When N = 25, The Probability P(t> 0.89) = [?] Round To Nearest Ten-thousandth.

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Understanding the Probability Distribution of the Standard Normal Variable

In probability theory, the standard normal distribution, also known as the z-distribution, is a widely used distribution that is used to model a large number of phenomena in various fields, including finance, engineering, and social sciences. The standard normal distribution is a continuous probability distribution with a mean of 0 and a standard deviation of 1. In this article, we will discuss how to calculate the probability of a standard normal variable exceeding a certain value, given a specific value of n.

The Standard Normal Distribution

The standard normal distribution is a symmetric distribution, meaning that it is symmetric around the mean. The distribution is characterized by the following probability density function (PDF):

f(z) = (1/√(2π)) * e(-z2/2)

where z is the standard normal variable, and e is the base of the natural logarithm.

Calculating the Probability

Given a value of n, we want to calculate the probability that a standard normal variable exceeds a certain value, say t. In this case, we are given that n = 25, and we want to calculate the probability P(t > 0.89).

To calculate this probability, we can use the cumulative distribution function (CDF) of the standard normal distribution, which is given by:

Φ(z) = (1/√(2π)) * ∫[0, z] e(-x2/2) dx

where Φ(z) is the CDF of the standard normal distribution, and x is the standard normal variable.

Using the Z-Table

To calculate the probability P(t > 0.89), we can use the z-table, which is a table that lists the values of the CDF of the standard normal distribution for various values of z. The z-table is typically used to find the probability that a standard normal variable falls within a certain range.

However, in this case, we want to find the probability that the standard normal variable exceeds a certain value, say 0.89. To do this, we can use the z-table to find the value of the CDF at z = 0.89, and then subtract this value from 1 to get the desired probability.

Calculating the Probability

Using the z-table, we find that the value of the CDF at z = 0.89 is approximately 0.8159. Therefore, the probability P(t > 0.89) is given by:

P(t > 0.89) = 1 - Φ(0.89) = 1 - 0.8159 = 0.1841

Rounding to the Nearest Ten-Thousandth

Rounding the probability to the nearest ten-thousandth, we get:

P(t > 0.89) = 0.1841

In this article, we discussed how to calculate the probability of a standard normal variable exceeding a certain value, given a specific value of n. We used the cumulative distribution function (CDF) of the standard normal distribution to calculate the probability, and then used the z-table to find the value of the CDF at the desired value of z. We also rounded the probability to the nearest ten-thousandth.

  • [1] Johnson, N. L., & Kotz, S. (1970). Distributions in statistics: Continuous univariate distributions-1. Wiley.
  • [2] Evans, M., Hastings, N., & Peacock, B. (2000). Statistical distributions. Wiley.
  • [3] Mood, A. M., Graybill, F. A., & Boes, D. C. (1974). Introduction to the theory of statistics. McGraw-Hill.
  • [1] Wikipedia: Standard normal distribution
  • [2] Khan Academy: Standard normal distribution
  • [3] Mathway: Standard normal distribution
    Frequently Asked Questions (FAQs) About the Standard Normal Distribution

In our previous article, we discussed how to calculate the probability of a standard normal variable exceeding a certain value, given a specific value of n. In this article, we will answer some frequently asked questions (FAQs) about the standard normal distribution.

Q: What is the standard normal distribution?

A: The standard normal distribution, also known as the z-distribution, is a widely used distribution that is used to model a large number of phenomena in various fields, including finance, engineering, and social sciences. The standard normal distribution is a continuous probability distribution with a mean of 0 and a standard deviation of 1.

Q: What is the difference between the standard normal distribution and the normal distribution?

A: The standard normal distribution and the normal distribution are both continuous probability distributions, but they have different means and standard deviations. The standard normal distribution has a mean of 0 and a standard deviation of 1, while the normal distribution has a mean of μ and a standard deviation of σ.

Q: How do I calculate the probability of a standard normal variable exceeding a certain value?

A: To calculate the probability of a standard normal variable exceeding a certain value, you can use the cumulative distribution function (CDF) of the standard normal distribution. The CDF is given by:

Φ(z) = (1/√(2π)) * ∫[0, z] e(-x2/2) dx

where Φ(z) is the CDF of the standard normal distribution, and x is the standard normal variable.

Q: What is the z-table, and how do I use it?

A: The z-table is a table that lists the values of the CDF of the standard normal distribution for various values of z. To use the z-table, you need to find the value of the CDF at the desired value of z, and then subtract this value from 1 to get the desired probability.

Q: Can I use the z-table to find the probability that a standard normal variable falls within a certain range?

A: Yes, you can use the z-table to find the probability that a standard normal variable falls within a certain range. To do this, you need to find the values of the CDF at the upper and lower bounds of the range, and then subtract the smaller value from the larger value to get the desired probability.

Q: What is the relationship between the standard normal distribution and the normal distribution?

A: The standard normal distribution is a special case of the normal distribution, where the mean is 0 and the standard deviation is 1. The normal distribution can be transformed into the standard normal distribution by subtracting the mean and dividing by the standard deviation.

Q: Can I use the standard normal distribution to model real-world phenomena?

A: Yes, the standard normal distribution can be used to model a wide range of real-world phenomena, including financial returns, engineering measurements, and social science data. However, you need to be careful to check the assumptions of the standard normal distribution before using it to model a particular phenomenon.

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

A: The standard normal distribution has a wide range of applications in various fields, including finance, engineering, and social sciences. Some common applications include:

  • Modeling financial returns and risk
  • Analyzing engineering measurements and quality control
  • Studying social science data and behavior
  • Predicting outcomes in sports and other competitive events

In this article, we answered some frequently asked questions (FAQs) about the standard normal distribution. We discussed the definition and properties of the standard normal distribution, how to calculate the probability of a standard normal variable exceeding a certain value, and some common applications of the standard normal distribution.

  • [1] Johnson, N. L., & Kotz, S. (1970). Distributions in statistics: Continuous univariate distributions-1. Wiley.
  • [2] Evans, M., Hastings, N., & Peacock, B. (2000). Statistical distributions. Wiley.
  • [3] Mood, A. M., Graybill, F. A., & Boes, D. C. (1974). Introduction to the theory of statistics. McGraw-Hill.
  • [1] Wikipedia: Standard normal distribution
  • [2] Khan Academy: Standard normal distribution
  • [3] Mathway: Standard normal distribution