Determine The Area Under The Standard Normal Curve That Lies Between:(a) $Z = -2.12$ And $Z = 2.12$(b) $Z = -1.65$ And $Z = 0$(c) $Z = -0.31$ And $Z = 0.04$(a) The Area That Lies Between $Z =
Introduction
The standard normal curve, 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 standard normal curve is used to model a wide range of phenomena, from the heights of individuals in a population to the returns on investments in a portfolio. In this article, we will explore how to determine the area under the standard normal curve that lies between specific z-scores.
The Standard Normal Curve
The standard normal curve is a probability distribution that is characterized by the following properties:
- It is symmetric about the mean, which is 0.
- It has a standard deviation of 1.
- It is continuous, meaning that it can take on any value within a given range.
- It is bell-shaped, with the majority of the data points clustered around the mean.
The standard normal curve is often represented by the following equation:
f(z) = (1/√(2π)) * e(-z2/2)
where f(z) is the probability density function of the standard normal curve, and z is the z-score.
Determining the Area Under the Standard Normal Curve
To determine the area under the standard normal curve that lies between specific z-scores, we can use a standard normal distribution table, also known as a z-table. The z-table provides the probability that a random variable will take on a value less than or equal to a given z-score.
Method 1: Using the z-Table
The z-table is a table that lists the probabilities of a random variable taking on a value less than or equal to a given z-score. To use the z-table, we need to find the z-score in the table and then look up the corresponding probability.
For example, let's say we want to find the area under the standard normal curve that lies between z = -2.12 and z = 2.12. We can use the z-table to find the probabilities of a random variable taking on a value less than or equal to z = -2.12 and z = 2.12.
| z-score | Probability |
|---|---|
| -2.12 | 0.0170 |
| 2.12 | 0.9830 |
The probability of a random variable taking on a value less than or equal to z = -2.12 is 0.0170, and the probability of a random variable taking on a value less than or equal to z = 2.12 is 0.9830.
To find the area under the standard normal curve that lies between z = -2.12 and z = 2.12, we can subtract the probability of a random variable taking on a value less than or equal to z = -2.12 from the probability of a random variable taking on a value less than or equal to z = 2.12.
Area = 0.9830 - 0.0170 = 0.9660
Therefore, the area under the standard normal curve that lies between z = -2.12 and z = 2.12 is 0.9660.
Method 2: Using a Calculator or Software
Alternatively, we can use a calculator or software to find the area under the standard normal curve that lies between specific z-scores. For example, we can use a calculator or software to find the area under the standard normal curve that lies between z = -2.12 and z = 2.12.
Using a calculator or software, we can find that the area under the standard normal curve that lies between z = -2.12 and z = 2.12 is 0.9660.
Example 1: Finding the Area Between z = -1.65 and z = 0
Let's say we want to find the area under the standard normal curve that lies between z = -1.65 and z = 0. We can use the z-table to find the probabilities of a random variable taking on a value less than or equal to z = -1.65 and z = 0.
| z-score | Probability |
|---|---|
| -1.65 | 0.0505 |
| 0 | 0.5000 |
The probability of a random variable taking on a value less than or equal to z = -1.65 is 0.0505, and the probability of a random variable taking on a value less than or equal to z = 0 is 0.5000.
To find the area under the standard normal curve that lies between z = -1.65 and z = 0, we can subtract the probability of a random variable taking on a value less than or equal to z = -1.65 from the probability of a random variable taking on a value less than or equal to z = 0.
Area = 0.5000 - 0.0505 = 0.4495
Therefore, the area under the standard normal curve that lies between z = -1.65 and z = 0 is 0.4495.
Example 2: Finding the Area Between z = -0.31 and z = 0.04
Let's say we want to find the area under the standard normal curve that lies between z = -0.31 and z = 0.04. We can use the z-table to find the probabilities of a random variable taking on a value less than or equal to z = -0.31 and z = 0.04.
| z-score | Probability |
|---|---|
| -0.31 | 0.3740 |
| 0.04 | 0.5244 |
The probability of a random variable taking on a value less than or equal to z = -0.31 is 0.3740, and the probability of a random variable taking on a value less than or equal to z = 0.04 is 0.5244.
To find the area under the standard normal curve that lies between z = -0.31 and z = 0.04, we can subtract the probability of a random variable taking on a value less than or equal to z = -0.31 from the probability of a random variable taking on a value less than or equal to z = 0.04.
Area = 0.5244 - 0.3740 = 0.1504
Therefore, the area under the standard normal curve that lies between z = -0.31 and z = 0.04 is 0.1504.
Conclusion
In conclusion, the standard normal curve 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 standard normal curve is used to model a wide range of phenomena, from the heights of individuals in a population to the returns on investments in a portfolio.
To determine the area under the standard normal curve that lies between specific z-scores, we can use a standard normal distribution table, also known as a z-table. The z-table provides the probability that a random variable will take on a value less than or equal to a given z-score.
