Question

If the Z-score of normal distribution is 2.5, the mean of the distribution is 45 and the standard deviation is 3, then the value of X for a normal distribution is:

• A. 42
• B. 52.5
• C. 47
• D. 3

Hint:

A Z-score of 2.5 means the value \(X\) is exactly 2.5 standard deviations above the mean.
Using simple mental arithmetic:
\[ \text{Mean} + 2.5 \times \text{S.D.} = 45 + 2.5(3) = 45 + 7.5 = 52.5 \]

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
In a normal distribution, any raw score \(X\) can be converted into a standardized score, known as a Z-score.
The Z-score represents how many standard deviations the raw score lies above or below the mean.
Key Formula or Approach:
The standardization formula is:
\[ Z = \frac{X - \mu}{\sigma} \]
We can rearrange this formula to solve directly for the raw score \(X\):
\[ X = \mu + Z \cdot \sigma \]
Where:
- \(X\) is the raw score.
- \(\mu\) is the mean of the distribution.
- \(\sigma\) is the standard deviation.
- \(Z\) is the standard normal score.

Step 2: Detailed Explanation:
Let us identify the given values from the problem statement:
- Z-score, \(Z = 2.5\)
- Mean, \(\mu = 45\)
- Standard deviation, \(\sigma = 3\)
Now, substitute these values into the rearranged formula to solve for \(X\):
\[ X = 45 + (2.5 \times 3) \]
Perform the multiplication first:
\[ 2.5 \times 3 = 7.5 \]
Add this value to the mean:
\[ X = 45 + 7.5 = 52.5 \]
Therefore, the value of the raw score \(X\) is 52.5.

Step 3: Final Answer:
The correct option is (B).