The Length Of A Local Bus Journey From Anrai's Home To School Is A Normally Distributed Random Variable T 1 T_1 T 1 ​ With A Mean Of 31 Minutes And A Standard Deviation Of 5 Minutes. Sketch The Following Events On Three Separate Diagrams:a) $P(T_1

Introduction

In probability theory, a normal distribution is a continuous probability distribution that is commonly observed in many real-world phenomena. In this article, we will explore the concept of normal distribution using the example of a local bus journey from Anrai's home to school. We will discuss the probability of certain events occurring within a given time frame and sketch the corresponding diagrams.

Understanding Normal Distribution

A normal distribution is characterized by a bell-shaped curve, where the majority of the data points are concentrated around the mean, and the probability of extreme values decreases as we move away from the mean. In this case, the length of the bus journey, denoted by $T_1$, is a normally distributed random variable with a mean of 31 minutes and a standard deviation of 5 minutes.

Sketching the Distribution

To sketch the distribution of $T_1$, we need to understand the following:

• The mean, $\mu$, is the average value of the distribution, which is 31 minutes in this case.
• The standard deviation, $\sigma$, measures the spread or dispersion of the distribution, which is 5 minutes in this case.
• The probability density function (PDF) of a normal distribution is given by the formula:

$$f(x) = \frac{1}{\sigma \sqrt{2\pi}} \exp\left(-\frac{(x-\mu)^2}{2\sigma^2}\right)$$

Using this formula, we can calculate the probability of $T_1$ taking on a value within a given range.

Event a) $P(T_1 < 25)$

To sketch the event $P(T_1 < 25)$, we need to find the probability that the length of the bus journey is less than 25 minutes. We can do this by calculating the z-score of 25 minutes and then using a standard normal distribution table or calculator to find the corresponding probability.

The z-score is given by the formula:

$$z = \frac{x-\mu}{\sigma}$$

Plugging in the values, we get:

$$z = \frac{25-31}{5} = -1.2$$

Using a standard normal distribution table or calculator, we find that the probability of $T_1$ being less than 25 minutes is approximately 0.1151.

Event b) $P(T_1 > 35)$

To sketch the event $P(T_1 > 35)$, we need to find the probability that the length of the bus journey is greater than 35 minutes. We can do this by calculating the z-score of 35 minutes and then using a standard normal distribution table or calculator to find the corresponding probability.

The z-score is given by the formula:

$$z = \frac{x-\mu}{\sigma}$$

Plugging in the values, we get:

$$z = \frac{35-31}{5} = 0.8$$

Using a standard normal distribution table or calculator, we find that the probability of $T_1$ being greater than 35 minutes is approximately 0.2119.

Event c) $P(30 < T_1 < 32)$

To sketch the event $P(30 < T_1 < 32)$, we need to find the probability that the length of the bus journey is between 30 and 32 minutes. We can do this by calculating the z-scores of 30 and 32 minutes and then using a standard normal distribution table or calculator to find the corresponding probabilities.

The z-scores are given by the formula:

$$z = \frac{x-\mu}{\sigma}$$

Plugging in the values, we get:

$$z_{30} = \frac{30-31}{5} = -0.2$$

$$z_{32} = \frac{32-31}{5} = 0.2$$

Using a standard normal distribution table or calculator, we find that the probability of $T_1$ being between 30 and 32 minutes is approximately 0.1359.

Conclusion

In this article, we have explored the concept of normal distribution using the example of a local bus journey from Anrai's home to school. We have discussed the probability of certain events occurring within a given time frame and sketched the corresponding diagrams. We have also calculated the z-scores and used a standard normal distribution table or calculator to find the corresponding probabilities.

References

• [1] Johnson, R. A., & Bhattacharyya, G. (2014). Statistics: Principles and Methods. John Wiley & Sons.
• [2] Moore, D. S., & McCabe, G. P. (2013). Introduction to the Practice of Statistics. W.H. Freeman and Company.

Mathematical Formulas

• Normal Distribution Formula

$$f(x) = \frac{1}{\sigma \sqrt{2\pi}} \exp\left(-\frac{(x-\mu)^2}{2\sigma^2}\right)$$

• Z-Score Formula

$$z = \frac{x-\mu}{\sigma}$$

Probability Tables

• Standard Normal Distribution Table

z	P(z < z)
-3.09	0.0010
-2.58	0.0050
-1.96	0.0250
-1.64	0.0500
-1.28	0.1000
-0.84	0.2000
-0.52	0.3000
-0.22	0.4000
0.00	0.5000
0.22	0.6000
0.52	0.7000
0.84	0.8000
1.28	0.9000
1.64	0.9500
1.96	0.9750
2.58	0.9950
3.09	0.9990

Code

import numpy as np
from scipy.stats import norm
mu = 31
sigma = 5

z_25 = (25 - mu) / sigma
z_35 = (35 - mu) / sigma
z_30 = (30 - mu) / sigma
z_32 = (32 - mu) / sigma

prob_25 = norm.cdf(z_25)
prob_35 = 1 - norm.cdf(z_35)
prob_30_32 = norm.cdf(z_32) - norm.cdf(z_30)
print("P(T_1 < 25) =", prob_25)
print("P(T_1 > 35) =", prob_35)
print("P(30 < T_1 < 32) =", prob_30_32)

$$$$$$

Introduction

In our previous article, we explored the concept of normal distribution using the example of a local bus journey from Anrai's home to school. We discussed the probability of certain events occurring within a given time frame and sketched the corresponding diagrams. In this article, we will answer some frequently asked questions related to normal distribution and provide additional insights into the topic.

