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To determine the least squares regression line for the data provided, we need to calculate the linear relationship between the "Cost of advertising" (independent variable, $ x $) and "Revenue" (dependent variable, $ y $). Let's go through the steps needed to find the regression line in the form:

\[
y = a + bx
\]

Where:
- $ a $ is the y-intercept,
- $ b $ is the slope of the line.

### Steps to Find the Least Squares Regression Line
1. **Calculate the Means**: Find the mean of $ x $ (Cost of advertising) and the mean of $ y $ (Revenue).
2. **Calculate the Slope $ b $**:
   \[
   b = \frac{\sum{(x_i - \bar{x})(y_i - \bar{y})}}{\sum{(x_i - \bar{x})^2}}
   \]
3. **Calculate the Intercept $ a $**:
   \[
   a = \bar{y} - b\bar{x}
   \]

Let me proceed with the calculations using the given data in the table.The least squares regression line for the given data is:

\[
\text{Revenue} = 51809.524 + 105.119 \times \text{Cost of advertising}
\]

This equation means that for each dollar increase in the cost of advertising, the revenue increases by approximately $105.119, and the base revenue when advertising cost is zero is approximately $51,809.52.

Would you like more details or have any questions?

Here are 5 related questions to expand on this topic:
1. How can we use this regression line to predict revenue for a different advertising cost?
2. What does the slope tell us about the relationship between advertising cost and revenue?
3. How can we determine how well the regression line fits the data?
4. What is the coefficient of determination ($ R^2 $) for this regression line?
5. How can we calculate the residuals for this dataset?

**Tip:** The least squares regression line is useful for making predictions, but it assumes a linear relationship between the variables. Always visualize your data to ensure this assumption holds.

The data shows the cost of advertising (in $) and the revenue made over a period of weeks. Determine the least squares regression line equation in terms of Cost and Revenue (3 significant figures).

Learn how to calculate the least squares regression line for a dataset showing advertising costs and corresponding revenues over 8 weeks. This involves calculating the slope and intercept of the linear relationship using the least squares method, suitable for students in Grades 11-12.

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