Introduction
In this article we will discuss step by step procedure to construct a plus four confidence interval for population proportion.
The plus four confidence interval for the population proportion can be used when the confidence coefficient is more than 90% and the sample size of the population is at least 10.
Plus Four 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 $p$. The estimator of the population proportion of success based on plus four rule is $\hat{p}=\frac{X+2}{n+4}$.
Let $C=1-\alpha$ be the confidence coefficient. We wish to construct $100(1-\alpha)$% plus four confidence interval estimate of a population proportion $p$.
The standard error of estimate of $\hat{p}$ is
$$ \begin{aligned} SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n+4}}, \end{aligned} $$
The margin of error for proportion is
$$ \begin{aligned} E = Z_{\alpha/2} \sqrt{\frac{\hat{p}(1-\hat{p})}{n+4}}, \end{aligned} $$
where $Z_{\alpha/2}$ is the table value from normal statistical table.
$100(1-\alpha)$% plus four confidence interval for population proportion is
$$ \begin{aligned} \hat{p} - E \leq p \leq \hat{p} + E. \end{aligned} $$
Assumptions
a. The sample size is at least 10, i.e., $n\geq 10$.
b. The sample is a random sample.
Step by step procedure
Step by step procedure to find the plus four 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 plus four estimate of population proportion of success is $\hat{p} =\frac{X+2}{n+4}$.
Step 3 Specify the formula
$100(1-\alpha)$% plus four confidence interval to estimate the population proportion is
$$ \begin{aligned} \hat{p} - E \leq p \leq \hat{p} + E. \end{aligned} $$
where $E=Z_{\alpha/2} \sqrt{\frac{\hat{p}(1-\hat{p})}{n+4}}$.
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
$$ \begin{aligned} E = Z_{\alpha/2} \sqrt{\frac{\hat{p}*(1-\hat{p})}{n+4}} \end{aligned} $$
Step 6 Determine the confidence interval
$100(1-\alpha)$% plus four confidence interval estimate for population proportion is
$$ \begin{aligned} \hat{p} - E \leq p \leq \hat{p} + E \end{aligned} $$
Equivalently, $100(1-\alpha)$% plus four confidence interval estimate of population proportion is $\hat{p} \pm E$ or $(\hat{p} -E, \hat{p} +E)$.
Thus $100(1-\alpha)$% plus four confidence interval estimate of population proportion $p$ is
$$ \begin{aligned} \bigg(\hat{p}-Z_{\alpha/2} \sqrt{\frac{\hat{p}*(1-\hat{p})}{n+4}}, \hat{p}+Z_{\alpha/2} \sqrt{\frac{\hat{p}*(1-\hat{p})}{n+4}}\bigg). \end{aligned} $$
Reference
You can read step by step tutorial on Plus Four Confidence Interval for Proportion examples,tutorial will help you to understand step by step examples to construct a plus four confidence interval for population proportion.
You can also use calculator to compute the plus four confidence interval for population proportion.
Plus Four Confidence Interval for Proportion examples
Plus Four Confidence Interval for Proportion Calculator
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