Answer :
Certainly! Let's tackle the problem step by step to determine the relationship between advertisement expenditure and sales revenue and then estimate the sales revenue when the advertisement expenditure is Rs. 40,000.
### Step-by-Step Solution
Step 1: Convert the Data
First, we need to reshape the given data for analysis, and calculate the midpoints for the ranges.
1. Advertisement Expenditure (Midpoints):
| Range | Midpoint |
|-------------|----------|
| 0 - 40 | 20 |
| 40 - 80 | 60 |
| 80 - 120 | 100 |
| 120 - 160 | 140 |
| 160 - 200 | 180 |
2. Sales Revenue (Midpoints):
| Range | Midpoint |
|-------------|----------|
| 0.50 | 0.50 |
| 50 - 100 | 75.00 |
| 100 - 150 | 125.00 |
| 150 - 200 | 175.00 |
| 200 - 250 | 225.00 |
Step 2: Create a List of Frequencies
Convert the table into a format that lists frequencies for each combination of advertisement expenditure and sales revenue.
| Ad Expenditure | Sales Revenue | Frequency |
|----------------|---------------|-----------|
| 20 | 0.50 | 12 |
| 20 | 75.00 | 6 |
| 20 | 125.00 | 8 |
| 60 | 0.50 | 2 |
| 60 | 75.00 | 18 |
| 60 | 125.00 | 4 |
| 60 | 175.00 | 5 |
| 60 | 225.00 | 1 |
| 100 | 75.00 | 8 |
| 100 | 125.00 | 10 |
| 100 | 175.00 | 2 |
| 100 | 225.00 | 4 |
| 140 | 75.00 | 1 |
| 140 | 125.00 | 10 |
| 140 | 175.00 | 2 |
| 140 | 225.00 | 1 |
| 180 | 125.00 | 1 |
| 180 | 175.00 | 2 |
| 180 | 225.00 | 3 |
Step 3: Flatten the Data
Next, we need to convert these frequencies into a flat dataset of expenditures and sales for correlation and regression analysis.
Flattened Data:
We need to repeat the values according to their frequencies.
For example:
- Ad Expenditure of 20 with Sales Revenue 0.50 is repeated 12 times.
- Ad Expenditure of 20 with Sales Revenue 75.00 is repeated 6 times.
- Continue this for all combinations...
Flattened Data (Partial View):
| Ad Expenditure | Sales Revenue |
|----------------|---------------|
| 20 | 0.50 |
| 20 | 0.50 |
| ... (12 times) |
| 20 | 75.00 |
| 20 | 75.00 |
| ... (6 times) |
| (Continue this until all combinations are flattened) |
Step 4: Calculate the Correlation Coefficient
The correlation coefficient [tex]\( r \)[/tex] measures the strength and direction of the linear relationship between two variables.
1. Calculate the mean of advertisement expenditure and mean of sales revenue.
2. Calculate the covariance of advertisement expenditure and sales revenue.
3. Calculate the standard deviations of advertisement expenditure and sales revenue.
4. Use the formula:
[tex]\[ r = \frac{\text{Cov}(X, Y)}{\sigma_X \cdot \sigma_Y} \][/tex]
Step 5: Perform Linear Regression
To predict the sales revenue when advertisement expenditure is Rs. 40,000 (or 400 units):
1. Use the linear regression formula:
[tex]\[ Y = a + bX \][/tex]
where [tex]\( Y \)[/tex] is the sales revenue, [tex]\( X \)[/tex] is the advertisement expenditure, [tex]\( a \)[/tex] is the intercept, and [tex]\( b \)[/tex] is the slope.
2. Calculate [tex]\( a \)[/tex] and [tex]\( b \)[/tex] using least squares estimation:
- [tex]\( b = \frac{n(\sum XY) - (\sum X)(\sum Y)}{n(\sum X^2) - (\sum X)^2} \)[/tex]
- [tex]\( a = \frac{\sum Y - b(\sum X)}{n} \)[/tex]
Step 6: Predict Sales Revenue for Rs. 40,000
1. Convert Rs. 40,000 to the appropriate scale (units):
- 40,000 (or 400 units because it’s scaled in 00 Rs.)
2. Use the regression equation to predict:
[tex]\[ \text{Estimated Sales Revenue} = a + b \times 400 \][/tex]
Step 7: Hypothesis Testing
To test the significance of the correlation:
1. Determine the null hypothesis [tex]\( H_0 \)[/tex] that there's no significant relationship ([tex]\( r = 0 \)[/tex]).
2. Use a t-test to test the significance of the correlation coefficient:
[tex]\[ t = \frac{r\sqrt{n-2}}{\sqrt{1-r^2}} \][/tex]
with [tex]\( n - 2 \)[/tex] degrees of freedom, where [tex]\( n \)[/tex] is the number of data points.
3. Compare the calculated [tex]\( t \)[/tex]-value with the critical value from the t-distribution table at the chosen significance level (e.g., 0.05).
Final Steps:
1. Manually calculate these values based on the frequencies.
2. Once the significant result is established, use the slope and intercept from your regression calculation to find the predicted sales revenue for an ad expenditure of Rs. 40,000.
