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. $P(z \geq -1.4)$What Is $P (0.6 \leq Z \leq 2.12)$?A. 16% B. 26% C. 73% D. 98%Use The
Introduction
Probability distributions are a crucial concept in statistics and mathematics, used to describe the likelihood of different outcomes in a random experiment. The standard normal distribution, also known as the z-distribution, is a specific type of probability distribution that is widely used in statistical analysis. In this article, we will explore the concept of probability distributions and the standard normal distribution, and provide solutions to two problems related to these concepts.
What is a Probability Distribution?
A probability distribution is a function that assigns a probability to each possible outcome of a random experiment. It is a mathematical description of the likelihood of different outcomes, and is used to make predictions and decisions based on data. Probability distributions can be discrete or continuous, depending on the type of data being analyzed.
The Standard Normal Distribution
The standard normal distribution, also known as the z-distribution, is a specific type of probability distribution that is widely used in statistical analysis. It is a continuous distribution that is symmetric about the mean, with a standard deviation of 1. The standard normal distribution is used to model a wide range of phenomena, including the heights of people, the scores of students on a test, and the prices of stocks.
Solving the First Problem
The first problem asks us to find the equivalent of $P (z \geq 1.4)$. To solve this problem, we need to understand the concept of the cumulative distribution function (CDF) of the standard normal distribution. The CDF of the standard normal distribution is a function that gives the probability that a random variable takes on a value less than or equal to a given value.
Using the symmetry of the standard normal distribution, we can rewrite $P (z \geq 1.4)$ as $1 - P (z \leq 1.4)$. This is because the probability that a random variable takes on a value greater than or equal to a given value is equal to 1 minus the probability that it takes on a value less than or equal to that value.
Therefore, the equivalent of $P (z \geq 1.4)$ is $1 - P (z \leq 1.4)$.
Solving the Second Problem
The second problem asks us to find the value of $P (0.6 \leq z \leq 2.12)$. To solve this problem, we need to use the CDF of the standard normal distribution. We can use a standard normal distribution table or calculator to find the probabilities.
Using a standard normal distribution table, we find that $P (z \leq 2.12) = 0.9834$ and $P (z \leq 0.6) = 0.2743$. Therefore, $P (0.6 \leq z \leq 2.12) = P (z \leq 2.12) - P (z \leq 0.6) = 0.9834 - 0.2743 = 0.7091$.
Converting this probability to a percentage, we get $P (0.6 \leq z \leq 2.12) = 70.91%$.
However, this is not among the answer choices. We need to find the closest answer choice.
The closest answer choice is C. 73%.
Conclusion
In this article, we have explored the concept of probability distributions and the standard normal distribution. We have provided solutions to two problems related to these concepts, and have used the cumulative distribution function (CDF) of the standard normal distribution to find the probabilities.
We have also seen how to use a standard normal distribution table or calculator to find the probabilities, and have converted the probabilities to percentages.
The final answer to the first problem is B. .
The final answer to the second problem is C. 73%.
References
- Kendall, M. G., & Stuart, A. (1977). The advanced theory of statistics. Macmillan.
- Johnson, N. L., & Kotz, S. (1970). Distributions in statistics: Continuous univariate distributions. Houghton Mifflin.
- Evans, M., Hastings, N., & Peacock, B. (2000). Statistical distributions. John Wiley & Sons.
Frequently Asked Questions (FAQs) on Probability Distributions and Standard Normal Distribution =============================================================================================
Q: What is a probability distribution?
A: A probability distribution is a function that assigns a probability to each possible outcome of a random experiment. It is a mathematical description of the likelihood of different outcomes, and is used to make predictions and decisions based on data.
Q: What is the standard normal distribution?
A: The standard normal distribution, also known as the z-distribution, is a specific type of probability distribution that is widely used in statistical analysis. It is a continuous distribution that is symmetric about the mean, with a standard deviation of 1.
Q: How do I find the probability of a value in a standard normal distribution?
A: To find the probability of a value in a standard normal distribution, you can use a standard normal distribution table or calculator. The table or calculator will give you the probability that a random variable takes on a value less than or equal to a given value.
Q: What is the cumulative distribution function (CDF) of a standard normal distribution?
A: The cumulative distribution function (CDF) of a standard normal distribution is a function that gives the probability that a random variable takes on a value less than or equal to a given value.
Q: How do I use the CDF to find the probability of a value in a standard normal distribution?
A: To use the CDF to find the probability of a value in a standard normal distribution, you can use the following formula:
P(z ≤ z0) = Φ(z0)
where Φ is the CDF of the standard normal distribution, and z0 is the value for which you want to find the probability.
Q: What is the relationship between the probability of a value in a standard normal distribution and the probability of a value in a normal distribution with a different mean and standard deviation?
A: The probability of a value in a standard normal distribution is equal to the probability of a value in a normal distribution with a different mean and standard deviation, scaled by the standard deviation of the normal distribution.
Q: How do I find the probability of a range of values in a standard normal distribution?
A: To find the probability of a range of values in a standard normal distribution, you can use the following formula:
P(a ≤ z ≤ b) = Φ(b) - Φ(a)
where Φ is the CDF of the standard normal distribution, and a and b are the lower and upper bounds of the range, respectively.
Q: What is the relationship between the probability of a value in a standard normal distribution and the probability of a value in a binomial distribution?
A: The probability of a value in a standard normal distribution is equal to the probability of a value in a binomial distribution, scaled by the square root of the number of trials in the binomial distribution.
Q: How do I use the standard normal distribution to make predictions and decisions based on data?
A: To use the standard normal distribution to make predictions and decisions based on data, you can use the following steps:
- Collect data on the variable of interest.
- Calculate the mean and standard deviation of the data.
- Use the standard normal distribution to find the probability of a value in the data.
- Use the probability to make predictions and decisions based on the data.
Q: What are some common applications of the standard normal distribution?
A: The standard normal distribution has many common applications in statistics and data analysis, including:
- 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 analyze the relationship between a dependent variable and one or more independent variables.
- Time series analysis: The standard normal distribution is used to analyze and forecast time series data.
Conclusion
In this article, we have answered some frequently asked questions (FAQs) about probability distributions and the standard normal distribution. We have covered topics such as the definition of a probability distribution, the standard normal distribution, and how to use the cumulative distribution function (CDF) to find the probability of a value in a standard normal distribution. We have also discussed some common applications of the standard normal distribution, including hypothesis testing, confidence intervals, regression analysis, and time series analysis.