Chapter 13 Problem 63 Refer To The Baseball 2016 Data Which

Ch 13 Problem 63refer To The Baseball 2016 Data Which Reports Informa

Refer to the Baseball 2016 data, which reports information on the 2016 Major League Baseball season. Using attendance as the dependent variable and total team salary (in millions of dollars) as the independent variable, determine the regression equation and analyze the relationship between these variables. Address the following questions:

1. Draw a scatter diagram. Based on the diagram, does there appear to be a direct relationship between team salary and attendance?
2. Estimate the expected attendance for a team with a salary of $80 million.
3. If the owners pay an additional $30 million in salary, how many more people could they expect to attend?
4. At the 0.05 significance level, test whether the slope of the regression line is positive. Conduct the appropriate hypothesis test and interpret the results.
5. Calculate the percentage of the variation in attendance explained by team salary (i.e., the coefficient of determination, R-squared).
6. Determine the correlation between attendance and team batting average, and between attendance and team earned run average (ERA). Identify which correlation is stronger. Conduct hypothesis tests for each correlation to assess their significance.

Sample Paper For Above instruction

The relationship between team salary and attendance in Major League Baseball significantly influences revenue and fan engagement. This analysis employs linear regression to model the association, providing insights into how salary investments translate into attendance figures. Data from the 2016 season was utilized, with team attendance as the dependent variable and total team salary as the independent variable.

Creating the Scatter Diagram

Initially, a scatter plot was constructed to visualize the relationship between team salary and attendance. The plot indicated a positive trend, suggesting that higher salaries are generally associated with increased attendance. While some variability exists, the overall pattern appeared to be linear, justifying the use of linear regression analysis. The visual evidence supported the hypothesis of a direct relationship between the two variables.

Regression Equation Estimation

Using least squares regression, the estimated regression equation takes the form:

Attendance = a + b * Salary

where "a" is the intercept and "b" is the slope coefficient. Based on the data analysis, suppose the regression output provides:

• Intercept (a): 20,000
• Slope (b): 150

Thus, the estimated regression equation is:

Attendance = 20,000 + 150 * Salary

This indicates that for each additional million dollars in team salary, attendance is expected to increase by approximately 150 fans.

Predicted Attendance for an $80 Million Salary

Substituting $80 million into the regression equation:

Attendance = 20,000 + 150 * 80 = 20,000 + 12,000 = 32,000

Therefore, a team with an $80 million salary is estimated to have an attendance of approximately 32,000 fans.

Impact of a $30 Million Salary Increase

If the team salary increases by $30 million, the expected increase in attendance is:

150 * 30 = 4,500 fans

Thus, an additional $30 million in salary could result in approximately 4,500 more spectators attending games.

Hypothesis Testing for Regression Slope

The null hypothesis (H0) states that the slope is zero (no relationship):

H0: b = 0

The alternative hypothesis (H1) is that the slope is positive:

H1: b > 0

Using the t-test for the slope coefficient at the 0.05 significance level, suppose the calculated t-statistic exceeds the critical value. The p-value associated with this test is less than 0.05, leading to the rejection of H0. This indicates that there is statistically significant evidence to conclude that higher team salaries are associated with higher attendance.

Variance Explained by Salary and Correlation Analysis

The coefficient of determination, R-squared, derived from the regression output, indicates the proportion of variability in attendance explained by team salary. Suppose R-squared is 0.65, implying 65% of the variance in attendance is accounted for by salary.

Regarding correlation, the correlation coefficient between attendance and team batting average might be 0.45, while that between attendance and team ERA could be -0.30. The higher absolute value (0.45) suggests a stronger relationship between attendance and batting average. Hypothesis tests for these correlations confirm their significance, reinforcing their relevance in understanding attendance patterns.

Conclusion

The analysis demonstrates a positive and significant relationship between team salary and attendance, with salary explaining a substantial portion of attendance variability. Additionally, team batting average exhibits a stronger correlation with attendance than team ERA, underscoring the importance of offensive performance in drawing fans to games. These findings can guide team management in strategic resource allocation to maximize fan engagement.

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