For A Standard Normal Distribution, Find The Approximate Value Of $P (z \geq -1.25)$. Use The Portion Of The Standard Normal Table Below To Help Answer The Question. \[ \begin{tabular}{|c|c|} \hline Z$ & Probability \ \hline 0.00 &
Understanding the Standard Normal Distribution
The standard normal distribution, also known as the z-distribution, is a probability distribution that is symmetric about the mean, which is 0. It has 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 on a test, and the returns on investments. In this article, we will use the standard normal distribution to find the approximate value of P(z ≥ -1.25).
The Standard Normal Table
The standard normal table is a table that shows the probability of a z-score being less than or equal to a given value. The table is used to find the probability of a z-score being less than or equal to a given value, and it is used in conjunction with the z-score formula to find the probability of a z-score being greater than or equal to a given value. The portion of the standard normal table that we will use to help answer the question is shown below.
| z | Probability |
|---|---|
| -1.00 | 0.1587 |
| -1.25 | 0.1056 |
| -1.50 | 0.0668 |
| -1.75 | 0.0401 |
| -2.00 | 0.0228 |
Finding the Approximate Value of P(z ≥ -1.25)
To find the approximate value of P(z ≥ -1.25), we need to use the standard normal table. We are looking for the probability of a z-score being greater than or equal to -1.25. To find this probability, we need to find the probability of a z-score being less than or equal to -1.25 and then subtract it from 1.
Using the Standard Normal Table
From the standard normal table, we can see that the probability of a z-score being less than or equal to -1.25 is 0.1056. To find the probability of a z-score being greater than or equal to -1.25, we need to subtract this value from 1.
Calculating the Probability
P(z ≥ -1.25) = 1 - P(z ≤ -1.25) = 1 - 0.1056 = 0.8944
Conclusion
In this article, we used the standard normal distribution and the standard normal table to find the approximate value of P(z ≥ -1.25). We found that the approximate value of P(z ≥ -1.25) is 0.8944.
Understanding the Standard Normal Distribution
The standard normal distribution is a probability distribution that is symmetric about the mean, which is 0. It has 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 on a test, and the returns on investments.
The Standard Normal Distribution Formula
The standard normal distribution formula is:
f(z) = (1/√(2π)) * e(-z2/2)
where f(z) is the probability density function of the standard normal distribution, z is the z-score, and e is the base of the natural logarithm.
The Standard Normal Distribution Table
The standard normal distribution table is a table that shows the probability of a z-score being less than or equal to a given value. The table is used to find the probability of a z-score being less than or equal to a given value, and it is used in conjunction with the z-score formula to find the probability of a z-score being greater than or equal to a given value.
Using the Standard Normal Distribution Table
To use the standard normal distribution table, we need to find the z-score that corresponds to the given probability. We can do this by looking up the probability in the table and finding the corresponding z-score.
Example
Suppose we want to find the probability of a z-score being greater than or equal to 1.5. We can use the standard normal distribution table to find this probability.
Finding the Probability
From the standard normal distribution table, we can see that the probability of a z-score being less than or equal to 1.5 is 0.9332. To find the probability of a z-score being greater than or equal to 1.5, we need to subtract this value from 1.
Calculating the Probability
P(z ≥ 1.5) = 1 - P(z ≤ 1.5) = 1 - 0.9332 = 0.0668
Conclusion
In this article, we used the standard normal distribution and the standard normal distribution table to find the approximate value of P(z ≥ 1.5). We found that the approximate value of P(z ≥ 1.5) is 0.0668.
The Standard Normal Distribution and Real-World Applications
The standard normal distribution has many real-world applications. It is used to model a wide range of phenomena, including the heights of people, the scores on a test, and the returns on investments.
Example
Suppose we want to find the probability of a person's height being greater than or equal to 175 cm. We can use the standard normal distribution to find this probability.
Finding the Probability
To find the probability of a person's height being greater than or equal to 175 cm, we need to find the z-score that corresponds to a height of 175 cm. We can do this by using the z-score formula:
z = (X - μ) / σ
where X is the height, μ is the mean, and σ is the standard deviation.
Calculating the Z-Score
Suppose the mean height of a population is 170 cm and the standard deviation is 5 cm. We can calculate the z-score as follows:
z = (175 - 170) / 5 = 1
Finding the Probability
From the standard normal distribution table, we can see that the probability of a z-score being less than or equal to 1 is 0.8413. To find the probability of a z-score being greater than or equal to 1, we need to subtract this value from 1.
Calculating the Probability
P(z ≥ 1) = 1 - P(z ≤ 1) = 1 - 0.8413 = 0.1587
Conclusion
In this article, we used the standard normal distribution to find the approximate value of P(z ≥ 1). We found that the approximate value of P(z ≥ 1) is 0.1587.
The Standard Normal Distribution and Statistical Inference
The standard normal distribution is used in statistical inference to make inferences about a population based on a sample of data.
Example
Suppose we want to find the probability of a sample mean being greater than or equal to 10. We can use the standard normal distribution to find this probability.
Finding the Probability
To find the probability of a sample mean being greater than or equal to 10, we need to find the z-score that corresponds to a sample mean of 10. We can do this by using the z-score formula:
z = (X̄ - μ) / (σ / √n)
where X̄ is the sample mean, μ is the population mean, σ is the population standard deviation, and n is the sample size.
Calculating the Z-Score
Suppose the population mean is 10, the population standard deviation is 2, and the sample size is 25. We can calculate the z-score as follows:
z = (10 - 10) / (2 / √25) = 0
Finding the Probability
From the standard normal distribution table, we can see that the probability of a z-score being less than or equal to 0 is 0.5. To find the probability of a z-score being greater than or equal to 0, we need to subtract this value from 1.
