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 $\big((\hat{p}_1-\hat{p}_2) -E, (\hat{p}_1-\hat{p}_2) +E\big)$.

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