Inference for Proportions¶

One-Sample z-test for Population Proportion¶

z-test for Population Proportion¶

Used to test whether the population proportion, \(p\), has changed A random sample of \(n\) individuals from the population with a sample proportion of \(\hat{p}\) is used to try to prove the case

Conditions for A z-test for Population Proportion¶

• Items in the sample satisfy the independence condition
  • Random sampling/assignment
  • If without replacement, \(n < 0.1N\)

• Sample size is large enough such that sampling distribution is approximately a normal distribution

  • \(np_0 \geq 10\)
  • \(n(1-p_0) \geq 10\)
  • \(p_0\) is the population proportion in the null hypothesis, \(p = p_0\)

• \(z = \dfrac{statistics - parameter}{standard\ error\ of\ the\ statistic}\)

• \(standard\ error\ of\ the\ statistic = \sqrt{\dfrac{p_0(1-p_0)}n}\)

Confidence Interval¶

\(confidence\ interval = \hat{p} \pm z \cdot \sqrt{\dfrac{p_0(1-p_0)}n}\)