The theater director recorded the number of seats occupied at past performances and the total revenue per performance.

After plotting her results, the director noticed that the relationship between the two variables was fairly linear, so she used the data to calculate the following least squares regression equation for predicting the total revenue, in dollars, from the number of seats occupied.

\[\hat y = -325+14 x\]

**What is the residual of a performance with a revenue of \[\$700\] and \[70\] seats occupied ?**

\[\$\]

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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.

Learn how to calculate the residual in a linear regression scenario using the least squares regression equation. This problem involves predicting revenue based on seats occupied and determining the residual when actual revenue is $700 with 70 seats. Explore the accuracy of regression models and implications of residuals in predictions.

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