Exact sampling distribution of a sample proportion
The population has \(4\) successful packages and \(3\) unsuccessful packages
A sample of \(n=3\) 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 |
|---|---|---|---|
| \(0\) | \(0\) | \(1\) | \(\frac{1}{35}\) |
| \(1\) | \(\frac{1}{3}\) | \(12\) | ? |
| \(2\) | \(\frac{2}{3}\) | \(18\) | \(\frac{18}{35}\) |
| \(3\) | \(1\) | \(4\) | \(\frac{4}{35}\) |