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 positive Z-score means the value \(X\) is above the mean.
Since \(Z = 2.5\), \(X\) must be exactly \(2.5\) standard deviations above the mean of \(45\).
Using simple mental math: \(45 + 2.5 \times 3 = 45 + 7.5 = 52.5\).

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
A Z-score (or standard score) indicates how many standard deviations a particular data point \(X\) lies above or below the mean \(\mu\) of a normal distribution.
Standardizing a normal variable allows us to compare different normal distributions using the standard normal distribution table.
Key Formula or Approach:
The relationship between a raw score \(X\), the population mean \(\mu\), the population standard deviation \(\sigma\), and the standard Z-score is given by:
\[ Z = \frac{X - \mu}{\sigma} \]
We can rearrange this formula to solve for the raw value \(X\):
\[ X = \mu + Z \cdot \sigma \]

Step 2: Detailed Explanation:
From the problem description, we are given the following values:
- Mean, \(\mu = 45\)
- Standard deviation, \(\sigma = 3\)
- Z-score, \(Z = 2.5\)
Now, let us substitute these values into the rearranged formula to find \(X\):
\[ X = 45 + (2.5 \cdot 3) \]
First, calculate the product of the Z-score and the standard deviation:
\[ 2.5 \cdot 3 = 7.5 \]
Next, add this product to the mean:
\[ X = 45 + 7.5 = 52.5 \]
Thus, the value of the raw observation \(X\) is 52.5.

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