Question

If X follows Normal distribution with mean 50 and variance 3600 then sample mean $\bar{X}$ based on a large sample of 100 observations follows Normal distribution with parameters}

• A. mean 50, variance 3600
• B. mean 60, variance 60
• C. mean 50, variance 36
• D. mean 50, variance 60

Hint:

The variance of the sample mean decreases as the sample size $n$ increases, specifically by a factor of $1/n$.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The sampling distribution of the sample mean $\bar{X}$ derived from a normally distributed population is itself normally distributed.
Key Formula or Approach:
If $X \sim N(\mu, \sigma^2)$, then the sample mean of $n$ independent observations follows:
\[ \bar{X} \sim N\left(\mu, \frac{\sigma^2}{n}\right) \]

Step 2: Detailed Explanation:
We are given the population parameters:
Population mean, $\mu = 50$
Population variance, $\sigma^2 = 3600$
Sample size, $n = 100$
Let us compute the parameters for the distribution of the sample mean $\bar{X}$:
1. The mean of $\bar{X}$ is:
\[ E(\bar{X}) = \mu = 50 \]
2. The variance of $\bar{X}$ is:
\[ \text{Var}(\bar{X}) = \frac{\sigma^2}{n} = \frac{3600}{100} = 36 \]
Thus, $\bar{X}$ follows a Normal distribution with parameters mean 50 and variance 36.

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