Question
Assume the mean \( =100 \) and the standard deviation \( =25: x=60, z= \)
Ask by Brewer Bartlett. in the United States
Jan 25,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The z-score for \( x = 60 \) is **-1.6**.
Solution
To calculate the **z-score** for a given value \( x \) using the mean (\( \mu \)) and standard deviation (\( \sigma \)), you can use the following formula:
\[
z = \frac{x - \mu}{\sigma}
\]
**Given:**
- Mean (\( \mu \)) = 100
- Standard Deviation (\( \sigma \)) = 25
- Value (\( x \)) = 60
**Calculation:**
\[
z = \frac{60 - 100}{25} = \frac{-40}{25} = -1.6
\]
**Result:**
\[
z = -1.6
\]
This z-score of **-1.6** indicates that the value \( x = 60 \) is 1.6 standard deviations **below** the mean.
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Simplify this solution Bonus Knowledge
To find the z-score, you can use the formula: \( z = \frac{(x - \mu)}{\sigma} \). Plugging in the values: \( z = \frac{(60 - 100)}{25} = \frac{-40}{25} = -1.6 \). So, the z-score for \( x = 60 \) is \(-1.6\). This means that 60 is 1.6 standard deviations below the mean. In practical terms, z-scores help to understand how far a data point is from the average relative to the variation in the dataset. A z-score of -1.6 indicates that 60 is less than the average value of 100, which can be a signal to investigate why that measurement is lower – perhaps it signifies an anomaly or a unique situation.
