The scores on a standardized test are normally distributed with a mean of 95 and standard deviation of 20. What test score is 0.6 standard deviations below the mean?
Inverse Normal Distribution
The method used for finding the corresponding z-critical value in a normal distribution using the known probability is said to be an inverse normal distribution. The inverse normal distribution is a continuous probability distribution with a family of two parameters.
Mean, Median, Mode
It is a descriptive summary of a data set. It can be defined by using some of the measures. The central tendencies do not provide information regarding individual data from the dataset. However, they give a summary of the data set. The central tendency or measure of central tendency is a central or typical value for a probability distribution.
Z-Scores
A z-score is a unit of measurement used in statistics to describe the position of a raw score in terms of its distance from the mean, measured with reference to standard deviation from the mean. Z-scores are useful in statistics because they allow comparison between two scores that belong to different normal distributions.
![### 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**.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa2daf47f-1d43-4c0e-a218-05866800687b%2F8aa6ac32-8abc-42bd-88d9-1bc99ad85d47%2Ftkwmpsi.jpeg&w=3840&q=75)
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