To analyze the relationship between sales revenue and advertising expenditure, follow these detailed steps to find the correlation coefficient, interpret its result, test its significance, and estimate the sales revenue when the advertising expenditure is Rs. 50 lakhs.
### Step-by-Step Solution:
#### Step 1: Calculate the Correlation Coefficient (r)
The Pearson correlation coefficient (r) quantifies the linear relationship between two variables. For the given data:
[tex]\[ r = -0.119 \][/tex]
Interpretation:
- Correlation Coefficient ([tex]\( r \)[/tex]): A correlation coefficient of -0.119 suggests a very weak negative linear relationship between sales revenue and advertising expenditure. This means that as the advertising expenditure increases, the sales revenue slightly tends to decrease, but this trend is very weak.
#### Step 2: Test the Significance of the Correlation Coefficient
To test the significance of the correlation coefficient, calculate the probable error (PE):
[tex]\[ \text{PE}(r) = 0.082 \][/tex]
Interpretation:
- Probable Error ([tex]\( \text{PE} \)[/tex]): The probable error helps to determine the reliability of the correlation coefficient. The rule of thumb is if [tex]\( |r| > 6 \times \text{PE} \)[/tex], the correlation is considered significant.
- Calculating [tex]\( 6 \times \text{PE} \)[/tex]:
[tex]\[
6 \times 0.082 = 0.492
\][/tex]
- Since [tex]\( |-0.119| = 0.119 \)[/tex] which is less than [tex]\( 0.492 \)[/tex], the correlation coefficient is not significant.
Conclusion:
- Given that [tex]\( |r| \leq 6 \times \text{PE} \)[/tex], we conclude that the correlation is not statistically significant. Thus, there is no strong evidence to support a meaningful linear relationship between sales revenue and advertising expenditure.
#### Step 3: Estimate Sales Revenue for Advertising Expenditure of Rs. 50 lakhs
Using the linear regression equation provided:
[tex]\[ \hat{Y} = 172.52 - 0.5165X \][/tex]
where:
- [tex]\( X \)[/tex] is the advertising expenditure (in lakhs Rs.)
- [tex]\( \hat{Y} \)[/tex] is the estimated sales revenue (in lakhs Rs.)
Plugging in the value [tex]\( X = 50 \)[/tex]:
[tex]\[ \hat{Y} = 172.52 - 0.5165 \times 50 \][/tex]
[tex]\[ \hat{Y} = 172.52 - 25.825 \][/tex]
[tex]\[ \hat{Y} = 146.695 \][/tex]
Conclusion:
- Estimated Sales Revenue: When the advertising expenditure is Rs. 50 lakhs, the estimated sales revenue is Rs. 146.695 lakhs.
### Summary
1. Correlation Coefficient ([tex]\( r \)[/tex]) is -0.119, indicating a very weak negative linear relationship.
2. Statistical Significance: The correlation is not statistically significant since [tex]\( |r| \leq 6 \times \text{PE} \)[/tex].
3. Estimated Sales Revenue: For an advertising expenditure of Rs. 50 lakhs, the estimated sales revenue is Rs. 146.695 lakhs.
