Question

For the population 5,6,7      take sample size 2 and apply central limit theorem and prove that mean of sample means is equal to population mean and standard deviation of sampling distribution of statistic is equal to population standard deviation divide by the square root of sample size.

Answer 1

Suppose X is a random variable with a distribution that may be known or unknown (it can be any distribution).
Using a subscript that matches the random variable.

suppose:

= the mean of X

= the standard deviation of X

If you draw random samples of size n, then as n increases, the random samples X-bar which consists of sample means, tend to be normally distributed.

X-bar ~ N(, /)

The central limit theorem for sample means says that if you keep drawing larger and larger samples (such as rolling one, two, five, and finally, ten dice) and calculating their means, the sample means form their own normal distribution (the sampling distribution). The normal distribution has the same mean as the original distribution and a variance that equals the original variance divided by, the sample size. The variable n is the number of values that are averaged together, not the number of times the experiment is done.

To put it more formally, if you draw random samples of size n, the distribution of the random variable X-bar which consists of sample means, is called the sampling distribution of the mean. The sampling distribution of the mean approaches a normal distribution as the sample size n increases.

Hence, Mean of sample means is equal to population mean and standard deviation of sampling distribution of statistic is equal to population standard deviation divide by the square root of sample size.