Population Proportion

Description:

• True population proportion $p$ is unknown
• $\bar{p}=\frac{X}{n}$
  • where $X$ is binomial random variable denotes the number of successes in the sample
  • $X\sim B(n,p).E[X]=np$ and ${\sigma }_X=\sqrt{npq}$
• $E[\bar{p}]=p$ and $\sigma (\bar{p})=\sqrt{\frac{pq}{n}}$

Inteference about $p_1-p_2$:

• The difference between 2 population proportions is $p_1-p_2$
• Let ${\bar{p}}_1,{\bar{p}}_2$ be the saple proportion for a random sample from population 1 and 2
• $E[{\bar{p}}_1-{\bar{p}}_2]=p_1-p_2$ and ${\sigma }_{{\bar{p}}_1-{\bar{p}}_2}=\sqrt{\frac{p_1(1-p_1)}{n_1}+\frac{p_2(1-p_2)}{n_2}}$
• If $n_1{\bar{p}}_1\ge 5$ and $n_1{\bar{q}}_1\ge 5$ and $n_2{\bar{p}}_2\ge 5$ and $n_2{\bar{q}}_2\ge 5$
  • The sampling distribution of ${\bar{p}}_1-{\bar{p}}_2$ can be approximated by normal distribution
  • $CI(\alpha \%)={\bar{p}}_1-{\bar{p}}_2\pm z_{\alpha /2}\sqrt{\frac{p_1(1-p_1)}{n_1}+\frac{p_2(1-p_2)}{n_2}}$