To find the residual for a performance with a revenue of $\$700$ and $70$ seats occupied, we need to follow these steps:

Step 1: Calculate the Predicted Revenue ($\hat{y}$)

Using the least squares regression equation: $\hat{y} = -325 + 14x$ where $x = 70$ (the number of seats occupied), we can substitute the value of $x$ into the equation to find the predicted revenue.

Step 2: Calculate the Residual

The residual is the difference between the actual revenue and the predicted revenue: $\text{Residual} = \text{Actual Revenue} - \text{Predicted Revenue}$

Let's calculate it.The residual for the performance with a revenue of $700 and 70 seats occupied is $\$45$.

This means the actual revenue was $45 higher than the predicted revenue based on the regression equation.

Would you like to explore this further or have any other questions?

Here are five related questions:

1. How would the residual change if the actual revenue was $600 instead of $700?
2. What does a positive residual indicate about the performance's revenue prediction?
3. How would the regression equation change if the number of seats occupied increases?
4. What are some limitations of using a linear regression model in predicting revenue?
5. How is the slope $14$ in the regression equation interpreted in this context?

Tip: Understanding residuals can help assess the accuracy of a regression model and identify any patterns or discrepancies in the data.