For A Standard Normal Distribution, If P ( Z ≤ A ) = 0.7116 P(z \leq A) = 0.7116 P ( Z ≤ A ) = 0.7116 , What Is The Value Of P ( Z ≥ A P(z \geq A P ( Z ≥ A ]?A. 0.2884B. 0.7116C. 0.7884
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 various fields, including finance, engineering, and social sciences.
Properties of the Standard Normal Distribution
One of the key properties of the standard normal distribution is that it is symmetric about the mean of 0. This means that the probability of a value being less than or equal to a certain value (a) is equal to the probability of a value being greater than or equal to -a. Mathematically, this can be expressed as:
P(z ≤ a) = P(z ≥ -a)
Given Information
We are given that P(z ≤ a) = 0.7116. We need to find the value of P(z ≥ a).
Using the Symmetry Property
Since the standard normal distribution is symmetric about the mean of 0, we can use the symmetry property to find the value of P(z ≥ a). We know that:
P(z ≤ a) = P(z ≥ -a)
Since P(z ≤ a) = 0.7116, we can set up the following equation:
0.7116 = P(z ≥ -a)
Finding the Value of P(z ≥ a)
To find the value of P(z ≥ a), we need to use the fact that the total probability of all possible values of z is equal to 1. Mathematically, this can be expressed as:
P(z ≤ a) + P(z ≥ a) = 1
We can substitute the value of P(z ≤ a) = 0.7116 into this equation to get:
0.7116 + P(z ≥ a) = 1
Solving for P(z ≥ a)
To solve for P(z ≥ a), we can subtract 0.7116 from both sides of the equation:
P(z ≥ a) = 1 - 0.7116 P(z ≥ a) = 0.2884
Conclusion
In conclusion, we have used the symmetry property of the standard normal distribution to find the value of P(z ≥ a) given that P(z ≤ a) = 0.7116. The value of P(z ≥ a) is 0.2884.
Answer
The correct answer is A. 0.2884.
Additional Information
It's worth noting that the standard normal distribution is a continuous distribution, and the probability of a value being exactly equal to a certain value (a) is zero. Therefore, we can only talk about the probability of a value being less than or equal to a certain value (a) or greater than or equal to a certain value (a).
References
- [1] Moore, D. S., & McCabe, G. P. (2017). Introduction to the practice of statistics. W.H. Freeman and Company.
- [2] Johnson, R. A., & Bhattacharyya, G. K. (2010). Statistics: Principles and methods. John Wiley & Sons.
Table of Contents
- Understanding the Standard Normal Distribution
- Properties of the Standard Normal Distribution
- Given Information
- Using the Symmetry Property
- Finding the Value of P(z ≥ a)
- Solving for P(z ≥ a)
- Conclusion
- Answer
- Additional Information
- References
- Table of Contents
Q&A: Standard Normal Distribution =====================================
Q1: What is the standard normal distribution?
A1: 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 various fields, including finance, engineering, and social sciences.
Q2: What is the significance of the standard normal distribution?
A2: The standard normal distribution is significant because it provides a way to standardize and compare data from different populations. By converting data to z-scores, we can compare the relative position of data points in different distributions.
Q3: How is the standard normal distribution used in real-life applications?
A3: The standard normal distribution is used in a wide range of real-life applications, including:
- Finance: to model stock prices and returns
- Engineering: to model the behavior of complex systems
- Social sciences: to model the behavior of populations and societies
- Medicine: to model the behavior of diseases and treatments
Q4: What is the difference between the standard normal distribution and the normal distribution?
A4: 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 is a more general distribution that can have any mean and standard deviation.
Q5: How do I calculate the z-score of a value in a normal distribution?
A5: To calculate the z-score of a value in a normal distribution, you need to know the mean and standard deviation of the distribution. The z-score is calculated using the following formula:
z = (X - μ) / σ
where X is the value, μ is the mean, and σ is the standard deviation.
Q6: What is the 68-95-99.7 rule in the standard normal distribution?
A6: The 68-95-99.7 rule states that in a standard normal distribution, about 68% of the data points fall within 1 standard deviation of the mean, about 95% of the data points fall within 2 standard deviations of the mean, and about 99.7% of the data points fall within 3 standard deviations of the mean.
Q7: How do I use the standard normal distribution to find the probability of a value?
A7: To find the probability of a value in a standard normal distribution, you can use a z-table or a calculator to find the probability that a value is less than or equal to a certain z-score.
Q8: What is the relationship between the standard normal distribution and the cumulative distribution function (CDF)?
A8: The standard normal distribution is related to the CDF through the following equation:
P(Z ≤ z) = Φ(z)
where Φ(z) is the CDF of the standard normal distribution.
Q9: How do I use the standard normal distribution to find the confidence interval of a population parameter?
A9: To find the confidence interval of a population parameter using the standard normal distribution, you need to know the sample mean, sample standard deviation, and sample size. You can then use the following formula to calculate the confidence interval:
CI = (X̄ - z * σ / √n, X̄ + z * σ / √n)
where X̄ is the sample mean, z is the z-score, σ is the population standard deviation, and n is the sample size.
Q10: What are some common applications of the standard normal distribution in statistics?
A10: Some common applications of the standard normal distribution in statistics include:
- Hypothesis testing
- Confidence intervals
- Regression analysis
- Time series analysis
Conclusion
In conclusion, the standard normal distribution is a fundamental concept in statistics that has a wide range of applications in various fields. By understanding the properties and uses of the standard normal distribution, you can apply statistical techniques to real-world problems and make informed decisions.
References
- [1] Moore, D. S., & McCabe, G. P. (2017). Introduction to the practice of statistics. W.H. Freeman and Company.
- [2] Johnson, R. A., & Bhattacharyya, G. K. (2010). Statistics: Principles and methods. John Wiley & Sons.
Table of Contents
- Q&A: Standard Normal Distribution
- Q1: What is the standard normal distribution?
- Q2: What is the significance of the standard normal distribution?
- Q3: How is the standard normal distribution used in real-life applications?
- Q4: What is the difference between the standard normal distribution and the normal distribution?
- Q5: How do I calculate the z-score of a value in a normal distribution?
- Q6: What is the 68-95-99.7 rule in the standard normal distribution?
- Q7: How do I use the standard normal distribution to find the probability of a value?
- Q8: What is the relationship between the standard normal distribution and the cumulative distribution function (CDF)?
- Q9: How do I use the standard normal distribution to find the confidence interval of a population parameter?
- Q10: What are some common applications of the standard normal distribution in statistics?
- Conclusion
- References
- Table of Contents