Question

If the scores on a test have a mean of 36 and a standard deviation of 4, what is the z-score for a score of 26?

• A. 2.5
• B. 25
• C. -2.5
• D. -0.25

Hint:

If the raw score is less than the mean, the z-score must be negative.
Since 26 is less than 36, options (A) and (B) can be immediately eliminated.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
A z-score (standard score) measures how many standard deviations a raw score \( X \) lies above or below the population mean \( \mu \).
Key Formula or Approach:
The formula for calculating a z-score is:
\[ z = \frac{X - \mu}{\sigma} \] Where:
- \( X \) is the raw score.
- \( \mu \) is the population mean.
- \( \sigma \) is the population standard deviation.

Step 2: Detailed Explanation:
From the problem description, we are given:
- Population mean (\( \mu \)) = \( 36 \)
- Population standard deviation (\( \sigma \)) = \( 4 \)
- Raw score (\( X \)) = \( 26 \)
Substituting these values into the z-score formula:
\[ z = \frac{26 - 36}{4} \] Simplify the numerator:
\[ z = \frac{-10}{4} \] Calculate the division:
\[ z = -2.5 \] The negative sign indicates that the raw score of 26 lies \( 2.5 \) standard deviations below the mean of 36.

Step 3: Final Answer:
The z-score for a score of 26 is -2.5.