Introduction
Confidence interval can be used to estimate the population parameter with the help of an interval with some degree of confidence. One such a parameter that can be estimated is a population proportion.
In this article we will discuss step by step procedure to construct a confidence interval for population proportion.
Confidence Interval for Proportion
Let $X$ be the observed number of individuals possessing certain attributes (number of successes) in a random sample of size $n$ from a large population with population proportion $p$. Then $\hat{p}=\frac{X}{n}$ be the observed proportion of successes.
Let $C=1-\alpha$ be the confidence coefficient. We wish to construct $100(1-\alpha)$% confidence interval estimate of a population proportion $p$.
The margin of error for proportion is
$$ \begin{aligned} E = Z_{\alpha/2} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}, \end{aligned} $$
where $Z_{\alpha/2}$ is the table value from normal statistical table.
$100(1-\alpha)$% confidence interval for population proportion is
$$ \begin{aligned} \hat{p} - E \leq p \leq \hat{p} + E. \end{aligned} $$
Assumptions
a. $np\geq 10$ and $n(1-p)\geq 10$.
b. The sample is a random sample.
Step by step procedure
Step by step procedure to find the confidence interval for proportion is as follows :
Step 1 Specify the confidence level $(1-\alpha)$
Step 2 Given information
Specify the given information, sample size $n$, observed number of successes $X$. The estimate of population proportion of success is $\hat{p} =\frac{X}{n}$.
Step 3 Specify the formula
$100(1-\alpha)$% confidence interval to estimate the population proportion is
$$ \begin{aligned} \hat{p} - E \leq p \leq \hat{p} + E. \end{aligned} $$
where $E=Z_{\alpha/2} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$.
Step 4 Determine the critical value
Find the critical value $Z_{\alpha/2}$ from the normal statistical table that corresponds to the desired confidence level.
Step 5 Compute the margin of error
The margin of error for proportion is
$$ \begin{aligned} E = Z_{\alpha/2} \sqrt{\frac{\hat{p}*(1-\hat{p})}{n}} \end{aligned} $$
Step 6 Determine the confidence interval
$100(1-\alpha)$% confidence interval estimate for population proportion is
$$ \begin{aligned} \hat{p} - E \leq p \leq \hat{p} + E \end{aligned} $$
Equivalently, $100(1-\alpha)$% confidence interval estimate of population proportion is $\hat{p} \pm E$ or $(\hat{p} -E, \hat{p} +E)$.
Thus $100(1-\alpha)$% confidence interval estimate of population proportion $p$ is
$$ \begin{aligned} \bigg(\hat{p}-Z_{\alpha/2} \sqrt{\frac{\hat{p}*(1-\hat{p})}{n}}, \hat{p}+Z_{\alpha/2} \sqrt{\frac{\hat{p}*(1-\hat{p})}{n}}\bigg). \end{aligned} $$