Alternatively, we can use a calculator or software to find the area under the standard normal curve that lies between specific z-scores. For example, we can use a calculator or software to find the area under the standard normal curve that lies between z = -2.12 and z = 2.12.
In this article, we have explored how to determine the area under the standard normal curve that lies between specific z-scores. We have used the z-table to find the probabilities of a random variable taking on a value less than or equal to a given z-score, and we have used a calculator or software to find the area under the standard normal curve that lies between specific z-scores.
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, N. L., & Kotz, S. (1970). Continuous univariate distributions. Houghton Mifflin Company.
Frequently Asked Questions: Determining the Area Under the Standard Normal Curve ====================================================================================
Q: What is the standard normal curve?
A: The standard normal curve, 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 model a wide range of phenomena, from the heights of individuals in a population to the returns on investments in a portfolio.
Q: How do I determine the area under the standard normal curve that lies between specific z-scores?
A: To determine the area under the standard normal curve that lies between specific z-scores, you can use a standard normal distribution table, also known as a z-table. The z-table provides the probability that a random variable will take on a value less than or equal to a given z-score.
Q: What is the z-table?
A: The z-table is a table that lists the probabilities of a random variable taking on a value less than or equal to a given z-score. It is used to determine the area under the standard normal curve that lies between specific z-scores.
Q: How do I use the z-table to find the area under the standard normal curve that lies between specific z-scores?
A: To use the z-table, you need to find the z-score in the table and then look up the corresponding probability. The probability of a random variable taking on a value less than or equal to a given z-score is listed in the table.
Q: Can I use a calculator or software to find the area under the standard normal curve that lies between specific z-scores?
A: Yes, you can use a calculator or software to find the area under the standard normal curve that lies between specific z-scores. Many calculators and software programs have built-in functions to calculate the area under the standard normal curve.
Q: What is the difference between the area under the standard normal curve and the probability of a random variable taking on a value less than or equal to a given z-score?
A: The area under the standard normal curve is the total area under the curve, while the probability of a random variable taking on a value less than or equal to a given z-score is the area under the curve to the left of the z-score.
Q: How do I find the area under the standard normal curve that lies between two z-scores?
A: To find the area under the standard normal curve that lies between two z-scores, you can subtract the probability of a random variable taking on a value less than or equal to the lower z-score from the probability of a random variable taking on a value less than or equal to the higher z-score.
Q: What is the significance of the standard normal curve in statistics and probability theory?
A: The standard normal curve is a fundamental concept in statistics and probability theory. It is used to model a wide range of phenomena, from the heights of individuals in a population to the returns on investments in a portfolio.
Q: Can I use the standard normal curve to model real-world phenomena?
A: Yes, you can use the standard normal curve to model real-world phenomena. The standard normal curve is a continuous probability distribution that can be used to model a wide range of phenomena.
Q: What are some common applications of the standard normal curve?
A: Some common applications of the standard normal curve include:
- Modeling the heights of individuals in a population
- Modeling the returns on investments in a portfolio
- Modeling the scores on a standardized test
- Modeling the time it takes to complete a task
Q: How do I choose the right z-score to use in a problem?
A: To choose the right z-score to use in a problem, you need to consider the specific problem and the data that is available. You should choose a z-score that is relevant to the problem and that will provide the most accurate results.
Q: What are some common mistakes to avoid when using the standard normal curve?
A: Some common mistakes to avoid when using the standard normal curve include:
- Using the wrong z-score
- Not considering the specific problem and data
- Not using the correct probability distribution
- Not considering the assumptions of the standard normal curve
Q: How do I troubleshoot common problems when using the standard normal curve?
A: To troubleshoot common problems when using the standard normal curve, you should:
- Check the z-score and make sure it is correct
- Check the data and make sure it is relevant to the problem
- Check the probability distribution and make sure it is correct
- Check the assumptions of the standard normal curve and make sure they are met
Conclusion
In conclusion, the standard normal curve 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 standard normal curve is used to model a wide range of phenomena, from the heights of individuals in a population to the returns on investments in a portfolio.
To determine the area under the standard normal curve that lies between specific z-scores, you can use a standard normal distribution table, also known as a z-table. The z-table provides the probability that a random variable will take on a value less than or equal to a given z-score.
Alternatively, you can use a calculator or software to find the area under the standard normal curve that lies between specific z-scores. Many calculators and software programs have built-in functions to calculate the area under the standard normal curve.
In this article, we have explored how to determine the area under the standard normal curve that lies between specific z-scores. We have used the z-table to find the probabilities of a random variable taking on a value less than or equal to a given z-score, and we have used a calculator or software to find the area under the standard normal curve that lies between specific z-scores.
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, N. L., & Kotz, S. (1970). Continuous univariate distributions. Houghton Mifflin Company.