Question

If $Y$ follows Normal distribution with mean 4 and variance 16, then the standard Normal deviate is

• A. $(Y - 4)/16$
• B. $(Y - 16)/4$
• C. $(Y - 4)/4$
• D. $(Y - 16)/16$

Hint:

Ensure you divide by the standard deviation ($\sigma = \sqrt{\text{Variance}}$) rather than the variance ($\sigma^2$) when constructing the standard normal deviate.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
Standardization is the process of transforming a normal random variable into a standard normal variable (with mean 0 and variance 1).
Key Formula or Approach:
If a random variable $Y$ follows a Normal distribution $Y \sim N(\mu, \sigma^2)$, then the standard normal deviate $Z$ is defined as:
\[ Z = \frac{Y - \mu}{\sigma} \]
where $\sigma$ is the standard deviation.

Step 2: Detailed Explanation:
We are given:
Population mean, $\mu = 4$
Population variance, $\sigma^2 = 16$
First, calculate the standard deviation $\sigma$:
\[ \sigma = \sqrt{\sigma^2} = \sqrt{16} = 4 \]
Now substitute the values of $\mu$ and $\sigma$ into the standardization formula:
\[ Z = \frac{Y - 4}{4} \]
Therefore, the standard normal deviate is $(Y - 4)/4$.

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