Question

The same question continues to ask: Identify the probability of each sample and describe the sampling distribution of sample means. Each sample has a probability:(type an integer or a simplified fraction)

Answer 1

A simple random sample is a subset of a statistical population in which each member of the subset has an equal probability of being chosen. So each and every sample data has a probability equal to (1/population size) of getting selected.
Now when we create a sample, it's usually not of size 1, so say we have a random sample of size 'n', called a sample proportion. Now when we are looking at samples, we look for characteristics in the samples, whether the sample falls into one particular group or the other. So, say in our sample, some 'x' observations fall under our category of interest. Hence, the sample mean will basically be p=x/n. Now our sample proportion will have a probability distribution. This probability distribution is approximately normal if

np(1-p)≥10, with standard deviation $\sigma = \sqrt {(p(1-p)/n)}$

So our sample proportion is normally distributed with probability distribution

, x is sample data, if and only if the condition is satisfied.

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Also the distribution of the sample means is a normal distribution as long as the condition stated above is satisfied. This is also formally stated as the Central Limit Theorem.

P.S. If in the first part of the question, we were looking for the probability with which a sample of size n might be selected from a population of size N,then the probability would simply be p = n/N.