To find the value below which the least expensive 10% of personal computers fall, we need to determine the corresponding z-score that represents the 10th percentile in a standard normal distribution.
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Identify the z-score for the 10th percentile:
From z-tables or using a standard normal distribution calculator, the z-score that corresponds to the 10th percentile is approximately \(-1.2816\). -
Use the z-score formula to find the value (X):
The z-score formula is:
\[ z = \frac{X - \mu}{\sigma} \]
Where:
- \( z \) is the z-score.
- \( \mu \) is the mean (average price).
- \( \sigma \) is the standard deviation.
- \( X \) is the value we want to find.
Given:
- \( \mu = 949 \)
- \( \sigma = 100 \)
- \( z = -1.2816 \)
We can rearrange the formula to solve for \( X \):
\[ X = z \cdot \sigma + \mu \]
- Plug in the values:
\[ X = -1.2816 \cdot 100 + 949 \]
\[ X = -128.16 + 949 \]
\[ X = 820.84 \]
- Round to the nearest dollar:
The least expensive 10% of personal computers cost less than approximately $821.
Therefore, the answer is $821.