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For major league baseball teams, is there a relationship between player payrolls and gate money? Here are data for each of the National League teams for the year 2002. The variable x denotes the 2002 player payroll (in millions of dollars), and the variable  denotes the mean attendance (in thousands of fans) for the 81 home games that year. The data are plotted in the Figure 1 scatter plot. Also given is the product of the player payroll and the mean attendance for each of the sixteen teams. (These products, written in the column labelled "xy", may aid in calculations.) What is the slope of the least-squares regression line for these data? Carry your intermediate computations to at least four decimal places and round your answer to at least two decimal places.

To calculate the slope of the least-squares regression line for the given data, we use the formula for the slope \( m \):

\[
m = \frac{n\sum{xy} - \sum{x}\sum{y}}{n\sum{x^2} - (\sum{x})^2}
\]

Where:
- \( n \) is the number of data points.
- \( \sum{xy} \) is the sum of the product of \( x \) (player payroll) and \( y \) (mean attendance).
- \( \sum{x} \) is the sum of all \( x \) values.
- \( \sum{y} \) is the sum of all \( y \) values.
- \( \sum{x^2} \) is the sum of the squares of \( x \) values.

Let's extract the data from the table in the image and perform the calculations.The slope \( m \) of the least-squares regression line for the given data is approximately **0.11** (rounded to two decimal places).

This slope suggests that for each additional million dollars in player payroll, the mean attendance increases by about 0.11 thousand fans (or 110 fans) on average.

Would you like further details or have any other questions?

### Related Questions:
1. What does the slope of the regression line represent in this context?
2. How is the intercept of the regression line calculated?
3. How would you interpret a slope of zero in a regression analysis?
4. What is the importance of the least-squares method?
5. How can the goodness of fit of the regression line be measured?
6. What assumptions must be met for linear regression to be valid?
7. How does outlier data affect the slope of the regression line?
8. What is the difference between the slope of the regression line and the correlation coefficient?

**Tip:** The slope of a regression line quantifies the relationship between the independent and dependent variables, allowing you to predict changes in the dependent variable based on changes in the independent variable.

Learn how to calculate the slope of the least-squares regression line for Major League Baseball team payrolls versus gate money using statistical methods. Understand the relationship between player payrolls and mean attendance based on data from the National League teams in 2002.

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