Understanding Normal Distributions

Oct 13, 2025 by windailys.com 146 views

**The scores on a test are normally distributed with a mean of 200 and a standard deviation of 30. Find the score that is $$1 \frac{1}{2}$$ standard deviations above the mean.**

A score of $\square$ is $$1 \frac{1}{2}$$ standard deviations above the mean.

Discussion category: mathematics

Understanding the Problem

The problem states that the scores on a test are normally distributed with a mean of 200 and a standard deviation of 30. We are asked to find the score that is 1.5 standard deviations above the mean.

What is a Normal Distribution?

A normal distribution is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. In a normal distribution, the mean, median, and mode are all equal.

Calculating the Score

To find the score that is 1.5 standard deviations above the mean, we need to use the z-score formula:

z = (X - μ) / σ

where:

z is the z-score
X is the score we are trying to find
μ is the mean (200)
σ is the standard deviation (30)

We are given that the z-score is 1.5. We can plug in the values and solve for X:

1.5 = (X - 200) / 30

Solving for X

To solve for X, we can multiply both sides of the equation by 30:

45 = X - 200

Next, we can add 200 to both sides of the equation:

X = 245

The Final Answer

Therefore, the score that is 1.5 standard deviations above the mean is 245.

Q&A

Q: What is the mean of the test scores? A: The mean of the test scores is 200.

Q: What is the standard deviation of the test scores? A: The standard deviation of the test scores is 30.

Q: What is the z-score that we are given? A: The z-score that we are given is 1.5.

Q: How do we calculate the score that is 1.5 standard deviations above the mean? A: We use the z-score formula: z = (X - μ) / σ

Q: What is the final answer? A: The final answer is 245.

Conclusion

In this article, we discussed how to find the score that is 1.5 standard deviations above the mean in a normal distribution. We used the z-score formula to solve for the score and found that the final answer is 245.

Frequently Asked Questions

Q: What is a normal distribution? A: A normal distribution is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean.

Q: How do we calculate the z-score? A: We use the z-score formula: z = (X - μ) / σ

Q: What is the z-score that we are given? A: The z-score that we are given is 1.5.

Q: How do we solve for X? A: We multiply both sides of the equation by 30 and then add 200 to both sides of the equation.

Q: What is the final answer? A: The final answer is 245.

Related Topics

Understanding Normal Distributions
Calculating Z-Scores
Solving for X in the Z-Score Formula

References

"Statistics for Dummies" by Deborah J. Rumsey
"Mathematics for Dummies" by Mary Jane Sterling
"Statistics and Probability with Applications" by Jim H. Smith

About the Author

The author of this article is a mathematics expert with a passion for teaching and explaining complex concepts in a simple and easy-to-understand way.