This approach will let you analyze the presence of a relationship and predict the expected sales revenue.
### Step-by-Step Solution
Step 1: Convert the Data
First, we need to reshape the given data for analysis, and calculate the midpoints for the ranges.
1. Advertisement Expenditure (Midpoints):
| Range | Midpoint |
|-------------|----------|
| 0 - 40 | 20 |
| 40 - 80 | 60 |
| 80 - 120 | 100 |
| 120 - 160 | 140 |
| 160 - 200 | 180 |
2. Sales Revenue (Midpoints):
| Range | Midpoint |
|-------------|----------|
| 0.50 | 0.50 |
| 50 - 100 | 75.00 |
| 100 - 150 | 125.00 |
| 150 - 200 | 175.00 |
| 200 - 250 | 225.00 |
Step 2: Create a List of Frequencies
Convert the table into a format that lists frequencies for each combination of advertisement expenditure and sales revenue.
| Ad Expenditure | Sales Revenue | Frequency |
|----------------|---------------|-----------|
| 20 | 0.50 | 12 |
| 20 | 75.00 | 6 |
| 20 | 125.00 | 8 |
| 60 | 0.50 | 2 |
| 60 | 75.00 | 18 |
| 60 | 125.00 | 4 |
| 60 | 175.00 | 5 |
| 60 | 225.00 | 1 |
| 100 | 75.00 | 8 |
| 100 | 125.00 | 10 |
| 100 | 175.00 | 2 |
| 100 | 225.00 | 4 |
| 140 | 75.00 | 1 |
| 140 | 125.00 | 10 |
| 140 | 175.00 | 2 |
| 140 | 225.00 | 1 |
| 180 | 125.00 | 1 |
| 180 | 175.00 | 2 |
| 180 | 225.00 | 3 |
Step 3: Flatten the Data
Next, we need to convert these frequencies into a flat dataset of expenditures and sales for correlation and regression analysis.
Flattened Data:
We need to repeat the values according to their frequencies.
For example:
- Ad Expenditure of 20 with Sales Revenue 0.50 is repeated 12 times.
- Ad Expenditure of 20 with Sales Revenue 75.00 is repeated 6 times.
- Continue this for all combinations...
Flattened Data (Partial View):
| Ad Expenditure | Sales Revenue |
|----------------|---------------|
| 20 | 0.50 |
| 20 | 0.50 |
| ... (12 times) |
| 20 | 75.00 |
| 20 | 75.00 |
| ... (6 times) |
| (Continue this until all combinations are flattened) |
Step 4: Calculate the Correlation Coefficient
The correlation coefficient [tex]\( r \)[/tex] measures the strength and direction of the linear relationship between two variables.
1. Calculate the mean of advertisement expenditure and mean of sales revenue.
2. Calculate the covariance of advertisement expenditure and sales revenue.
3. Calculate the standard deviations of advertisement expenditure and sales revenue.
4. Use the formula:
[tex]\[ r = \frac{\text{Cov}(X, Y)}{\sigma_X \cdot \sigma_Y} \][/tex]
Step 5: Perform Linear Regression
To predict the sales revenue when advertisement expenditure is Rs. 40,000 (or 400 units):
1. Use the linear regression formula:
[tex]\[ Y = a + bX \][/tex]
where [tex]\( Y \)[/tex] is the sales revenue, [tex]\( X \)[/tex] is the advertisement expenditure, [tex]\( a \)[/tex] is the intercept, and [tex]\( b \)[/tex] is the slope.
2. Calculate [tex]\( a \)[/tex] and [tex]\( b \)[/tex] using least squares estimation:
- [tex]\( b = \frac{n(\sum XY) - (\sum X)(\sum Y)}{n(\sum X^2) - (\sum X)^2} \)[/tex]
- [tex]\( a = \frac{\sum Y - b(\sum X)}{n} \)[/tex]
Step 6: Predict Sales Revenue for Rs. 40,000
1. Convert Rs. 40,000 to the appropriate scale (units):
- 40,000 (or 400 units because it’s scaled in 00 Rs.)
2. Use the regression equation to predict:
[tex]\[ \text{Estimated Sales Revenue} = a + b \times 400 \][/tex]
Step 7: Hypothesis Testing
To test the significance of the correlation:
1. Determine the null hypothesis [tex]\( H_0 \)[/tex] that there's no significant relationship ([tex]\( r = 0 \)[/tex]).
2. Use a t-test to test the significance of the correlation coefficient:
[tex]\[ t = \frac{r\sqrt{n-2}}{\sqrt{1-r^2}} \][/tex]
with [tex]\( n - 2 \)[/tex] degrees of freedom, where [tex]\( n \)[/tex] is the number of data points.
3. Compare the calculated [tex]\( t \)[/tex]-value with the critical value from the t-distribution table at the chosen significance level (e.g., 0.05).
Final Steps:
1. Manually calculate these values based on the frequencies.
2. Once the significant result is established, use the slope and intercept from your regression calculation to find the predicted sales revenue for an ad expenditure of Rs. 40,000.
This approach will let you analyze the presence of a relationship and predict the expected sales revenue.
