For A Standard Normal Distribution, If $P (z \leq A) = 0.7116$, What Is The Value Of $P (z \geq A)$?A. 0.2116 B. 0.2884 C. 0.7116 D. 0.7884

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The standard normal distribution, also known as the z-distribution, is a probability distribution that is symmetric about the mean, with a mean of 0 and 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 the properties of the standard normal distribution and how to calculate probabilities using this distribution.

Properties of the Standard Normal Distribution

The standard normal distribution has several important properties that make it a useful tool for statistical analysis. Some of the key properties include:

  • Symmetry: The standard normal distribution is symmetric about the mean, which means that the probability of a value being above the mean is equal to the probability of a value being below the mean.
  • Mean: The mean of the standard normal distribution is 0, which means that the distribution is centered around 0.
  • Standard Deviation: The standard deviation of the standard normal distribution is 1, which means that the distribution is spread out over a range of 2 standard deviations (i.e., from -1 to 1).
  • Probability Density Function: The probability density function (PDF) of the standard normal distribution is given by the equation:

f(z)=12πe−z22f(z) = \frac{1}{\sqrt{2\pi}}e^{-\frac{z^2}{2}}

Calculating Probabilities using the Standard Normal Distribution

One of the most important applications of the standard normal distribution is calculating probabilities. To calculate a probability using the standard normal distribution, we need to know the value of the z-score, which is the number of standard deviations away from the mean. The z-score can be calculated using the following formula:

z=x−μσz = \frac{x - \mu}{\sigma}

where x is the value of interest, μ is the mean, and σ is the standard deviation.

Once we have the z-score, we can use a standard normal distribution table or calculator to find the probability. The probability is given by the area under the standard normal distribution curve to the left of the z-score.

Solving the Problem

Now, let's solve the problem given in the question. We are given that $P (z \leq a) = 0.7116$, and we need to find the value of $P (z \geq a)$.

Since the standard normal distribution is symmetric, we know that the probability of a value being above the mean is equal to the probability of a value being below the mean. Therefore, we can write:

P(z≥a)=1−P(z≤a)P (z \geq a) = 1 - P (z \leq a)

Substituting the given value, we get:

P(z≥a)=1−0.7116=0.2884P (z \geq a) = 1 - 0.7116 = 0.2884

Therefore, the value of $P (z \geq a)$ is 0.2884.

Conclusion

In this article, we have explored the properties of the standard normal distribution and how to calculate probabilities using this distribution. We have also solved a problem that illustrates the symmetry of the standard normal distribution. The standard normal distribution is a fundamental concept in statistics, and understanding its properties and how to calculate probabilities using this distribution is essential for any statistician or data analyst.

References

  • Kendall, M. G., & Stuart, A. (1973). The Advanced Theory of Statistics. Macmillan.
  • Johnson, N. L., & Kotz, S. (1970). Distributions in Statistics: Continuous Univariate Distributions. Wiley.
  • Evans, M., Hastings, N., & Peacock, J. (2000). Statistical Distributions. John Wiley & Sons.
    Frequently Asked Questions about the Standard Normal Distribution ====================================================================

The standard normal distribution is a fundamental concept in statistics, and it is used to model a wide range of phenomena. However, it can be a challenging topic to understand, especially for those who are new to statistics. In this article, we will answer some of the most frequently asked questions 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 probability distribution that is symmetric about the mean, with a mean of 0 and a standard deviation of 1. It is a fundamental concept in statistics and is used to model a wide range of phenomena.

Q: What are the properties of the standard normal distribution?

A: The standard normal distribution has several important properties that make it a useful tool for statistical analysis. Some of the key properties include:

  • Symmetry: The standard normal distribution is symmetric about the mean, which means that the probability of a value being above the mean is equal to the probability of a value being below the mean.
  • Mean: The mean of the standard normal distribution is 0, which means that the distribution is centered around 0.
  • Standard Deviation: The standard deviation of the standard normal distribution is 1, which means that the distribution is spread out over a range of 2 standard deviations (i.e., from -1 to 1).
  • Probability Density Function: The probability density function (PDF) of the standard normal distribution is given by the equation:

f(z)=12πe−z22f(z) = \frac{1}{\sqrt{2\pi}}e^{-\frac{z^2}{2}}

Q: How do I calculate a z-score?

A: To calculate a z-score, you need to know the value of interest, the mean, and the standard deviation. The z-score can be calculated using the following formula:

z=x−μσz = \frac{x - \mu}{\sigma}

where x is the value of interest, μ is the mean, and σ is the standard deviation.

Q: How do I use a standard normal distribution table or calculator to find a probability?

A: To use a standard normal distribution table or calculator to find a probability, you need to know the z-score. The probability is given by the area under the standard normal distribution curve to the left of the z-score.

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. The normal distribution is a probability distribution that is symmetric about the mean, with a mean of μ and a standard deviation of σ. The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1.

Q: How do I use the standard normal distribution to solve problems?

A: To use the standard normal distribution to solve problems, you need to follow these steps:

  1. Understand the problem: Read the problem carefully and understand what is being asked.
  2. Identify the distribution: Determine whether the problem involves a standard normal distribution or a normal distribution.
  3. Calculate the z-score: Calculate the z-score using the formula:

z=x−μσz = \frac{x - \mu}{\sigma}

  1. Use a standard normal distribution table or calculator: Use a standard normal distribution table or calculator to find the probability.
  2. Interpret the results: Interpret the results in the context of the problem.

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

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

  • Hypothesis testing: The standard normal distribution is used to test hypotheses about the mean of a population.
  • Confidence intervals: The standard normal distribution is used to construct confidence intervals for the mean of a population.
  • Regression analysis: The standard normal distribution is used in regression analysis to model the relationship between a dependent variable and one or more independent variables.

Conclusion

In this article, we have answered some of the most frequently asked questions about the standard normal distribution. The standard normal distribution is a fundamental concept in statistics, and it is used to model a wide range of phenomena. Understanding the properties and how to calculate probabilities using this distribution is essential for any statistician or data analyst.