1. Find The Corresponding Area Between $z = 0$ And $z = 0.96$. Your Answer: 2. What Is The Area Under The Standard Normal Curve To The Right Of $z = -2.67$? Your Answer: 3. If The Area Under The

Feb 28, 2025 by ADMIN 223 views

**Understanding the Standard Normal Curve: Calculating Areas and Probabilities**

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 real-world phenomena and to make predictions about future events. In this article, we will explore how to calculate areas and probabilities under the standard normal curve.

1. Find the corresponding area between $z = 0$ and $z = 0.96$

To find the area between $z = 0$ and $z = 0.96$ , we need to use a standard normal distribution table or calculator. The area under the standard normal curve to the left of $z = 0$ is 0.5, since the curve is symmetric about the mean. To find the area between $z = 0$ and $z = 0.96$ , we need to subtract the area to the left of $z = 0$ from the area to the left of $z = 0.96$ .

Using a standard normal distribution table or calculator, we find that the area to the left of $z = 0.96$ is approximately 0.8343. Therefore, the area between $z = 0$ and $z = 0.96$ is:

0.8343 - 0.5 = 0.3343

So, the area between $z = 0$ and $z = 0.96$ is approximately 0.3343.

2. What is the area under the standard normal curve to the right of $z = -2.67$ ?

To find the area under the standard normal curve to the right of $z = -2.67$ , we need to use a standard normal distribution table or calculator. The area under the standard normal curve to the left of $z = -2.67$ is approximately 0.0034. Since the curve is symmetric about the mean, the area to the right of $z = -2.67$ is equal to 1 minus the area to the left of $z = -2.67$ .

Therefore, the area under the standard normal curve to the right of $z = -2.67$ is:

1 - 0.0034 = 0.9966

So, the area under the standard normal curve to the right of $z = -2.67$ is approximately 0.9966.

3. If the area under the standard normal curve to the left of $z = 1.23$ is 0.8902, what is the area under the standard normal curve to the right of $z = 1.23$ ?

To find the area under the standard normal curve to the right of $z = 1.23$ , we need to use the fact that the area under the standard normal curve is equal to 1. Since the area to the left of $z = 1.23$ is 0.8902, the area to the right of $z = 1.23$ is equal to 1 minus the area to the left of $z = 1.23$ .

Therefore, the area under the standard normal curve to the right of $z = 1.23$ is:

1 - 0.8902 = 0.1098

So, the area under the standard normal curve to the right of $z = 1.23$ is approximately 0.1098.

4. If the area under the standard normal curve to the left of $z = -0.56$ is 0.7127, what is the area under the standard normal curve to the right of $z = -0.56$ ?

To find the area under the standard normal curve to the right of $z = -0.56$ , we need to use the fact that the area under the standard normal curve is equal to 1. Since the area to the left of $z = -0.56$ is 0.7127, the area to the right of $z = -0.56$ is equal to 1 minus the area to the left of $z = -0.56$ .

Therefore, the area under the standard normal curve to the right of $z = -0.56$ is:

1 - 0.7127 = 0.2873

So, the area under the standard normal curve to the right of $z = -0.56$ is approximately 0.2873.

5. If the area under the standard normal curve to the left of $z = 2.11$ is 0.9821, what is the area under the standard normal curve to the right of $z = 2.11$ ?

To find the area under the standard normal curve to the right of $z = 2.11$ , we need to use the fact that the area under the standard normal curve is equal to 1. Since the area to the left of $z = 2.11$ is 0.9821, the area to the right of $z = 2.11$ is equal to 1 minus the area to the left of $z = 2.11$ .

Therefore, the area under the standard normal curve to the right of $z = 2.11$ is:

1 - 0.9821 = 0.0179

So, the area under the standard normal curve to the right of $z = 2.11$ is approximately 0.0179.

Conclusion

In this article, we have explored how to calculate areas and probabilities under the standard normal curve. We have used standard normal distribution tables and calculators to find the areas under the curve to the left and right of various z-scores. We have also used the fact that the area under the standard normal curve is equal to 1 to find the areas to the right of various z-scores. These calculations are essential in statistics and probability theory, and are used to make predictions about future events and to model real-world phenomena.

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). Distributions in statistics: Continuous univariate distributions. John Wiley & Sons.

Q: What is the standard normal curve?

A: The standard normal curve is a continuous probability distribution that is symmetric about the mean, with a standard deviation of 1. It is also known as the z-distribution.

Q: What is the mean of the standard normal curve?

A: The mean of the standard normal curve is 0.

Q: What is the standard deviation of the standard normal curve?

A: The standard deviation of the standard normal curve is 1.

Q: What is the area under the standard normal curve to the left of z = 0?

A: The area under the standard normal curve to the left of z = 0 is 0.5, since the curve is symmetric about the mean.

Q: How do I find the area under the standard normal curve to the left of a given z-score?

A: To find the area under the standard normal curve to the left of a given z-score, you can use a standard normal distribution table or calculator. The table or calculator will give you the area to the left of the z-score.

Q: How do I find the area under the standard normal curve to the right of a given z-score?

A: To find the area under the standard normal curve to the right of a given z-score, you can use the fact that the area under the standard normal curve is equal to 1. You can subtract the area to the left of the z-score from 1 to find the area to the right of the z-score.

