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 $$ \begin{aligned} \bigg(\hat{p}-Z_{\alpha/2} \sqrt{\frac{\hat{p}*(1-\hat{p})}{n}}, \hat{p}+Z_{\alpha/2} \sqrt{\frac{\hat{p}*(1-\hat{p})}{n}}\bigg). \end{aligned} $$

References

  • Agresti, A. (2018). Categorical Data Analysis (3rd ed.). Wiley.
  • Wilson, E. B. (1927). “Probable inference, the law of succession, and statistical inference.” Journal of the American Statistical Association, 22(158), 209-212.
  • Brown, L. D., Cai, T. T., & DasGupta, A. (2001). “Interval estimation for a binomial proportion.” Statistical Science, 16(2), 101-133.