A Study Of College Football Games Shows That The Number Of H ✓ Solved

A Study Of College Football Games Shows That the Number Of Holding Pen

A study of college football games shows that the number of holding penalties assessed has a mean of 2.3 penalties per game and a standard deviation of 0.9 penalties per game. What is the probability that, for a sample of 40 college games to be played next week, the mean number of holding penalties will be 2.05 penalties per game or less? Carry your intermediate computations to at least four decimal places and round your final answer accordingly.

Sample Paper For Above instruction

To determine the probability that the average number of holding penalties per game in a sample of 40 college football games is 2.05 penalties or less, we apply concepts from probability theory involving the sampling distribution of the sample mean.

Understanding the problem

Given data:

  • Population mean, μ = 2.3 penalties per game
  • Population standard deviation, σ = 0.9 penalties per game
  • Sample size, n = 40 games
  • Sample mean, x̄ = 2.05 penalties per game

Objective: Find the probability that the sample mean (x̄) ≤ 2.05 penalties per game.

Applying the Central Limit Theorem

The Central Limit Theorem states that for sufficiently large n, the sampling distribution of the sample mean will be approximately normal, regardless of the population's distribution shape. Since n = 40 is generally considered large enough, we proceed assuming a normal distribution with mean μ and standard error of the mean:

Standard error of the mean (SE) = σ / √n = 0.9 / √40 ≈ 0.9 / 6.3246 ≈ 0.1421

Calculating the Z-score

The Z-score corresponding to the sample mean of 2.05 is:

Z = (x̄ - μ) / SE = (2.05 - 2.3) / 0.1421 ≈ (-0.25) / 0.1421 ≈ -1.7599

Finding the probability

Using standard normal distribution tables or a calculator, we find the probability associated with Z = -1.7599:

P(Z ≤ -1.7599) ≈ 0.0394 (to four decimal places)

Final answer

The probability that the mean number of holding penalties in a sample of 40 games is 2.05 or less is approximately 0.0394, or 3.94%.

Conclusion

This low probability indicates that observing such a low average penalty count (2.05) across 40 games is unlikely, assuming the population parameters hold true.

References

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