gives \( 0.761 \).

Answer:

0.761