Which Of The Following Statements Is Equivalent To $P(z \geq 1.7)$?A. $P(z \geq -1.7)$ B. $ 1 − P ( Z ≥ − 1.7 ) 1-P(z \geq -1.7) 1 − P ( Z ≥ − 1.7 ) [/tex] C. $P(z \leq 1.7)$ D. $1-P(z \geq 1.7)$
Introduction
The standard normal distribution, also known as the z-distribution, is a fundamental concept in statistics and probability theory. It is a continuous probability distribution that is symmetric about the mean, with a standard deviation of 1. The z-distribution is used to standardize values from different normal distributions, allowing for comparisons and calculations to be made. In this article, we will explore the concept of the standard normal distribution and its applications, with a focus on understanding the equivalent statements of a given probability expression.
The Standard Normal Distribution
The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. The probability density function (PDF) of the standard normal distribution is given by:
The standard normal distribution is used to standardize values from different normal distributions by converting them to z-scores. A z-score is a measure of how many standard deviations an observation is away from the mean. The z-score is calculated using the following formula:
where X is the value, μ is the mean, and σ is the standard deviation.
Understanding the Given Probability Expression
The given probability expression is:
This expression represents the probability that a random variable z is greater than or equal to 1.7. To understand this expression, we need to consider the area under the standard normal curve to the right of z = 1.7.
Equivalent Statements
Now, let's examine the equivalent statements of the given probability expression:
A. $P(z \geq -1.7)$
This statement is not equivalent to the given probability expression. The probability expression $P(z \geq -1.7)$ represents the probability that a random variable z is greater than or equal to -1.7. This is not the same as the given probability expression, which represents the probability that a random variable z is greater than or equal to 1.7.
B. $1-P(z \geq -1.7)$
This statement is also not equivalent to the given probability expression. The probability expression $1-P(z \geq -1.7)$ represents the probability that a random variable z is less than -1.7. This is not the same as the given probability expression, which represents the probability that a random variable z is greater than or equal to 1.7.
C. $P(z \leq 1.7)$
This statement is not equivalent to the given probability expression. The probability expression $P(z \leq 1.7)$ represents the probability that a random variable z is less than or equal to 1.7. This is not the same as the given probability expression, which represents the probability that a random variable z is greater than or equal to 1.7.
D. $1-P(z \geq 1.7)$
This statement is equivalent to the given probability expression. The probability expression $1-P(z \geq 1.7)$ represents the probability that a random variable z is less than 1.7. This is equivalent to the given probability expression, which represents the probability that a random variable z is greater than or equal to 1.7.
Conclusion
In conclusion, the equivalent statement of the given probability expression $P(z \geq 1.7)$ is $1-P(z \geq 1.7)$. This statement represents the probability that a random variable z is less than 1.7, which is equivalent to the given probability expression.
Key Takeaways
- The standard normal distribution is a continuous probability distribution that is symmetric about the mean, with a standard deviation of 1.
- The z-distribution is used to standardize values from different normal distributions, allowing for comparisons and calculations to be made.
- The given probability expression $P(z \geq 1.7)$ represents the probability that a random variable z is greater than or equal to 1.7.
- The equivalent statement of the given probability expression is $1-P(z \geq 1.7)$, which represents the probability that a random variable z is less than 1.7.
References
- Moore, D. S., & McCabe, G. P. (2013). Introduction to the practice of statistics. W.H. Freeman and Company.
- Ross, S. M. (2014). Introduction to probability models. Academic Press.
- Johnson, R. A., & Bhattacharyya, G. K. (2014). Statistics: Principles and methods. John Wiley & Sons.
Frequently Asked Questions (FAQs) on the Standard Normal Distribution and Its Applications =====================================================================================
Q: What is the standard normal distribution?
A: The standard normal distribution, also known as the z-distribution, is a continuous probability distribution that is symmetric about the mean, with a standard deviation of 1. It is used to standardize values from different normal distributions, allowing for comparisons and calculations to be made.
Q: What is the formula for the standard normal distribution?
A: The probability density function (PDF) of the standard normal distribution is given by:
Q: How is the z-score calculated?
A: The z-score is calculated using the following formula:
where X is the value, μ is the mean, and σ is the standard deviation.
Q: What is the difference between a z-score and a standard score?
A: A z-score and a standard score are the same thing. They are both measures of how many standard deviations an observation is away from the mean.
Q: How is the standard normal distribution used in real-life applications?
A: The standard normal distribution is used in a wide range of real-life applications, including:
- Finance: To calculate the probability of stock prices or returns
- Engineering: To calculate the probability of failure or success of a system
- Medicine: To calculate the probability of a disease or condition
- Social Sciences: To calculate the probability of a behavior or outcome
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 is a more general distribution that can have any mean and standard deviation.
Q: How is the standard normal distribution used in hypothesis testing?
A: The standard normal distribution is used in hypothesis testing to calculate the probability of observing a particular value or set of values, given a certain hypothesis. This is done using the z-score and the standard normal distribution.
Q: What is the difference between a one-tailed and a two-tailed test?
A: A one-tailed test is a test where the alternative hypothesis is in one direction (e.g. greater than or less than). A two-tailed test is a test where the alternative hypothesis is in both directions (e.g. greater than or less than).
Q: How is the standard normal distribution used in confidence intervals?
A: The standard normal distribution is used in confidence intervals to calculate the margin of error and the confidence level. This is done using the z-score and the standard normal distribution.
Q: What is the relationship between the standard normal distribution and the t-distribution?
A: The standard normal distribution is a special case of the t-distribution, where the degrees of freedom are infinite. The t-distribution is a more general distribution that can have any degrees of freedom.
Q: How is the standard normal distribution used in regression analysis?
A: The standard normal distribution is used in regression analysis to calculate the probability of observing a particular value or set of values, given a certain regression equation. This is done using the z-score and the standard normal distribution.
Q: What is the difference between a linear and a non-linear regression?
A: A linear regression is a regression where the relationship between the independent variable and the dependent variable is linear. A non-linear regression is a regression where the relationship between the independent variable and the dependent variable is non-linear.
Q: How is the standard normal distribution used in time series analysis?
A: The standard normal distribution is used in time series analysis to calculate the probability of observing a particular value or set of values, given a certain time series model. This is done using the z-score and the standard normal distribution.
Q: What is the relationship between the standard normal distribution and the ARIMA model?
A: The standard normal distribution is used in ARIMA models to calculate the probability of observing a particular value or set of values, given a certain ARIMA model. This is done using the z-score and the standard normal distribution.
Q: How is the standard normal distribution used in machine learning?
A: The standard normal distribution is used in machine learning to calculate the probability of observing a particular value or set of values, given a certain machine learning model. This is done using the z-score and the standard normal distribution.
Q: What is the difference between a supervised and an unsupervised learning?
A: A supervised learning is a learning where the model is trained on labeled data. An unsupervised learning is a learning where the model is trained on unlabeled data.
Q: How is the standard normal distribution used in deep learning?
A: The standard normal distribution is used in deep learning to calculate the probability of observing a particular value or set of values, given a certain deep learning model. This is done using the z-score and the standard normal distribution.
Q: What is the relationship between the standard normal distribution and the neural network?
A: The standard normal distribution is used in neural networks to calculate the probability of observing a particular value or set of values, given a certain neural network model. This is done using the z-score and the standard normal distribution.