Question

"Using the following data with x as the independent variable and y as the dependent variable, answer the items. x -17 -25 21 11 27 y 67 256 162 12 350 a. Compute the correlation coefficient.

Answer 1

The correlation coefficient for the given data set is approximately
`\( r \approx 0.439 \).`

Step-by-step explanation:

To compute the correlation coefficient
`(\( r \)),` we use the following formula:

`\[ r = (n(\sum xy) - (\sum x)(\sum y))/(√([n\sum x^2 - (\sum x)^2][n\sum y^2 - (\sum y)^2])) \]`

Where
`\( n \)` is the number of data points,
`\( \sum xy \)` is the sum of the product of
`\( x \)` and
`\( y \)` ,
`\( \sum x \) and \( \sum y \)` are the sums of
`\( x \) and \( y \)` respectively,
`\( \sum x^2 \)` and
`\( \sum y^2 \)` are the sums of the squares of
`\( x \) and \( y \)` respectively.

For the given data:

`\[ n = 5, \sum x = 17 - 25 + 21 + 11 + 27 = 51, \sum y = 67 + 256 + 162 + 12 + 350 = 847, \]`

`\[ \sum xy = (17 * 67) + (-25 * 256) + (21 * 162) + (11 * 12) + (27 * 350) = 5626, \]`

`\[ \sum x^2 = 17^2 + (-25)^2 + 21^2 + 11^2 + 27^2 = 1410, \]`

`\[ \sum y^2 = 67^2 + 256^2 + 162^2 + 12^2 + 350^2 = 197961. \]`

Substituting these values into the correlation coefficient formula, we get
`\( r \approx 0.439 \).` The correlation coefficient ranges from -1 to 1, where 1 indicates a perfect positive correlation, -1 indicates a perfect negative correlation, and 0 indicates no correlation. In this case,
`\( r \approx 0.439 \)` suggests a moderate positive correlation between
`\( x \) and \( y \)` in the given data set.