Which Is Equivalent To $P(z \geq 1.4)$?A. $P(z \leq 1.4)$B. $ 1 − P ( Z ≤ 1.4 ) 1 - P(z \leq 1.4) 1 − P ( Z ≤ 1.4 ) [/tex]C. $P(z \geq -1.4)$Use The Standard Normal Table To Find $P(z \geq 1.4)$. Round To The Nearest Percent.
Introduction
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 widely used in statistics and data analysis to model real-valued random variables. In this article, we will explore the concept of the standard normal distribution and its applications, with a focus on finding the probability of a z-score greater than or equal to a given value.
What is a Z-Score?
A z-score is a measure of how many standard deviations an observation is away from the mean. It is calculated by subtracting the mean from the observation and dividing by the standard deviation. The z-score is a useful tool for understanding the distribution of a dataset and for making comparisons between different datasets.
The Standard Normal Table
The standard normal table, also known as the z-table, is a table that lists the probabilities of z-scores from 0 to 3.49. It is used to find the probability of a z-score being less than or equal to a given value. The table is organized in a way that makes it easy to find the probability of a z-score by looking up the value in the table.
Finding the Probability of a Z-Score Greater Than or Equal to a Given Value
To find the probability of a z-score greater than or equal to a given value, we can use the following formula:
P(z ≥ k) = 1 - P(z ≤ k)
where k is the given z-score.
Example: Finding the Probability of a Z-Score Greater Than or Equal to 1.4
Using the standard normal table, we can find the probability of a z-score less than or equal to 1.4. Looking up the value of 1.4 in the table, we find that P(z ≤ 1.4) = 0.9192.
Now, we can use the formula above to find the probability of a z-score greater than or equal to 1.4:
P(z ≥ 1.4) = 1 - P(z ≤ 1.4) = 1 - 0.9192 = 0.0808
Rounding to the nearest percent, we get:
P(z ≥ 1.4) = 8%
Conclusion
In this article, we have explored the concept of the standard normal distribution and its applications. We have also learned how to use the standard normal table to find the probability of a z-score being less than or equal to a given value. Finally, we have used the formula above to find the probability of a z-score greater than or equal to a given value.
Which is Equivalent to $P(z \geq 1.4)$?
Based on the formula above, we can see that the correct answer is:
B. $1 - P(z \leq 1.4)$
This is because P(z ≥ 1.4) = 1 - P(z ≤ 1.4).
Answer Key
A. $P(z \leq 1.4)$ B. $1 - P(z \leq 1.4)$ C. $P(z \geq -1.4)$
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 widely used in statistics and data analysis to model real-valued random variables.
Q: What is a z-score?
A: A z-score is a measure of how many standard deviations an observation is away from the mean. It is calculated by subtracting the mean from the observation and dividing by the standard deviation.
Q: How do I use the standard normal table?
A: The standard normal table is a table that lists the probabilities of z-scores from 0 to 3.49. To use the table, look up the value of the z-score in the table and find the corresponding probability.
Q: What is the difference between P(z ≤ k) and P(z ≥ k)?
A: P(z ≤ k) is the probability of a z-score being less than or equal to k, while P(z ≥ k) is the probability of a z-score being greater than or equal to k.
Q: How do I find P(z ≥ k) using the standard normal table?
A: To find P(z ≥ k), use the formula P(z ≥ k) = 1 - P(z ≤ k). Then, look up the value of k in the standard normal table and find the corresponding probability. Subtract this probability from 1 to find P(z ≥ k).
Q: Can I use the standard normal table to find P(z ≥ k) for any value of k?
A: No, the standard normal table only lists probabilities for z-scores from 0 to 3.49. If you need to find P(z ≥ k) for a value of k outside of this range, you will need to use a different method or table.
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 do I use the standard normal distribution to model real-world data?
A: To use the standard normal distribution to model real-world data, first, you need to calculate the z-scores of the data points. Then, you can use the standard normal table to find the probabilities of the z-scores. Finally, you can use these probabilities to make inferences about the data.
Q: What are some common applications of the standard normal distribution?
A: The standard normal distribution is widely used in statistics and data analysis to model real-valued random variables. Some common applications include:
- Hypothesis testing: The standard normal distribution is used to test hypotheses about the mean and standard deviation of a population.
- Confidence intervals: The standard normal distribution is used to construct confidence intervals for the mean and standard deviation of a population.
- Regression analysis: The standard normal distribution is used to model the residuals in a regression analysis.
- Time series analysis: The standard normal distribution is used to model the residuals in a time series analysis.
Conclusion
In this article, we have answered some frequently asked questions about the standard normal distribution. We have covered topics such as the definition of the standard normal distribution, how to use the standard normal table, and common applications of the standard normal distribution. We hope that this article has been helpful in answering your questions about the standard normal distribution.