The choir sells tickets to its performances for an assortment of prices and gives some tickets away. The choir director wants to look at the relationship between the number of seats occupied at each performance and the ticket revenue for that performance. The data show a linear pattern with the summary statistics shown below:
mean	standard deviation
\[x=\] number of seats occupied	
\[\bar{x}=75.8\]	
\[s_x=14.8\]
\[y=\] ticket revenue (dollars)	
\[\bar{y}=696\]	
\[s_y=177.6\]
\[r=0.81\]
Find the equation of the least-squares regression line for predicting the ticket revenue from the number of seats occupied.
Round your entries to the nearest hundredth.

To find the equation of the least-squares regression line for predicting the ticket revenue (\(y\)) from the number of seats occupied (\(x\)), we use the formula for the regression line:

\[
y = a + bx
\]

Where:
- \(b\) is the slope of the line, calculated as:
  \[
  b = r \times \frac{s_y}{s_x}
  \]
- \(a\) is the y-intercept, calculated as:
  \[
  a = \bar{y} - b \times \bar{x}
  \]

### Step 1: Calculate the slope (\(b\))

We are given:
- \(r = 0.81\) (correlation coefficient)
- \(s_x = 14.8\) (standard deviation of \(x\))
- \(s_y = 177.6\) (standard deviation of \(y\))

Using the formula for the slope \(b\):

\[
b = 0.81 \times \frac{177.6}{14.8}
\]

Let me calculate this value:

\[
b = 0.81 \times 12.0 = 9.72
\]

### Step 2: Calculate the intercept (\(a\))

Now, we can calculate \(a\) using the means \(\bar{x} = 75.8\) and \(\bar{y} = 696\):

\[
a = 696 - 9.72 \times 75.8
\]

Let me calculate this value:

\[
a = 696 - 737.78 = -41.78
\]

### Final Equation

Thus, the equation of the least-squares regression line is:

\[
y = -41.78 + 9.72x
\]

This is the equation that predicts ticket revenue based on the number of seats occupied.

Would you like more details or further clarifications? Here are 5 related questions to deepen understanding:

1. What is the meaning of the slope \(b = 9.72\) in the context of this problem?
2. How would the equation change if the correlation \(r\) were negative?
3. Can the regression line be used to predict revenue for extreme seat numbers (very high or low)? Why?
4. What is the residual and how would you calculate it for a particular performance?
5. How does the standard deviation \(s_x\) and \(s_y\) affect the regression line?

**Tip**: The correlation coefficient \(r\) indicates the strength and direction of a linear relationship. An \(r\) value of 0.81 suggests a strong positive linear relationship between the number of seats occupied and ticket revenue.

The choir sells tickets to its performances for an assortment of prices and gives some tickets away. The choir director wants to look at the relationship between the number of seats occupied at each performance and the ticket revenue for that performance. The data show a linear pattern with the summary statistics shown below:
mean	standard deviation
[x=] number of seats occupied
[\bar{x}=75.8]
[s_x=14.8]
[y=] ticket revenue (dollars)
[\bar{y}=696]
[s_y=177.6]
[r=0.81]
Find the equation of the least-squares regression line for predicting the ticket revenue from the number of seats occupied.
Round your entries to the nearest hundredth.

Explore how to find the least-squares regression line to predict ticket revenue from the number of seats occupied using a linear relationship. With a correlation of r = 0.81, and summary statistics such as \bar{x} = 75.8, \bar{y} = 696, this problem demonstrates key concepts in linear regression. Suitable for students in Grades 10-12.

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