Exact sampling distribution of a sample proportion

A sample proportion is a zero–one sample mean

Assign value 1 to a success and 0 to a failure. The sum of a zero–one sample is its number of successes; divide that sum by the sample size to find its mean. Thus a sample proportion is a sample mean for binary data. General numerical sample means require more than this special case

A sample of six zero–one records contains two ones. Find its sample mean

Exact sampling without replacement

Consider six distinct labelled records: two have value 1 and four have value 0. Select an unordered sample of size three uniformly without replacement. There are twenty equally likely samples. Group them by the number of ones; divide that number by three to find each sample mean. Counts of mean values need not be equal

Six distinct records contain two ones and four zeros. If three are selected without replacement, how many samples have no ones?

Six distinct records contain two ones and four zeros. If three are selected without replacement, how many samples have exactly one one?

Six distinct zero–one records contain two ones and four zeros. Three are selected without replacement. What is the probability that their sample mean is two thirds?

The full distribution

Six distinct zero–one records contain two ones and four zeros. Three are selected uniformly without replacement. The possible sample means are 0,1/3,2/3 with probabilities 1/5,3/5,1/5. Their weighted mean is 1/3, equal to the population mean. This describes all possible samples under the stated selection rule, rather than one observed sample or an empirical histogram

Construct the exact table, ungraded

Six distinct zero–one records contain two ones and four zeros. Select three without replacement. List or count all twenty labelled samples, group them by sample mean, and explain why the rows are not equally likely. Verify that the total row count is twenty and the probability masses sum to one

Model: count disjoint groups

Six distinct zero–one records contain two ones and four zeros. Select three without replacement. For zero, one or two ones, the counts and expectation are calculated below. Divide each count by the twenty possible samples. Every three-record subset belongs to exactly one group, so the counts sum to twenty

Keep dependence and the census endpoint

These draws are dependent because records are not replaced. Do not apply an independent-draw variance formula automatically. Selecting the entire finite population gives exactly its mean every time, so the sampling distribution has zero spread. Repeated independent draws with replacement are a different model, taught in the next lesson