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In a survey on political attitudes, researchers measure trust in government on a scale, where scores are normally distributed with a population mean of *µ* = 50 and a population standard deviation of *σ* = 15.  If we transform this distribution into z-scores, then the distribution of z-scores will have a mean of ________ and a standard deviation of ________.

In the context of the given problem, you want to transform a normally distributed set of scores into z-scores. 

The formula for transforming a score \( X \) from a normal distribution with mean \( \mu \) and standard deviation \( \sigma \) into a z-score \( Z \) is:

\[ Z = \frac{X - \mu}{\sigma} \]

Here, the scores are normally distributed with:
- Mean (\(\mu\)) = 50
- Standard deviation (\(\sigma\)) = 15

When you transform a normal distribution into z-scores, the resulting distribution of z-scores will always have:
- A mean of 0
- A standard deviation of 1

This is because the z-score transformation re-centers the distribution around 0 and rescales it to have a standard deviation of 1.

So, the distribution of z-scores will have a mean of **0** and a standard deviation of **1**.

Would you like further details or have any other questions?

Here are some related questions you might find useful:
1. How do you calculate a z-score for a given value?
2. What is the significance of a z-score in statistics?
3. How does the Central Limit Theorem relate to the distribution of sample means?
4. What are the properties of a normal distribution?
5. How do you interpret a z-score in the context of hypothesis testing?
6. What are some practical applications of z-scores?
7. Can you explain the concept of standard deviation in more detail?
8. How do you use z-scores to compare scores from different distributions?

**Tip:** When working with normal distributions, understanding the properties and transformations, such as z-scores, can greatly simplify the analysis and comparison of data sets.

Explore how to transform a normal distribution into z-scores with a detailed explanation. Learn about the properties of z-scores, including their mean and standard deviation transformations. Suitable for advanced high school students and anyone interested in statistical distributions.

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