Using the Portion of the Standard Normal Table

Oct 13, 2025 by recoveryshake.com 88 views

The standard normal distribution, also known as the z-distribution, is a probability distribution that is symmetric about the mean of 0 and has a standard deviation of 1. It is a fundamental concept in statistics and is used to model a wide range of phenomena. In this article, we will explore how to find the approximate value of the probability that a standard normal random variable takes on a value less than or equal to 0.42.

The Standard Normal Table

The standard normal table is a table that lists the probabilities of a standard normal random variable taking on values less than or equal to a given value. The table is typically organized in a way that the values of z are listed in the first column, and the corresponding probabilities are listed in the second column. The table is usually divided into two parts: the left-hand side of the table lists the probabilities for negative values of z, and the right-hand side lists the probabilities for positive values of z.

Finding the Approximate Value of P(z ≤ 0.42)

To find the approximate value of P(z ≤ 0.42), we can use the standard normal table. Since the value of z is positive, we will look at the right-hand side of the table. We will start by looking at the row that corresponds to the value of z = 0.40. The probability listed in this row is 0.6554. Since the value of z that we are interested in is 0.42, which is less than 0.40, we know that the probability of z taking on a value less than or equal to 0.42 is greater than 0.6554.

Using the Portion of the Standard Normal Table

To find the approximate value of P(z ≤ 0.42), we can use the portion of the standard normal table that lists the probabilities for values of z between 0.40 and 0.50. This portion of the table is as follows:

z	Probability
0.40	0.6554
0.41	0.6571
0.42	0.6588
0.43	0.6605
0.44	0.6622
0.45	0.6639
0.46	0.6656
0.47	0.6673
0.48	0.6690
0.49	0.6707
0.50	0.6724

Interpolating the Value of P(z ≤ 0.42)

Since the value of z that we are interested in is 0.42, which is between 0.40 and 0.50, we can use the portion of the table to interpolate the value of P(z ≤ 0.42). We can do this by finding the difference between the probabilities listed in the rows that correspond to the values of z = 0.40 and z = 0.42, and then multiplying this difference by the fraction of the distance between 0.40 and 0.42 that 0.42 is from 0.40.

Calculating the Approximate Value of P(z ≤ 0.42)

To calculate the approximate value of P(z ≤ 0.42), we can use the following formula:

P(z ≤ 0.42) = P(z ≤ 0.40) + (0.42 - 0.40) * (P(z ≤ 0.42) - P(z ≤ 0.40)) / (0.42 - 0.40)

Using the values listed in the portion of the table, we can calculate the approximate value of P(z ≤ 0.42) as follows:

P(z ≤ 0.42) = 0.6554 + (0.42 - 0.40) * (0.6588 - 0.6554) / (0.42 - 0.40) = 0.6554 + 0.02 * 0.0034 / 0.02 = 0.6554 + 0.00017 = 0.65557

Conclusion

In this article, we have explored how to find the approximate value of the probability that a standard normal random variable takes on a value less than or equal to 0.42. We have used the standard normal table to help answer the question, and have interpolated the value of P(z ≤ 0.42) using the portion of the table that lists the probabilities for values of z between 0.40 and 0.50. The approximate value of P(z ≤ 0.42) is 0.65557.

References

Moore, D. S., & McCabe, G. P. (2011). Introduction to the practice of statistics. W.H. Freeman and Company.
Johnson, R. A., & Bhattacharyya, G. K. (2010). Statistics: Principles and methods. John Wiley & Sons.

Discussion

The standard normal distribution is a fundamental concept in statistics, and is used to model a wide range of phenomena. The standard normal table is a useful tool for finding the probabilities of a standard normal random variable taking on values less than or equal to a given value. In this article, we have explored how to use the standard normal table to find the approximate value of P(z ≤ 0.42). We have also discussed the importance of interpolating the value of P(z ≤ 0.42) using the portion of the table that lists the probabilities for values of z between 0.40 and 0.50.

Future Work

In future work, we can explore other applications of the standard normal distribution, such as finding the probabilities of a standard normal random variable taking on values greater than or equal to a given value. We can also discuss the importance of using the standard normal table in real-world applications, such as in finance and engineering.

Conclusion

In the previous article, we explored how to find the approximate value of the probability that a standard normal random variable takes on a value less than or equal to 0.42 using the standard normal table. In this article, we will answer some frequently asked questions about the standard normal distribution and the standard normal table.

Q: What is the standard normal distribution?

A: The standard normal distribution, also known as the z-distribution, is a probability distribution that is symmetric about the mean of 0 and has a standard deviation of 1. It is a fundamental concept in statistics and is used to model a wide range of phenomena.

Q: What is the standard normal table?

A: The standard normal table is a table that lists the probabilities of a standard normal random variable taking on values less than or equal to a given value. The table is typically organized in a way that the values of z are listed in the first column, and the corresponding probabilities are listed in the second column.

Q: How do I use the standard normal table to find the probability of a standard normal random variable taking on a value less than or equal to a given value?

A: To use the standard normal table to find the probability of a standard normal random variable taking on a value less than or equal to a given value, you need to look up the value of z in the table and find the corresponding probability. If the value of z is positive, you will look at the right-hand side of the table. If the value of z is negative, you will look at the left-hand side of the table.

Q: What if the value of z is not listed in the standard normal table?

A: If the value of z is not listed in the standard normal table, you can use interpolation to estimate the probability. This involves finding the probabilities listed in the rows that correspond to the values of z that are closest to the value of z you are interested in, and then using these probabilities to estimate the probability of interest.

Q: What are some common applications of the standard normal distribution?

A: The standard normal distribution has many applications in statistics and other fields. Some common applications include:

Modeling the distribution of scores on a standardized test
Modeling the distribution of heights or weights in a population
Modeling the distribution of stock prices or returns
Modeling the distribution of weather patterns or climate variables

Q: What are some common mistakes to avoid when using the standard normal table?

A: Some common mistakes to avoid when using the standard normal table include:

Not checking the units of the values of z listed in the table
Not checking the sign of the values of z listed in the table
Not using interpolation correctly when the value of z is not listed in the table
Not considering the assumptions of the standard normal distribution when using it to model a real-world phenomenon

Q: What are some resources for learning more about the standard normal distribution and the standard normal table?

A: Some resources for learning more about the standard normal distribution and the standard normal table include:

Textbooks on statistics and probability
Online tutorials and courses on statistics and probability
Research articles and papers on the standard normal distribution and its applications
Software packages and programming languages that implement the standard normal distribution and its applications

Conclusion

In conclusion, the standard normal distribution and the standard normal table are fundamental concepts in statistics and probability. They have many applications in real-world phenomena, and are used to model a wide range of distributions. By understanding how to use the standard normal table and avoiding common mistakes, you can use the standard normal distribution to model and analyze real-world phenomena.

References

Moore, D. S., & McCabe, G. P. (2011). Introduction to the practice of statistics. W.H. Freeman and Company.
Johnson, R. A., & Bhattacharyya, G. K. (2010). Statistics: Principles and methods. John Wiley & Sons.
Casella, G., & Berger, R. L. (2002). Statistical inference. Duxbury Press.

Discussion

The standard normal distribution and the standard normal table are fundamental concepts in statistics and probability. They have many applications in real-world phenomena, and are used to model a wide range of distributions. By understanding how to use the standard normal table and avoiding common mistakes, you can use the standard normal distribution to model and analyze real-world phenomena.