Question

question 9
12 pts
based on the data shown below, calculate the correlation coefficient (to three decimal places)

x	y
5	20
6	17.8
7	17.1
8	15.8
9	16.3

Explanation:

Step1: Identify the formula for correlation coefficient

The Pearson correlation coefficient \( r \) is given by:

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

where \( n \) is the number of data points.

Step2: Calculate the necessary sums

First, list the values of \( x \), \( y \), \( xy \), \( x^2 \), and \( y^2 \):

\( x \)	\( y \)	\( xy \)	\( x^2 \)	\( y^2 \)
6	17.8	\( 6\times17.8 = 106.8 \)	\( 6^2 = 36 \)	\( 17.8^2 = 316.84 \)
7	17.1	\( 7\times17.1 = 119.7 \)	\( 7^2 = 49 \)	\( 17.1^2 = 292.41 \)
8	15.8	\( 8\times15.8 = 126.4 \)	\( 8^2 = 64 \)	\( 15.8^2 = 249.64 \)
9	16.3	\( 9\times16.3 = 146.7 \)	\( 9^2 = 81 \)	\( 16.3^2 = 265.69 \)

Now, calculate the sums:

• \( n = 5 \)
• \( \sum x = 5 + 6 + 7 + 8 + 9 = 35 \)
• \( \sum y = 20 + 17.8 + 17.1 + 15.8 + 16.3 = 87 \)
• \( \sum xy = 100 + 106.8 + 119.7 + 126.4 + 146.7 = 599.6 \)
• \( \sum x^2 = 25 + 36 + 49 + 64 + 81 = 255 \)
• \( \sum y^2 = 400 + 316.84 + 292.41 + 249.64 + 265.69 = 1524.58 \)

Step3: Substitute into the formula

First, calculate the numerator:

$$ n\sum xy - \sum x \sum y = 5\times599.6 - 35\times87 = 2998 - 3045 = -47 $$

Next, calculate the denominator components:

• For \( x \): \( n\sum x^2 - (\sum x)^2 = 5\times255 - 35^2 = 1275 - 1225 = 50 \)
• For \( y \): \( n\sum y^2 - (\sum y)^2 = 5\times1524.58 - 87^2 = 7622.9 - 7569 = 53.9 \)

Then, the denominator is:

$$ \sqrt{50\times53.9} = \sqrt{2695} \approx 51.913 $$

Now, calculate \( r \):

$$ r = \frac{-47}{51.913} \approx -0.905 $$

Answer:

\( -0.905 \)