Let's compute the correlation coefficient [tex]\( r \)[/tex] using the given data. The correlation coefficient formula is:
[tex]\[
r = \frac{n\Sigma xy - (\Sigma x)(\Sigma y)}{\sqrt{[n \Sigma x^2 - (\Sigma x)^2][n \Sigma y^2 - (\Sigma y)^2]}}
\][/tex]
Given values:
[tex]\[
\Sigma x = 24.0
\][/tex]
[tex]\[
\Sigma x^2 = 85.48
\][/tex]
[tex]\[
\Sigma y = 53.5
\][/tex]
[tex]\[
\Sigma y^2 = 457.33
\][/tex]
[tex]\[
\Sigma xy = 190.02
\][/tex]
[tex]\[
n = 5
\][/tex]
### Step-by-Step Calculation
1. Calculate the numerator:
[tex]\[
\text{{numerator}} = n \Sigma xy - (\Sigma x)(\Sigma y)
\][/tex]
[tex]\[
\text{{numerator}} = 5 \cdot 190.02 - (24.0 \cdot 53.5)
\][/tex]
[tex]\[
\text{{numerator}} = 950.1 - 1284
\][/tex]
[tex]\[
\text{{numerator}} = -333.9
\][/tex]
2. Calculate the denominator:
[tex]\[
\text{{denominator}} = \sqrt{[n \Sigma x^2 - (\Sigma x)^2][n \Sigma y^2 - (\Sigma y)^2]}
\][/tex]
- First, compute [tex]\( n \Sigma x^2 - (\Sigma x)^2 \)[/tex]:
[tex]\[
n \Sigma x^2 - (\Sigma x)^2 = 5 \cdot 85.48 - (24.0)^2
\][/tex]
[tex]\[
= 427.4 - 576
\][/tex]
[tex]\[
= -148.6
\][/tex]
- Next, compute [tex]\( n \Sigma y^2 - (\Sigma y)^2 \)[/tex]:
[tex]\[
n \Sigma y^2 - (\Sigma y)^2 = 5 \cdot 457.33 - (53.5)^2
\][/tex]
[tex]\[
= 2286.65 - 2862.25
\][/tex]
[tex]\[
= -575.6
\][/tex]
- Now, compute the product:
[tex]\[
[n \Sigma x^2 - (\Sigma x)^2][n \Sigma y^2 - (\Sigma y)^2] = (-148.6) \times (-575.6)
\][/tex]
[tex]\[
= 85560.16
\][/tex]
- Finally, take the square root:
[tex]\[
\text{{denominator}} = \sqrt{85560.16}
\][/tex]
[tex]\[
= 292.4622
\][/tex]
3. Compute [tex]\( r \)[/tex]:
[tex]\[
r = \frac{\text{{numerator}}}{\text{{denominator}}}
\][/tex]
[tex]\[
r = \frac{-333.9}{292.4622}
\][/tex]
[tex]\[
r \approx -1.1417
\][/tex]
Therefore, the correlation coefficient [tex]\( r \)[/tex] rounded to four decimal places is:
[tex]\[
r \approx -1.1417
\][/tex]
