The Time It Takes To Drive From New York To Florida Is Normally Distributed With $\mu = 19$ Hours And $\sigma = 2.5$ Hours. What Is The Probability That The Next Time Someone Drives From NY To FL It Will Take Between 20 And 24 Hours?

Feb 28, 2025 by ADMIN 234 views

**The Time it Takes to Drive from New York to Florida: A Statistical Analysis**

Introduction

The time it takes to drive from New York to Florida is a random variable that can be modeled using a normal distribution. In this article, we will analyze the probability of driving from NY to FL within a specific time frame, given the mean and standard deviation of the distribution.

Understanding the Normal Distribution

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. In this case, the mean time it takes to drive from NY to FL is 19 hours, and the standard deviation is 2.5 hours.

Calculating the Probability

To calculate the probability that the next time someone drives from NY to FL it will take between 20 and 24 hours, we need to use the z-score formula:

z = (X - μ) / σ

where X is the value we are interested in (20 or 24 hours), μ is the mean (19 hours), and σ is the standard deviation (2.5 hours).

Calculating the z-score for 20 hours

z = (20 - 19) / 2.5 z = 1 / 2.5 z = 0.4

Calculating the z-score for 24 hours

z = (24 - 19) / 2.5 z = 5 / 2.5 z = 2

Using a Standard Normal Distribution Table

To find the probability that the time it takes to drive from NY to FL is between 20 and 24 hours, we need to use a standard normal distribution table (also known as a z-table). The z-table shows the probability that a random variable with a standard normal distribution will take on a value less than or equal to a given z-score.

Finding the probability for z = 0.4

Using the z-table, we find that the probability that a random variable with a standard normal distribution will take on a value less than or equal to 0.4 is approximately 0.6554.

Finding the probability for z = 2

Using the z-table, we find that the probability that a random variable with a standard normal distribution will take on a value less than or equal to 2 is approximately 0.9772.

Calculating the Final Probability

To find the probability that the time it takes to drive from NY to FL is between 20 and 24 hours, we need to subtract the probability that the time is less than 20 hours from the probability that the time is less than 24 hours.

P(20 < X < 24) = P(X < 24) - P(X < 20) = 0.9772 - 0.6554 = 0.3218

Conclusion

In this article, we analyzed the probability of driving from NY to FL within a specific time frame, given the mean and standard deviation of the distribution. We used the z-score formula to calculate the z-scores for 20 and 24 hours, and then used a standard normal distribution table to find the probabilities that the time is less than or equal to these values. Finally, we calculated the final probability that the time it takes to drive from NY to FL is between 20 and 24 hours.

References

[1] Moore, D. S., & McCabe, G. P. (2017). Introduction to the practice of statistics. W.H. Freeman and Company.
[2] Ross, S. M. (2017). Introduction to probability models. Academic Press.

Mathematical Formulas

z = (X - μ) / σ
P(X < x) = Φ(z)
Φ(z) = 1 - Φ(-z)

Code

import numpy as np

def calculate_z_score(x, mu, sigma):
    return (x - mu) / sigma

def calculate_probability(z):
    return 1 - np.exp(-z**2 / 2) / np.sqrt(2 * np.pi)

# Calculate z-scores
z_20 = calculate_z_score(20, 19, 2.5)
z_24 = calculate_z_score(24, 19, 2.5)

# Calculate probabilities
p_20 = calculate_probability(z_20)
p_24 = calculate_probability(z_24)

# Calculate final probability
p_final = p_24 - p_20

print("The final probability is:", p_final)

Introduction

In our previous article, we analyzed the probability of driving from NY to FL within a specific time frame, given the mean and standard deviation of the distribution. We used the z-score formula to calculate the z-scores for 20 and 24 hours, and then used a standard normal distribution table to find the probabilities that the time is less than or equal to these values. In this article, we will answer some frequently asked questions related to the topic.

Q: What is the mean time it takes to drive from NY to FL?

A: The mean time it takes to drive from NY to FL is 19 hours.

Q: What is the standard deviation of the time it takes to drive from NY to FL?

A: The standard deviation of the time it takes to drive from NY to FL is 2.5 hours.

Q: How do I calculate the z-score for a given time?

A: To calculate the z-score for a given time, you can use the following formula:

z = (X - μ) / σ

where X is the time you are interested in, μ is the mean (19 hours), and σ is the standard deviation (2.5 hours).

Q: What is the probability that the time it takes to drive from NY to FL is less than 20 hours?

A: Using the z-table, we find that the probability that a random variable with a standard normal distribution will take on a value less than or equal to 0.4 is approximately 0.6554.

Q: What is the probability that the time it takes to drive from NY to FL is less than 24 hours?

A: Using the z-table, we find that the probability that a random variable with a standard normal distribution will take on a value less than or equal to 2 is approximately 0.9772.

Q: How do I calculate the final probability that the time it takes to drive from NY to FL is between 20 and 24 hours?

A: To calculate the final probability, you can subtract the probability that the time is less than 20 hours from the probability that the time is less than 24 hours.

P(20 < X < 24) = P(X < 24) - P(X < 20) = 0.9772 - 0.6554 = 0.3218

Q: Can I use this analysis to predict the time it will take to drive from NY to FL for a specific trip?

A: No, this analysis is based on historical data and is not a prediction of the time it will take to drive from NY to FL for a specific trip. There are many factors that can affect the time it takes to drive from NY to FL, such as traffic, road conditions, and weather.

Q: Can I use this analysis to determine the best time to drive from NY to FL?

A: Yes, you can use this analysis to determine the best time to drive from NY to FL. If you want to minimize the time it takes to drive from NY to FL, you should try to avoid driving during peak traffic hours (usually during rush hour) and try to drive during off-peak hours.

Q: Can I use this analysis to determine the best route to drive from NY to FL?

A: No, this analysis is based on the time it takes to drive from NY to FL and does not take into account the route you take. There are many factors that can affect the time it takes to drive from NY to FL, such as traffic, road conditions, and weather.

Conclusion

In this article, we answered some frequently asked questions related to the topic of the time it takes to drive from NY to FL. We hope that this article has been helpful in answering your questions and providing you with a better understanding of the topic.

References

[1] Moore, D. S., & McCabe, G. P. (2017). Introduction to the practice of statistics. W.H. Freeman and Company.
[2] Ross, S. M. (2017). Introduction to probability models. Academic Press.

Mathematical Formulas

z = (X - μ) / σ
P(X < x) = Φ(z)
Φ(z) = 1 - Φ(-z)

Code

import numpy as np

def calculate_z_score(x, mu, sigma):
    return (x - mu) / sigma

def calculate_probability(z):
    return 1 - np.exp(-z**2 / 2) / np.sqrt(2 * np.pi)

# Calculate z-scores
z_20 = calculate_z_score(20, 19, 2.5)
z_24 = calculate_z_score(24, 19, 2.5)

# Calculate probabilities
p_20 = calculate_probability(z_20)
p_24 = calculate_probability(z_24)

# Calculate final probability
p_final = p_24 - p_20

print("The final probability is:", p_final)

Note: The code is provided as an example and may not be the most efficient or accurate way to calculate the probability.