Plus Four Confidence Interval for Proportion Examples
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 $(0.1822,0.6512)$.
Reference
You can read step by step tutorial on Plus Four Confidence Interval for Proportion, tutorial will help you to understand step by step procedure 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
Plus Four Confidence Interval for Proportion Calculator
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