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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

The analysis of whether the mean attendance per game in 2002 exceeded that of 2001 involves formulating and testing appropriate statistical hypotheses. The primary goal is to determine if there is sufficient evidence to support the claim that the average attendance has increased, based on the sample data collected midway through the 2002 season. This process involves setting up null and alternative hypotheses, choosing a significance level, calculating the relevant test statistic, and making a decision based on the p-value or critical value approach.

First, we establish the hypotheses. The null hypothesis (H₀) posits that there is no change in the mean attendance per game from 2001 to 2002, meaning:

  • H₀: μ₂ ≤ μ₁, where μ₂ is the mean attendance in 2002 and μ₁ is the mean attendance in 2001, which is 3,530 people per game.

The alternative hypothesis (H₁) asserts that the mean attendance in 2002 is greater than in 2001, indicating an increase:

  • H₁: μ₂ > μ₁ (where μ₁ = 3,530).

This hypothesis test is a one-sided (right-tailed) test of the population mean. The test typically involves comparing the sample mean attendance in 2002, \(\bar{x}\), against the known population mean from 2001, 3,530, using the sample standard deviation and size to compute the test statistic. The choice of significance level, α = 0.01, is strict, reflecting a desire for strong evidence before concluding there has been an increase.

The test statistic for assessing the mean (assuming the population standard deviation is unknown and the sample size is sufficiently large or the population standard deviation is known) is generally a z-statistic or t-statistic. The formula for the z-test, under the assumption of a known population standard deviation, is:

\[ z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}} \]

where \(\bar{x}\) is the sample mean attendance, \(\mu_0 = 3530\) is the population mean in 2001, \(\sigma\) is the population standard deviation (if known), and \(n\) is the sample size.

If \(\sigma\) is unknown and the sample size is small, a t-test should be used instead with the sample standard deviation (s). The decision rule involves comparing the calculated test statistic with the critical value from the z or t-distribution at α = 0.01 for a right-tailed test.

Based on the findings, if the calculated p-value is less than 0.01, we reject the null hypothesis and conclude that there is statistically significant evidence that the mean attendance per game in 2002 exceeds 2001 levels. Conversely, if the p-value exceeds 0.01, we fail to reject the null hypothesis, indicating insufficient evidence to claim an increase.

In conclusion, the hypothesis testing process provides a formal method to assess whether the observed data from the 2002 season supports the claim that attendance has increased beyond the previous year's figure. The outcome relies heavily on the sample data collected midway through the season, the variability of attendance, and the sample size. A rigorous statistical conclusion about the increase in attendance can help the league’s management make informed decisions regarding their operations and marketing strategies.

References

  • New York Times. (2002). Attendance records in minor league baseball. Retrieved from https://www.nytimes.com
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