Question

Use a graphing calculator to find the value of the correlation coefficient r to determine if there is a strong correlation among the data. (1,7), (2,5), (3,-1),(4,3), (5,-5)

Answer 1

The correlation coefficient r is approximately -0.853. This indicates a strong negative correlation among the given data points.

To calculate the correlation coefficient r, we can use the following formula:

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

where:

- n is the number of data points,

-
`\(\sum\)` denotes the sum,

- x and y are the respective variables.

Given the data points (1,7), (2,5), (3,-1), (4,3), (5,-5), let's calculate \(r\):

`\[ n = 5 \]\[ \sum{x} = 1 + 2 + 3 + 4 + 5 = 15 \]\[ \sum{y} = 7 + 5 - 1 + 3 - 5 = 9 \]\[ \sum{xy} = (1 * 7) + (2 * 5) + (3 * -1) + (4 * 3) + (5 * -5) = 7 + 10 - 3 + 12 - 25 = 1 \]\[ \sum{x^2} = 1^2 + 2^2 + 3^2 + 4^2 + 5^2 = 1 + 4 + 9 + 16 + 25 = 55 \]\[ \sum{y^2} = 7^2 + 5^2 + (-1)^2 + 3^2 + (-5)^2 = 49 + 25 + 1 + 9 + 25 = 109 \]`

Now, plug these values into the formula:

`\[ r = ((5 * 1) - (15 * 9))/(√([(5 * 55) - 15^2][(5 * 109) - 9^2])) \]`

Calculate the numerator and denominator separately, and then the final value for r.

`\[ \text{Numerator} = n(\sum{xy}) - (\sum{x})(\sum{y}) = 5(1) - (15)(9) = 5 - 135 = -130 \]\[ \text{Denominator} = \sqrt{[n\sum{x^2} - (\sum{x})^2][n\sum{y^2} - (\sum{y})^2]} \]\[ = √([5 * 55 - 15^2][5 * 109 - 9^2]) \]\[ = √((275 - 225)(545 - 81)) \]\[ = √(50 * 464) \]\[ = √(23200) \]`

Now, substitute these values back into the correlation coefficient formula:

`\[ r = \frac{\text{Numerator}}{\text{Denominator}} = (-130)/(√(23200)) \]\[ r \approx (-130)/(152.315) \]\[ r \approx -0.853 \]`

Therefore, the correlation coefficient r is approximately -0.853. This indicates a strong negative correlation among the given data points.