Question

. Assume X ~ N (10, 4).

1. What is the (approximate) distribution of X if the sample size is 100? Briefly discuss the theorem underlying your answer.
2. What happens to the variance of as the sample size increases? Draw a diagram and explain.
3. What two values of (symmetric around the population mean) contain a) 75% and b) 95% of the distribution? Draw a diagram. Relate your answer to b) to the empirical rule.

Answer 1

Given

i. The approximate distribution of sample mean \bar{X} if the sample size n = 100,
is Normally distributed

According the Central Limit Theorem, consider an iid sample  

Then as the sample size increases, the sample mean converges in distribution to a standard normal

So the approximate distribution of \bar{X} for a sample size n is

In our case with

ii. As sample size increases, the variance of \bar{X} decreases

So as sample size n increases, variance decreases

Here is the sampling distribution of \bar{X} as a function of n,
You can see in the plot below that the curve becomes less wider as n increases,
hence the variance decreases

iii) The two values (symmetric around the population mean) that contain
of the population can be derived as follows:

Let L and U be the standardized lower and upper bounds of the interval, then as it is a symmetric interval,

\implies

(This is the symmetric interval)

The plot of the symmetric interval is given below

Hence for

a) 75% interval,

Now the interval is given by,

In our case with

a) 95% interval,

Now the interval is given by,

In our case with