Let's walk through the steps to find the value of the correlation coefficient \( r \) from the given data.
### Step 1: Calculate the Standard Deviations
To find the correlation coefficient, we first need to calculate the standard deviations of \( x \) and \( y \).
Given:
- Average \( x \) = 60
- Average \( y \) = 95
- Sum of squares of deviations for \( x \) = 920
- Sum of squares of deviations for \( y \) = 1050
- The sum of product of deviations = -545
We assume the sample size \( n \) is such that \( n-1 = 10 \), meaning \( n = 11 \). With \( n-1 \) as our value (which is typically used in the calculation of standard deviation for samples), we compute the standard deviations.
The formula for the standard deviation of \( x \) is:
[tex]\[ \sigma_x = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}} \][/tex]
For \( x \):
[tex]\[ \sigma_x = \sqrt{\frac{920}{10}} \][/tex]
[tex]\[ \sigma_x = \sqrt{92} \][/tex]
[tex]\[ \sigma_x \approx 9.59 \][/tex]
For \( y \):
[tex]\[ \sigma_y = \sqrt{\frac{\sum (y_i - \bar{y})^2}{n-1}} \][/tex]
[tex]\[ \sigma_y = \sqrt{\frac{1050}{10}} \][/tex]
[tex]\[ \sigma_y = \sqrt{105} \][/tex]
[tex]\[ \sigma_y \approx 10.25 \][/tex]
### Step 2: Calculate the Correlation Coefficient \( r \)
The formula for the correlation coefficient \( r \) is:
[tex]\[ r = \frac{\sum((x_i - \bar{x})(y_i - \bar{y}))}{\sqrt{\sum (x_i - \bar{x})^2 \sum (y_i - \bar{y})^2}} \][/tex]
Given the sum of product of deviations:
[tex]\[ \sum ((x_i - \bar{x})(y_i - \bar{y})) = -545 \][/tex]
We use this sum of product of deviations along with the standard deviations calculated above:
[tex]\[ r = \frac{\sum((x_i - \bar{x})(y_i - \bar{y}))}{(n-1) \sigma_x \sigma_y} \][/tex]
Substitute the values:
[tex]\[ r = \frac{-545}{10 \times 9.59 \times 10.25} \][/tex]
Calculate the denominator:
[tex]\[ 9.59 \times 10.25 = 98.2975 \][/tex]
[tex]\[ 10 \times 98.2975 = 982.975 \][/tex]
Now, compute \( r \):
[tex]\[ r = \frac{-545}{982.975} \][/tex]
[tex]\[ r \approx -0.5545 \][/tex]
### Conclusion
Therefore, the correlation coefficient [tex]\( r \approx -0.5545 \)[/tex].
