The Attendance At Baseball Games At A Certain Stadium Is Nor ✓ Solved

The Attendance At Baseball Games At A Certain Stadium Is Normally Dist

The attendance at baseball games at a certain stadium is normally distributed, with a mean of 23,000 and a standard deviation of 1,100. For any given game, perform the following analyses:

  1. Calculate the probability that attendance exceeds 25,500.
  2. Determine the probability that attendance is 24,000 or more.
  3. Find the probability that attendance falls between 21,000 and 25,000.
  4. Identify the attendance level that places a game in the top 10% of all games.
  5. Calculate the probability that attendance is less than 22,000.

Sample Paper For Above instruction

Introduction

The analysis of attendance at baseball games involves understanding the distribution patterns and calculating probabilities based on the normal distribution. With a mean attendance of 23,000 and a standard deviation of 1,100, researchers and stadium managers can evaluate various attendance scenarios, which are crucial for planning and operational decision-making.

Understanding the Normal Distribution Parameters

Given:

- Mean (μ) = 23,000

- Standard deviation (σ) = 1,100

The use of the normal distribution enables the computation of probabilities related to attendance figures.

Part A: Probability that Attendance is Greater Than 25,500

To find P(X > 25,500), first compute the z-score:

z = (X - μ) / σ = (25,500 - 23,000) / 1,100 ≈ 2.27

Using standard normal distribution tables or software:

P(Z > 2.27) ≈ 0.0116

Interpretation: There is approximately a 1.16% chance that audience attendance exceeds 25,500 at any given game.

Part B: Probability that Attendance is 24,000 or More

Calculate the z-score for 24,000:

z = (24,000 - 23,000) / 1,100 ≈ 0.91

Find P(X ≥ 24,000):

P(Z ≥ 0.91) ≈ 1 - P(Z ≤ 0.91) ≈ 1 - 0.8186 ≈ 0.1814

Interpretation: There is roughly an 18.14% probability that attendance will be at least 24,000.

Part C: Probability of Attendance Between 21,000 and 25,000

Calculate z-scores:

z₁ = (21,000 - 23,000) / 1,100 ≈ -1.82

z₂ = (25,000 - 23,000) / 1,100 ≈ 1.82

Find the probabilities:

P(Z between -1.82 and 1.82) = P(Z ≤ 1.82) - P(Z ≤ -1.82) ≈ 0.9656 - 0.0344 = 0.9312

Interpretation: About 93.12% of games have attendance between 21,000 and 25,000.

Part D: Attendance Level for Top 10% of All Games

Find the z-score corresponding to the 90th percentile:

From the standard normal table, z ≈ 1.28

Calculate the attendance:

X = μ + zσ = 23,000 + (1.28)(1,100) ≈ 23,000 + 1,408 ≈ 24,408

Interpretation: A game must have an attendance of approximately 24,408 or more to be in the top 10%.

Part E: Probability that Attendance is Less Than 22,000

Calculate z:

z = (22,000 - 23,000) / 1,100 ≈ -0.91

Using the cumulative distribution:

P(Z

Interpretation: There is about an 18.14% chance that attendance will be below 22,000.

Conclusion

This analysis demonstrates how normal distribution calculations provide valuable insights into attendance patterns at baseball games. These probabilities assist stadium management in planning for various scenarios, understanding streaks of high or low attendance, and setting benchmarks for exceptional games.

References

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