The Times Of The Runners In A Marathon Are Normally Distributed, With A Mean Of 3 Hours And 50 Minutes And A Standard Deviation Of 30 Minutes. What Is The Probability That A Randomly Selected Runner Has A Time Less Than Or Equal To 3 Hours And 20

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Introduction

In the world of sports, particularly in marathons, understanding the distribution of times is crucial for athletes, coaches, and organizers. The normal distribution, also known as the Gaussian distribution, is a fundamental concept in statistics that helps us understand how data is spread out. In this article, we will explore the normal distribution of marathon times and calculate the probability of a randomly selected runner having a time less than or equal to 3 hours and 20 minutes.

Understanding Normal Distribution

The normal distribution is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. In the context of marathon times, the mean time is 3 hours and 50 minutes, and the standard deviation is 30 minutes. This means that most runners will have times close to the mean, with fewer runners having times significantly above or below the mean.

Calculating the Probability

To calculate the probability of a randomly selected runner having a time less than or equal to 3 hours and 20 minutes, we need to use the z-score formula. The z-score formula is:

z = (X - μ) / σ

where X is the value we are interested in (3 hours and 20 minutes), μ is the mean (3 hours and 50 minutes), and σ is the standard deviation (30 minutes).

First, we need to convert the times to minutes:

3 hours and 20 minutes = 200 minutes 3 hours and 50 minutes = 230 minutes

Now, we can plug in the values into the z-score formula:

z = (200 - 230) / 30 z = -30 / 30 z = -1

Interpreting the Z-Score

The z-score tells us how many standard deviations away from the mean our value is. In this case, the z-score is -1, which means that the time of 3 hours and 20 minutes is 1 standard deviation below the mean.

Using a Standard Normal Distribution Table

To find the probability of a randomly selected runner having a time less than or equal to 3 hours and 20 minutes, we need to use a standard normal distribution table (also known as a z-table). The z-table shows the probability of a value being less than or equal to a given z-score.

Looking up the z-score of -1 in the z-table, we find that the probability is approximately 0.1587. This means that about 15.87% of runners will have a time less than or equal to 3 hours and 20 minutes.

Conclusion

In conclusion, the normal distribution of marathon times is a useful tool for understanding how data is spread out. By using the z-score formula and a standard normal distribution table, we can calculate the probability of a randomly selected runner having a time less than or equal to 3 hours and 20 minutes. This information can be useful for athletes, coaches, and organizers to better understand the distribution of times and make informed decisions.

References

  • Moore, D. S., & McCabe, G. P. (2013). Introduction to the practice of statistics. W.H. Freeman and Company.
  • Johnson, R. A., & Bhattacharyya, G. K. (2014). Statistics: Principles and methods. John Wiley & Sons.

Further Reading

  • For more information on normal distribution, see the article "Understanding Normal Distribution" by [Author].
  • For more information on z-scores, see the article "Calculating Z-Scores" by [Author].
  • For more information on standard normal distribution tables, see the article "Using a Standard Normal Distribution Table" by [Author].

Mathematical Formulas

  • z = (X - μ) / σ
  • P(X ≤ x) = P(Z ≤ z)

Mathematical Symbols

  • μ: mean
  • σ: standard deviation
  • X: value
  • Z: z-score
  • P: probability

Code

import numpy as np

# Define the mean and standard deviation
mean = 230  # minutes
std_dev = 30  # minutes

# Define the value of interest
value = 200  # minutes

# Calculate the z-score
z = (value - mean) / std_dev

# Print the z-score
print("Z-score:", z)

# Use a standard normal distribution table to find the probability
probability = 0.1587

# Print the probability
print("Probability:", probability)

Introduction

In our previous article, we explored the normal distribution of marathon times and calculated the probability of a randomly selected runner having a time less than or equal to 3 hours and 20 minutes. In this article, we will answer some frequently asked questions related to normal distribution and marathon times.

Q&A

Q: What is the normal distribution?

A: The normal distribution, also known as the Gaussian distribution, is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean.

Q: What is the mean and standard deviation of marathon times?

A: The mean time of marathon runners is 3 hours and 50 minutes, and the standard deviation is 30 minutes.

Q: How do I calculate the probability of a randomly selected runner having a time less than or equal to 3 hours and 20 minutes?

A: To calculate the probability, you need to use the z-score formula: z = (X - μ) / σ, where X is the value of interest (3 hours and 20 minutes), μ is the mean (3 hours and 50 minutes), and σ is the standard deviation (30 minutes). Then, use a standard normal distribution table to find the probability.

Q: What is the z-score?

A: The z-score tells us how many standard deviations away from the mean our value is. In this case, the z-score is -1, which means that the time of 3 hours and 20 minutes is 1 standard deviation below the mean.

Q: How do I use a standard normal distribution table?

A: To use a standard normal distribution table, look up the z-score in the table and find the corresponding probability. In this case, the z-score of -1 corresponds to a probability of approximately 0.1587.

Q: What is the probability of a randomly selected runner having a time less than or equal to 3 hours and 20 minutes?

A: The probability is approximately 0.1587, which means that about 15.87% of runners will have a time less than or equal to 3 hours and 20 minutes.

Q: Can I use a calculator or software to calculate the probability?

A: Yes, you can use a calculator or software such as Excel or Python to calculate the probability. However, for more accurate calculations, use a reliable statistical software or library.

Q: What are some real-world applications of normal distribution?

A: Normal distribution has many real-world applications, including finance, engineering, and sports. For example, in finance, the normal distribution is used to model stock prices and returns. In engineering, the normal distribution is used to model the distribution of measurements and errors. In sports, the normal distribution is used to model the distribution of player performance and game outcomes.

Q: Can I use normal distribution to model other types of data?

A: Yes, you can use normal distribution to model other types of data, such as heights, weights, and IQ scores. However, the normal distribution is not always the best model for all types of data. Other distributions, such as the Poisson distribution and the binomial distribution, may be more suitable for certain types of data.

Conclusion

In conclusion, the normal distribution of marathon times is a useful tool for understanding how data is spread out. By using the z-score formula and a standard normal distribution table, we can calculate the probability of a randomly selected runner having a time less than or equal to 3 hours and 20 minutes. This information can be useful for athletes, coaches, and organizers to better understand the distribution of times and make informed decisions.

References

  • Moore, D. S., & McCabe, G. P. (2013). Introduction to the practice of statistics. W.H. Freeman and Company.
  • Johnson, R. A., & Bhattacharyya, G. K. (2014). Statistics: Principles and methods. John Wiley & Sons.

Further Reading

  • For more information on normal distribution, see the article "Understanding Normal Distribution" by [Author].
  • For more information on z-scores, see the article "Calculating Z-Scores" by [Author].
  • For more information on standard normal distribution tables, see the article "Using a Standard Normal Distribution Table" by [Author].

Mathematical Formulas

  • z = (X - μ) / σ
  • P(X ≤ x) = P(Z ≤ z)

Mathematical Symbols

  • μ: mean
  • σ: standard deviation
  • X: value
  • Z: z-score
  • P: probability

Code

import numpy as np

# Define the mean and standard deviation
mean = 230  # minutes
std_dev = 30  # minutes

# Define the value of interest
value = 200  # minutes

# Calculate the z-score
z = (value - mean) / std_dev

# Print the z-score
print("Z-score:", z)

# Use a standard normal distribution table to find the probability
probability = 0.1587

# Print the probability
print("Probability:", probability)

Note: The code above is a simple example and is not intended to be used in production. For more accurate calculations, use a reliable statistical software or library.