Confidence interval for two proportions

Confidence interval for two proportions

Let $X_1$ be the observed number of individuals possessing certain attributes (number of successes) in a random sample of size $n_1$ from a large population with population proportion $p_1$ and let $X_2$ be the observed number of individuals possessing certain attributes (number of successes) in a random sample of size $n_2$ from a large population with population proportion $p_2$. The two sample are independent. Then $\hat{p_1}=\dfrac{X_1}{n_1}$ and $\hat{p_2}=\dfrac{X_2}{n_2}$.

Let $C=1-\alpha$ be the confidence coefficient. Our objective is to construct a $100(1-\alpha)$% confidence interval estimate of the difference between two population proportions $(p_1-p_2)$.

The margin of error for the difference $(p_1-p_2)$ is

$$ \begin{aligned} E = Z_{\alpha/2} \sqrt{\frac{\hat{p}_1*(1-\hat{p}_1)}{n_1}+\frac{\hat{p}_2*(1-\hat{p}_2)}{n_2}} \end{aligned} $$

where $Z_{\alpha/2}$ is the value from normal statistical table.

Then $100(1-\alpha)\%$ confidence interval for the difference $(p_1-p_2)$ is

$$ \begin{aligned} (\hat{p}_1-\hat{p}_2) - E \leq (p_1 -p_2) \leq (\hat{p}_1 -\hat{p}_2)+ E. \end{aligned} $$

Assumptions

a. The sample proportions are from the two random samples that are independent.

b. For each of the two samples $np\geq 5$ and $n(1-p)\geq 5$.

Step by step procedure

Step by step procedure to estimate the confidence interval for difference between two population proportions is as follows:

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

Step 2 Given information

Specify the given information, sample sizes $n_1$ and $n_2$. The observed number of successes $X_1$ and $X_2$. The estimates of population proportions are $\hat{p}_1 =\dfrac{X_1}{n_1}$ and $\hat{p}_2 = \dfrac{X_2}{n_2}$.

Step 3 Specify the formula

$100(1-\alpha)$% confidence interval for the difference $(p_1-p_2)$ is

$$ \begin{aligned} (\hat{p}_1-\hat{p}_2) - E \leq (p_1 -p_2) \leq (\hat{p}_1 -\hat{p}_2)+ E. \end{aligned} $$

where $E = Z_{\alpha/2} \sqrt{\frac{\hat{p}_1*(1-\hat{p}_1)}{n_1}+\frac{\hat{p}_2*(1-\hat{p}_2)}{n_2}}$.

Step 4 Determine the critical value

Determine 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 the difference $(p_1-p_2)$ is

$$ \begin{aligned} E = Z_{\alpha/2} \sqrt{\frac{\hat{p}_1*(1-\hat{p}_1)}{n_1}+\frac{\hat{p}_2*(1-\hat{p}_2)}{n_2}}. \end{aligned} $$

Step 6 Determine the confidence interval

Thus, $100(1-\alpha)$% confidence interval estimate for the difference $(p_1-p_2)$ is $(\hat{p}_1 -\hat{p}_2) \pm E$ or $\big((\hat{p}_1-\hat{p}_2) -E, (\hat{p}_1-\hat{p}_2) +E\big)$.

Do read my step by step tutorial on Confidence interval for two proportions examples,tutorial will help you to understand examples on confidence interval for difference between two population proportions.

Let me know in the comments if you have any questions on Confidence interval for two proportions and your thought on this article.