Question

Suppose a distribution has a mean of 100 and a standard deviation of 20. Further suppose that random samples of size n = 100 are taken with replacement from this distribution. The mean of the sampling distribution of sample means is mu subscript x with bar on top end subscript = and the standard deviation of the sampling distribution of sample means is sigma subscript top enclose x end subscript = .

Answer 1

Ans.

Central Limit Theorem (CLT)

The central limit theorem states that the sampling distribution of the mean of any independent, random variable will be normal or nearly normal, if the sample size is large enough.

How large is "large enough"? The answer depends on two factors.

• Requirements for accuracy. The more closely the sampling distribution needs to resemble a normal distribution, the more sample points will be required.
• The shape of the underlying population. The more closely the original population resembles a normal distribution, the fewer sample points will be required.

In practice, some statisticians say that a sample size of 30 is large enough when the population distribution is roughly bell-shaped. Others recommend a sample size of at least 40. But if the original population is distinctly not normal (e.g., is badly skewed, has multiple peaks, and/or has outliers), researchers like the sample size to be even larger.

Here, Mean () = 100

Standard Deviation () = 20

Sample size (n) = 100 > 40

Hence , Sampling Distribution :-

Mean of Sampling Distribution:-

Standard Deviation of Sampling Distribution:-