## 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
```
$$
\begin{aligned}
\bigg(\hat{p}-Z_{\alpha/2} \sqrt{\frac{\hat{p}*(1-\hat{p})}{n+4}}, \hat{p}+Z_{\alpha/2} \sqrt{\frac{\hat{p}*(1-\hat{p})}{n+4}}\bigg).
\end{aligned}
$$
```