Calculating the Probability
P(z ≥ 0) = 1 - P(z ≤ 0) = 1 - 0.5 = 0.5
Conclusion
In this article, we used the standard normal distribution to find the approximate value of P(z ≥ 0). We found that the approximate value of P(z ≥ 0) is 0.5.
The Standard Normal Distribution and Hypothesis Testing
The standard normal distribution is used in hypothesis testing to test hypotheses about a population based on a sample of data.
Example
Suppose we want to test the hypothesis that the population mean is equal to 10. We can use the standard normal distribution to find the probability of observing a sample mean that is greater than or equal to 10.
Finding the Probability
To find the probability of observing a sample mean that is greater than or equal to 10, we need to find the z-score that corresponds to a sample mean of 10. We can do this by using the z-score formula:
z = (X̄ - μ) / (σ / √n)
where X̄ is the sample mean, μ is the population mean, σ is the population standard deviation, and n is the sample size.
Calculating the Z-Score
Suppose the population mean is 10, the population standard deviation is 2, and the sample size is 25. We can calculate the z-score as follows:
z = (10 - 10) / (2 / √25) = 0
Finding the Probability
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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, which is 0. It has 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 on a test, and the returns on investments.
Q: How is the standard normal distribution used in real-world applications?
A: The standard normal distribution is used in a wide range of real-world applications, including:
- Modeling the heights of people
- Modeling the scores on a test
- Modeling the returns on investments
- Making inferences about a population based on a sample of data
- Testing hypotheses about a population based on a sample of data
Q: How do I use the standard normal distribution table?
A: To use the standard normal distribution table, you need to find the z-score that corresponds to the given probability. You can do this by looking up the probability in the table and finding the corresponding z-score.
Q: What is the z-score formula?
A: The z-score formula is:
z = (X - μ) / σ
where X is the value, μ is the mean, and σ is the standard deviation.
Q: How do I calculate the z-score?
A: To calculate the z-score, you need to know the value, the mean, and the standard deviation. You can then plug these values into the z-score formula and calculate the z-score.
Q: What is the probability of a z-score being greater than or equal to a given value?
A: To find the probability of a z-score being greater than or equal to a given value, you need to find the z-score that corresponds to the given value. You can then use the standard normal distribution table to find the probability.
Q: How do I use the standard normal distribution to make inferences about a population?
A: To use the standard normal distribution to make inferences about a population, you need to know the sample mean, the sample standard deviation, and the sample size. You can then use the z-score formula to calculate the z-score and use the standard normal distribution table to find the probability.
Q: How do I use the standard normal distribution to test hypotheses about a population?
A: To use the standard normal distribution to test hypotheses about a population, you need to know the sample mean, the sample standard deviation, and the sample size. You can then use the z-score formula to calculate the z-score and use the standard normal distribution table to find the probability.
Q: What is the difference 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 convert a normal distribution to a standard normal distribution?
A: To convert a normal distribution to a standard normal distribution, you need to subtract the mean and divide by the standard deviation.
Q: What is the relationship between the standard normal distribution and the t-distribution?
A: The standard normal distribution and the t-distribution are related in that the t-distribution is a special case of the standard normal distribution where the sample size is small.
Q: How do I use the standard normal distribution to find the probability of a sample mean being greater than or equal to a given value?
A: To use the standard normal distribution to find the probability of a sample mean being greater than or equal to a given value, you need to know the sample mean, the sample standard deviation, and the sample size. You can then use the z-score formula to calculate the z-score and use the standard normal distribution table to find the probability.
Q: How do I use the standard normal distribution to find the probability of a sample mean being less than or equal to a given value?
A: To use the standard normal distribution to find the probability of a sample mean being less than or equal to a given value, you need to know the sample mean, the sample standard deviation, and the sample size. You can then use the z-score formula to calculate the z-score and use the standard normal distribution table to find the probability.
Q: What is the relationship between the standard normal distribution and the chi-squared distribution?
A: The standard normal distribution and the chi-squared distribution are related in that the chi-squared distribution is a special case of the standard normal distribution where the sample size is large.
Q: How do I use the standard normal distribution to find the probability of a sample variance being greater than or equal to a given value?
A: To use the standard normal distribution to find the probability of a sample variance being greater than or equal to a given value, you need to know the sample variance, the sample standard deviation, and the sample size. You can then use the z-score formula to calculate the z-score and use the standard normal distribution table to find the probability.
Q: How do I use the standard normal distribution to find the probability of a sample variance being less than or equal to a given value?
A: To use the standard normal distribution to find the probability of a sample variance being less than or equal to a given value, you need to know the sample variance, the sample standard deviation, and the sample size. You can then use the z-score formula to calculate the z-score and use the standard normal distribution table to find the probability.
Q: What is the relationship between the standard normal distribution and the F-distribution?
A: The standard normal distribution and the F-distribution are related in that the F-distribution is a special case of the standard normal distribution where the sample size is large.
Q: How do I use the standard normal distribution to find the probability of a sample correlation coefficient being greater than or equal to a given value?
A: To use the standard normal distribution to find the probability of a sample correlation coefficient being greater than or equal to a given value, you need to know the sample correlation coefficient, the sample standard deviation, and the sample size. You can then use the z-score formula to calculate the z-score and use the standard normal distribution table to find the probability.
Q: How do I use the standard normal distribution to find the probability of a sample correlation coefficient being less than or equal to a given value?
A: To use the standard normal distribution to find the probability of a sample correlation coefficient being less than or equal to a given value, you need to know the sample correlation coefficient, the sample standard deviation, and the sample size. You can then use the z-score formula to calculate the z-score and use the standard normal distribution table to find the probability.