Answer:
To find the magnitude of the correlation (strength) between sales revenue (y) and advertising expenditure (x) based on the provided summarized data, we can use the formula for the Pearson correlation coefficient \( r \):
\[ r = \frac{n \Sigma xy - \Sigma x \Sigma y}{\sqrt{[n \Sigma x^2 - (\Sigma x)^2][n \Sigma y^2 - (\Sigma y)^2]}} \]
Given:
- \( \Sigma x = 680 \)
- \( \Sigma y = 730 \)
- \( \Sigma xy = 25892 \)
- \( \Sigma x^2 = 72500 \)
- \( \Sigma y^2 = 90500 \)
- \( n = 25 \) (number of months)
Let's calculate \( r \):
1. Calculate the numerator:
\[ n \Sigma xy - \Sigma x \Sigma y = 25 \cdot 25892 - 680 \cdot 730 \]
\[ = 647300 - 496400 \]
\[ = 150900 \]
2. Calculate the denominators separately:
- Denominator for \( x \):
\[ n \Sigma x^2 - (\Sigma x)^2 = 25 \cdot 72500 - (680)^2 \]
\[ = 1812500 - 462400 \]
\[ = 1350100 \]
- Denominator for \( y \):
\[ n \Sigma y^2 - (\Sigma y)^2 = 25 \cdot 90500 - (730)^2 \]
\[ = 2262500 - 532900 \]
\[ = 1729600 \]
3. Calculate the square root of the product of the denominators:
\[ \sqrt{(1350100)(1729600)} \]
\[ = \sqrt{233448160000} \]
\[ = 48317.09 \]
4. Now, calculate \( r \):
\[ r = \frac{150900}{48317.09} \]
\[ r \approx 3.12 \]
### Interpretation of the Result:
The calculated correlation coefficient \( r \approx 3.12 \) indicates a strong positive linear relationship between advertising expenditure and sales revenue.
- **Strength of Relationship:** The magnitude of \( r \) being significantly greater than zero suggests a strong positive correlation. This means that as advertising expenditure increases, sales revenue tends to increase as well.
- **Direction of Relationship:** The positive sign of \( r \) indicates that higher advertising expenditure is associated with higher sales revenue.
Therefore, based on the magnitude of \( r \) calculated, there is a strong positive correlation between advertising expenditure and sales revenue over the 25-month period.
Step-by-step explanation: