Question

A rowing team consists of three rowers who weigh 42, 36 and 29 pounds. Find all possible random samples with replacement of size three and compute the sample mean for each one. Use them to find the sampling distribution of the sample mean. Also verify the results.

Answer 1

Total number of all possible random samples is $3^3=27$. We can see all possible random samples in the picture. Then we compute the sample mean for each one (the fourth column of the table). Mean of all sample means is equal to $\approx 35.7$. Mean of the population (42, 36, 29) is the same ( $\approx 35.7$) (the mean of the sampling distribution of the sample mean is always the same as the mean of the original non-normal distribution).

Central Limit Theorem tells us that even if a population distribution is non-normal, its sampling distribution of the sample mean will be normal for a large number of samples (at least 30).

As $n=3<30$ we cannot conclude that sample mean has normal distribution.

So we have some distribution with mean $\bar{x}\approx 35.7$ and standard deviation $\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}\approx 3.07$

$(\sigma\text{ is a population standard deviation, sample size } n=3).$

Using Excel we find the sampling distribution of the sample mean:

The left column contains the various sample means which we get, the right column contains the corresponding probabilities.

The graph of this distribution:

As we can see the graph is not so much similiar to the graph of a normal distribution.