Q: What is the relationship between the standard normal curve and the normal distribution?

A: The standard normal curve is a special case of the normal distribution, where the mean is 0 and the standard deviation is 1. The normal distribution is a continuous probability distribution that is symmetric about the mean, with a standard deviation of σ.

Q: How do I use the standard normal curve to make predictions about future events?

A: You can use the standard normal curve to make predictions about future events by finding the area under the curve to the left of a given z-score. This will give you the probability of the event occurring.

Q: What are some common applications of the standard normal curve?

A: The standard normal curve has many applications in statistics and probability theory, including:

Hypothesis testing: The standard normal curve is used to test hypotheses about population means and proportions.
Confidence intervals: The standard normal curve is used to construct confidence intervals for population means and proportions.
Regression analysis: The standard normal curve is used to analyze the relationship between variables in a regression model.
Time series analysis: The standard normal curve is used to analyze time series data and make predictions about future events.

Q: What are some common mistakes to avoid when working with the standard normal curve?

A: Some common mistakes to avoid when working with the standard normal curve include:

Not using the correct z-score: Make sure to use the correct z-score when finding the area under the curve.
Not using the correct table or calculator: Make sure to use the correct table or calculator when finding the area under the curve.
Not understanding the concept of symmetry: Make sure to understand the concept of symmetry in the standard normal curve.
Not using the correct formula: Make sure to use the correct formula when finding the area under the curve.

Conclusion

In this article, we have answered some frequently asked questions about the standard normal curve. We have covered topics such as the mean and standard deviation of the standard normal curve, how to find the area under the curve to the left and right of a given z-score, and common applications and mistakes to avoid when working with the standard normal curve. We hope this article has been helpful in understanding the standard normal curve and how to use it in statistics and probability theory.

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). Distributions in statistics: Continuous univariate distributions. John Wiley & Sons.

1. Find The Corresponding Area Between $z = 0$ And $z = 0.96$. Your Answer: 2. What Is The Area Under The Standard Normal Curve To The Right Of $z = -2.67$? Your Answer: 3. If The Area Under The

1. Find the corresponding area between $z = 0$ and $z = 0.96$

2. What is the area under the standard normal curve to the right of $z = -2.67$ ?

3. If the area under the standard normal curve to the left of $z = 1.23$ is 0.8902, what is the area under the standard normal curve to the right of $z = 1.23$ ?

4. If the area under the standard normal curve to the left of $z = -0.56$ is 0.7127, what is the area under the standard normal curve to the right of $z = -0.56$ ?

5. If the area under the standard normal curve to the left of $z = 2.11$ is 0.9821, what is the area under the standard normal curve to the right of $z = 2.11$ ?

Conclusion

References

Further Reading

Q: What is the standard normal curve?

Q: What is the mean of the standard normal curve?

Q: What is the standard deviation of the standard normal curve?

Q: What is the area under the standard normal curve to the left of z = 0?

Q: How do I find the area under the standard normal curve to the left of a given z-score?

Q: How do I find the area under the standard normal curve to the right of a given z-score?

Q: What is the relationship between the standard normal curve and the normal distribution?

Q: How do I use the standard normal curve to make predictions about future events?

Q: What are some common applications of the standard normal curve?

Q: What are some common mistakes to avoid when working with the standard normal curve?

Conclusion

References

Further Reading

1. Find the corresponding area between z=0z = 0z=0 and z=0.96z = 0.96z=0.96

2. What is the area under the standard normal curve to the right of z=−2.67z = -2.67z=−2.67?

3. If the area under the standard normal curve to the left of z=1.23z = 1.23z=1.23 is 0.8902, what is the area under the standard normal curve to the right of z=1.23z = 1.23z=1.23?

4. If the area under the standard normal curve to the left of z=−0.56z = -0.56z=−0.56 is 0.7127, what is the area under the standard normal curve to the right of z=−0.56z = -0.56z=−0.56?

5. If the area under the standard normal curve to the left of z=2.11z = 2.11z=2.11 is 0.9821, what is the area under the standard normal curve to the right of z=2.11z = 2.11z=2.11?

Conclusion

References

Further Reading

Q: What is the standard normal curve?

Q: What is the mean of the standard normal curve?

Q: What is the standard deviation of the standard normal curve?

Q: What is the area under the standard normal curve to the left of z = 0?

Q: How do I find the area under the standard normal curve to the left of a given z-score?

Q: How do I find the area under the standard normal curve to the right of a given z-score?

Q: What is the relationship between the standard normal curve and the normal distribution?

Q: How do I use the standard normal curve to make predictions about future events?

Q: What are some common applications of the standard normal curve?

Q: What are some common mistakes to avoid when working with the standard normal curve?

Conclusion

References

Further Reading

1. Find the corresponding area between $z = 0$ and $z = 0.96$

2. What is the area under the standard normal curve to the right of $z = -2.67$ ?

3. If the area under the standard normal curve to the left of $z = 1.23$ is 0.8902, what is the area under the standard normal curve to the right of $z = 1.23$ ?

4. If the area under the standard normal curve to the left of $z = -0.56$ is 0.7127, what is the area under the standard normal curve to the right of $z = -0.56$ ?

5. If the area under the standard normal curve to the left of $z = 2.11$ is 0.9821, what is the area under the standard normal curve to the right of $z = 2.11$ ?