The National Association Of Professional Baseball League Inc
The National Association Of Professional Baseball League Inc Report
The National Association of Professional Baseball League, Inc., reported that attendance for 176 minor league baseball teams reached an all-time high during the 2001 season (New York Times, July 28, 2002). On a per game basis, the mean attendance for minor league baseball was 3530 people per game. Midway through the 2002 season, the president of the association asked for an attendance report that would hopefully show that the mean attendance for 2002 was exceeding the 2001 level.
Formulate hypotheses that could be used to determine whether the mean attendance per game in 2002 was greater than the previous year's level. At alpha equals .01, what is your conclusion?
Paper For Above instruction
Introduction
The focal point of this analysis is to determine whether the average attendance per game in the minor league baseball season of 2002 has increased beyond the 2001 level, which was reported as an all-time high of 3,530 spectators. This evaluation is statistically framed through hypothesis testing, a fundamental method in inferential statistics used to make decisions based on sample data relative to a population parameter. The process involves formulating null and alternative hypotheses, selecting an appropriate significance level, calculating the test statistic, and drawing conclusions based on the resulting p-value or critical value.
Formulating the Hypotheses
The primary objective here is to evaluate if the mean attendance in 2002 exceeds the 2001 figure. Therefore, the formulated hypotheses are:
- Null hypothesis (H0): μ_2002 ≤ 3530
- Alternative hypothesis (H1): μ_2002 > 3530
This setup aligns with a one-sided (right-tailed) test, which is appropriate when the research interest centers on detecting an increase in the mean attendance.
Choosing the Significance Level
The significance level (alpha) is specified as 0.01, reflecting a 1% probability of committing a Type I error—that is, rejecting the null hypothesis when it is actually true. This threshold indicates a strong criterion for evidence; only if the data strongly suggests an increase beyond the historical level will the null hypothesis be rejected.
Conducting the Hypothesis Test
To perform the hypothesis test, sample data from the 2002 season's attendances are necessary, including the sample mean, standard deviation, and sample size. Assuming the sample is randomly selected and sufficiently large, the test statistic follows a t-distribution for small samples or a normal distribution if the sample size is large and the population standard deviation is known or approximated.
The test statistic is calculated as:
\[ t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}, \]
where:
- \( \bar{x} \) = sample mean attendance in 2002,
- \( \mu_0 \) = 3530 (2001 attendance),
- \( s \) = sample standard deviation,
- \( n \) = sample size.
Once \( t \) is computed, it is compared against the critical value from the t-distribution table at \( \alpha = 0.01 \) for degrees of freedom \( n - 1 \) or the corresponding z-value if the population standard deviation is known.
Decision and Conclusion
If the calculated test statistic exceeds the critical value, or equivalently, if the p-value is less than 0.01, the null hypothesis is rejected, supporting the conclusion that the average attendance in 2002 is statistically greater than in 2001. Conversely, if the test statistic does not exceed the critical threshold, failure to reject H0 indicates insufficient evidence to claim an increase.
Given that actual sample data are not provided here, a hypothetical scenario can be considered for illustrative purposes:
Suppose the sample mean attendance in 2002 was 3,600 with a standard deviation of 400 across 100 games:
\[ t = \frac{3600 - 3530}{400 / \sqrt{100}} = \frac{70}{40} = 1.75. \]
Using a critical t-value for 99 degrees of freedom at \( \alpha = 0.01 \), approximately 2.626, we find \( 1.75
Discussion
Hypothesis testing provides a rigorous framework for evaluating claims about population parameters based on sample data. In this context, the test assesses whether the observed increase in attendance could reasonably be attributed to chance variation or signifies a real upward trend. The significance level critically influences the decision, with a lower alpha demanding stronger evidence for rejection. This approach ensures that decisions are made with quantifiable confidence, contributing to strategic planning for minor league baseball teams and the league's stakeholders.
Implications
Accurate assessment of attendance trends is vital for operational planning, resource allocation, and marketing strategies. If the null hypothesis is rejected, it indicates a successful effort in fan engagement, possibly justifying increased marketing expenditure or facility investments. If not rejected, efforts might shift toward targeted campaigns to push attendance higher or investigating other factors influencing attendance figures.
Conclusion
In conclusion, hypothesis testing is a crucial statistical tool applied to evaluate if the minor league baseball attendance in 2002 has truly surpassed the previous maximum, with the significance level set at 1%. Based on the example calculations, the evidence may not be sufficient to conclude an increase, but actual data should guide final decisions. Consistent monitoring and analysis of attendance data are essential for understanding and fostering growth within minor league baseball sectors.
References
- Hogg, R. V., & Tanis, E. A. (2015). Probability and Statistical Inference. Pearson.
- Moore, D. S., McCabe, G. P., & Craig, B. A. (2017). Introduction to the Practice of Statistics. W. H. Freeman.
- Wackerly, D., Mendenhall, W., & Scheaffer, R. (2014). Mathematical Statistics with Applications. Cengage Learning.
- New York Times. (2002). Minor League Baseball Attendance Records. https://www.nytimes.com
- Gelman, A., & Hill, J. (2006). Data Analysis Using Regression and Multilevel/Hierarchical Models. Cambridge University Press.
- Lehmann, E. L. (2005). Testing Statistical Hypotheses. Springer.
- Rumsey, D. J. (2016). Statistics For Dummies. Wiley Publishing.
- Bluman, A. G. (2017). Elementary Statistics: A step by step approach. McGraw-Hill Education.
- Wasserman, L. (2004). All of Statistics: A Concise Course in Statistical Inference. Springer.
- Schervish, M. J. (2012). Theory of Statistics. Springer.