To find the z-score of a person who scored 460 on a standardized exam with a mean score of 400 and a standard deviation of 20, follow these steps:
### Step-by-Step Solution:
Step 1: Understand the z-score formula
The z-score is a measure that describes the position of a score within a distribution in terms of how many standard deviations it is away from the mean. The formula for the z-score is:
[tex]\[
z = \frac{X - \mu}{\sigma}
\][/tex]
where:
- [tex]\( X \)[/tex] is the score for which we need to find the z-score.
- [tex]\( \mu \)[/tex] is the mean of the distribution.
- [tex]\( \sigma \)[/tex] is the standard deviation of the distribution.
Step 2: Identify the given values
From the problem, we know:
- The mean score ([tex]\( \mu \)[/tex]) is 400.
- The standard deviation ([tex]\( \sigma \)[/tex]) is 20.
- The score ([tex]\( X \)[/tex]) is 460.
Step 3: Substitute the values into the z-score formula
[tex]\[
z = \frac{460 - 400}{20}
\][/tex]
Step 4: Simplify the expression
Perform the subtraction in the numerator:
[tex]\[
460 - 400 = 60
\][/tex]
Now, divide the result by the standard deviation:
[tex]\[
z = \frac{60}{20} = 3
\][/tex]
Conclusion:
The z-score of a person who scored 460 on the exam is 3. This means that the person's score is 3 standard deviations above the mean score of the distribution.
