Algebra and Trigonometry (6th Edition)
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Author: Robert F. Blitzer
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![### Understanding Standardized Test Scores
Given Problem:
The scores on a standardized test are normally distributed with a mean of 95 and a standard deviation of 20. What test score is 0.6 standard deviations below the mean?
To solve this problem, we need to understand the concept of z-scores and how they represent standard deviations from the mean in a normal distribution.
#### Steps to Solve:
1. **Identify the mean (μ) and standard deviation (σ):**
- Mean (μ) = 95
- Standard Deviation (σ) = 20
2. **Calculate the z-score:**
- A z-score of 0.6 below the mean is represented as -0.6.
3. **Use the formula:**
\[
X = μ + (z \times σ)
\]
where \( X \) is the test score, \( μ \) is the mean, \( z \) is the z-score, and \( σ \) is the standard deviation.
\[
X = 95 + (-0.6 \times 20)
\]
4. **Perform the calculation:**
\[
X = 95 + (-12) = 95 - 12 = 83
\]
Therefore, a test score that is 0.6 standard deviations below the mean is **83**.](https://content.bartleby.com/qna-images/question/a2daf47f-1d43-4c0e-a218-05866800687b/8aa6ac32-8abc-42bd-88d9-1bc99ad85d47/tkwmpsi.jpeg)
Transcribed Image Text:### Understanding Standardized Test Scores
Given Problem:
The scores on a standardized test are normally distributed with a mean of 95 and a standard deviation of 20. What test score is 0.6 standard deviations below the mean?
To solve this problem, we need to understand the concept of z-scores and how they represent standard deviations from the mean in a normal distribution.
#### Steps to Solve:
1. **Identify the mean (μ) and standard deviation (σ):**
- Mean (μ) = 95
- Standard Deviation (σ) = 20
2. **Calculate the z-score:**
- A z-score of 0.6 below the mean is represented as -0.6.
3. **Use the formula:**
\[
X = μ + (z \times σ)
\]
where \( X \) is the test score, \( μ \) is the mean, \( z \) is the z-score, and \( σ \) is the standard deviation.
\[
X = 95 + (-0.6 \times 20)
\]
4. **Perform the calculation:**
\[
X = 95 + (-12) = 95 - 12 = 83
\]
Therefore, a test score that is 0.6 standard deviations below the mean is **83**.
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