## Plus Four Confidence Interval for Proportion

In this article we will discuss step by step examples to construct a plus four confidence interval for population proportion.

## Example 1

In a random sample of 60 students from a college, 32 opted mathematics as major subject. Using plus four method find a 95% confidence interval for the proportion of students who opted mathematics as a major subject.

### Solution

Given that sample size $n = 60$, observed $X = 32$.

The estimate of sample proportion of students who opted mathematics as a major subject based on plus four rule is `$\hat{p}=\dfrac{X+2}{n+4}=\dfrac{32+2}{60+4}=0.5312$`

.

### Step 1 Specify the confidence level $(1-\alpha)$

Confidence level is $1-\alpha = 0.95$. Thus, the level of significance is $\alpha = 0.05$.

### Step 2 Given information

Given that sample size $n =60$, observed number of students who opted mathematics as a major subject is $X=32$.

The estimate of the proportion of students who opted mathematics as a major subject based on plus four rule is `$\hat{p} =\dfrac{X+2}{n+4} =\dfrac{32+2}{60+4}=0.5312$`

.

### Step 3 Specify the formula

$100(1-\alpha)$% plus four confidence interval for population proportion is
```
$$
\begin{aligned}
\hat{p} - E \leq p \leq \hat{p} + E.
\end{aligned}
$$
```

where `$E=Z_{\alpha/2} \sqrt{\dfrac{\hat{p}(1-\hat{p})}{n+4}}$`

and $Z_{\alpha/2}$ is the $Z$ value providing an area of $\alpha/2$ in the upper tail of the standard normal probability distribution.

### Step 4 Determine the critical value

The critical value of $Z$ for given level of significance is $Z_{\alpha/2}$.

Thus `$Z_{\alpha/2} = Z_{0.025} = 1.96$`

.

### Step 5 Compute the margin of error

The margin of error for proportions is
```
$$
\begin{aligned}
E & = Z_{\alpha/2} \sqrt{\frac{\hat{p}*(1-\hat{p})}{n+4}}\\
& = 1.96 \sqrt{\frac{0.5312*(1-0.5312)}{60+4}}\\
& =0.1223.
\end{aligned}
$$
```

### Step 6 Determine the confidence interval

$95$% plus four confidence interval estimate for population proportion is
```
$$
\begin{aligned}
\hat{p} - E & \leq p \leq \hat{p} + E\\
0.5312 - 0.1223 & \leq p \leq 0.5312 + 0.1223\\
0.4089 & \leq p \leq 0.6535.
\end{aligned}
$$
```

Thus, $95$% plus four confidence interval estimate for population proportion $p$ of students who opted mathematics as a major subject is $(0.4089,0.6535)$.

## Example 2

A random sample of 20 college students was asked: “Have you smoked a cigarette in the past week?” eight students reported smoking within the past week. Use the plus-four method to find a 98% confidence interval for the true proportion of college students who smoke.

### Solution

Given that sample size $n = 20$, observed $X = 8$.

The estimate of sample proportion of students who smoked a cigarette in the past week based on plus four rule is `$\hat{p}=\dfrac{X+2}{n+4}=\dfrac{8+2}{20+4}=0.4167$`

.

### Step 1 Specify the confidence level $(1-\alpha)$

Confidence level is $1-\alpha = 0.98$. Thus, the level of significance is $\alpha = 0.02$.

### Step 2 Given information

Given that sample size $n =20$, observed number of students who smoked a cigarette in the past week is $X=8$.

The estimate of the proportion of students who smoked a cigarette in the past week based on plus four rule is `$\hat{p} =\dfrac{X+2}{n+4} =\dfrac{8+2}{20+4}=0.4167$`

.

### Step 3 Specify the formula

$100(1-\alpha)$% plus four confidence interval for population proportion is
```
$$
\begin{aligned}
\hat{p} - E \leq p \leq \hat{p} + E.
\end{aligned}
$$
```

where `$E=Z_{\alpha/2} \sqrt{\dfrac{\hat{p}(1-\hat{p})}{n+4}}$`

and $Z_{\alpha/2}$ is the $Z$ value providing an area of $\alpha/2$ in the upper tail of the standard normal probability distribution.

### Step 4 Determine the critical value

The critical value of $Z$ for given level of significance is $Z_{\alpha/2}$.

Thus `$Z_{\alpha/2} = Z_{0.01} = 2.33$`

.

### Step 5 Compute the margin of error

The margin of error for proportions is
```
$$
\begin{aligned}
E & = Z_{\alpha/2} \sqrt{\frac{\hat{p}*(1-\hat{p})}{n+4}}\\
& = 2.33 \sqrt{\frac{0.4167*(1-0.4167)}{20+4}}\\
& =0.2345.
\end{aligned}
$$
```

### Step 6 Determine the confidence interval

$98$% plus four confidence interval estimate for population proportion is
```
$$
\begin{aligned}
\hat{p} - E & \leq p \leq \hat{p} + E\\
0.4167 - 0.2345 & \leq p \leq 0.4167 + 0.2345\\
0.1822 & \leq p \leq 0.6512.
\end{aligned}
$$
```

Thus, $98$% plus four confidence interval estimate for population proportion $p$ of students who smoked a cigarette in the past week is $(0.1822,0.6512)$.