The Z-score tells you the number of standard deviations away from the mean (and in what direction) a data value is. The formula:  can be used to find the Z-score for a single member of the population. 

Notice  tells you the signed distance the data value (X) is from the mean. When you divide that by the size of each chunk (the standard deviation) you are measuring the distance in units of the size of the standard deviations (how many standard deviations fit in the distance between the data value and the mean) - which gives you the Z-score. 

The formula  can be used to find the value in the population when given the Z-score (signed number of standard deviations). 

Notice that   tells you how far from the mean the data value is (since Z is the number of standard deviations and  is the size of each standard deviation the product tells the total distance). So adding the distance from the mean to the mean gives the location along the number line for the data value. 

 

A certain population is normally distributed with a mean of 122 and a standard deviation of 18. 

What is the data value in the population (X) if the Z-score is 3.29 standard deviations?  


To find the data value in the population \( X \) when the Z-score is given, you can use the formula:

\[
X = \mu + Z \cdot \sigma
\]

Where:
- \( X \) is the data value,
- \( \mu \) is the mean of the population,
- \( Z \) is the Z-score,
- \( \sigma \) is the standard deviation.

Given:
- Mean \( \mu = 122 \),
- Standard deviation \( \sigma = 18 \),
- Z-score \( Z = 3.29 \),

Substitute the values into the formula:

\[
X = 122 + 3.29 \cdot 18
\]

Now calculate:

\[
X = 122 + 59.22
\]

\[
X = 181.22
\]

Therefore, the data value in the population \( X \) is approximately **181.22**.

Would you like more details or explanations on any of these steps?

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Here are 5 related questions for deeper exploration:

1. What is the Z-score if a data value in the population is 100, given the same mean and standard deviation?
2. How would the Z-score change if the standard deviation of the population increased to 25?
3. If the Z-score is negative, what does it mean in terms of the data value’s position relative to the mean?
4. How would you interpret a Z-score of 0 in any data set?
5. Can the Z-score ever be greater than 5? If so, under what conditions?

**Tip:** A Z-score tells you how extreme a value is compared to the mean of the dataset; Z-scores around 0 represent typical values, while values farther from 0 indicate more unusual or extreme data points.

A certain population is normally distributed with a mean of 122 and a standard deviation of 18. What is the data value in the population (X) if the Z-score is 3.29 standard deviations?

Learn how to calculate the data value in a normally distributed population with a mean of 122 and a standard deviation of 18, given a Z-score of 3.29. This solution covers key statistical concepts like normal distribution and Z-scores. Suitable for high school students (Grades 10-12).

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