Exact sampling distribution of a sample proportion
The population has \(7\) successful responses and \(2\) unsuccessful responses
A sample of \(n=2\) is selected without replacement. Let \(X\) be the number of successes in the sample
\[\hat p=\frac{X}{n}\]
The complete table below gives the possible values of \(X\) and the sample proportion
| \(x\) | \(\hat p\) | Number of samples | Probability |
|---|---|---|---|
| \(0\) | \(0\) | \(1\) | \(\frac{1}{36}\) |
| \(1\) | \(\frac{1}{2}\) | \(14\) | \(\frac{7}{18}\) |
| \(2\) | \(1\) | \(21\) | ? |