Q: What is normal distribution?

A: Normal distribution is a continuous probability distribution that is commonly observed in many real-world phenomena. It is characterized by a bell-shaped curve, where the majority of the data points are concentrated around the mean, and the probability of extreme values decreases as we move away from the mean.

Q: What are the key characteristics of a normal distribution?

A: The key characteristics of a normal distribution are:

• Mean: The average value of the distribution, which is denoted by μ.
• Standard Deviation: The spread or dispersion of the distribution, which is denoted by σ.
• Symmetry: The distribution is symmetric around the mean.
• Bell-Shaped Curve: The distribution has a bell-shaped curve, where the majority of the data points are concentrated around the mean.

Q: How do I calculate the probability of a normal distribution?

A: To calculate the probability of a normal distribution, you need to use the following formula:

$$P(x < a) = \Phi\left(\frac{a-\mu}{\sigma}\right)$$

where Φ is the cumulative distribution function (CDF) of the standard normal distribution.

Q: What is the z-score?

A: The z-score is a measure of how many standard deviations an observation is away from the mean. It is calculated using the following formula:

$$z = \frac{x-\mu}{\sigma}$$

Q: How do I use a z-table to find the probability of a normal distribution?

A: To use a z-table to find the probability of a normal distribution, you need to follow these steps:

1. Calculate the z-score using the formula: z = (x - μ) / σ
2. Look up the z-score in the z-table to find the corresponding probability
3. Use the probability to answer the question

Q: What is the difference between a normal distribution and a binomial distribution?

A: A normal distribution is a continuous probability distribution, while a binomial distribution is a discrete probability distribution. A normal distribution is used to model continuous data, such as heights or weights, while a binomial distribution is used to model discrete data, such as the number of heads in a coin toss.

Q: Can I use a normal distribution to model any type of data?

A: No, you cannot use a normal distribution to model any type of data. A normal distribution is only suitable for modeling continuous data that is symmetric around the mean. If your data is not symmetric or is discrete, you may need to use a different type of distribution, such as a binomial distribution or a Poisson distribution.

Q: How do I choose the right distribution for my data?

A: To choose the right distribution for your data, you need to follow these steps:

1. Examine the data to see if it is continuous or discrete
2. Check if the data is symmetric around the mean
3. Use a histogram or a density plot to visualize the data
4. Use statistical tests, such as the Shapiro-Wilk test, to determine if the data is normally distributed
5. Choose the distribution that best fits the data

Conclusion

In this article, we have answered some frequently asked questions related to normal distribution and provided additional insights into the topic. We have discussed the key characteristics of a normal distribution, how to calculate the probability of a normal distribution, and how to use a z-table to find the probability of a normal distribution. We have also discussed the difference between a normal distribution and a binomial distribution and how to choose the right distribution for your data.

References

• [1] Johnson, R. A., & Bhattacharyya, G. (2014). Statistics: Principles and Methods. John Wiley & Sons.
• [2] Moore, D. S., & McCabe, G. P. (2013). Introduction to the Practice of Statistics. W.H. Freeman and Company.

Mathematical Formulas

• Normal Distribution Formula

$$f(x) = \frac{1}{\sigma \sqrt{2\pi}} \exp\left(-\frac{(x-\mu)^2}{2\sigma^2}\right)$$

• Z-Score Formula

$$z = \frac{x-\mu}{\sigma}$$

Probability Tables

• Standard Normal Distribution Table

z	P(z < z)
-3.09	0.0010
-2.58	0.0050
-1.96	0.0250
-1.64	0.0500
-1.28	0.1000
-0.84	0.2000
-0.52	0.3000
-0.22	0.4000
0.00	0.5000
0.22	0.6000
0.52	0.7000
0.84	0.8000
1.28	0.9000
1.64	0.9500
1.96	0.9750
2.58	0.9950
3.09	0.9990

Code

import numpy as np
from scipy.stats import norm

mu = 31
sigma = 5

z_25 = (25 - mu) / sigma
z_35 = (35 - mu) / sigma
z_30 = (30 - mu) / sigma
z_32 = (32 - mu) / sigma

prob_25 = norm.cdf(z_25)
prob_35 = 1 - norm.cdf(z_35)
prob_30_32 = norm.cdf(z_32) - norm.cdf(z_30)
print("P(T_1 < 25) =", prob_25)
print("P(T_1 > 35) =", prob_35)
print("P(30 < T_1 < 32) =", prob_30_32)