## CI for difference between two population means (variances are known)

Let $X_1, X_2, \cdots, X_{n_1}$ be a random sample of size $n_1$ from a population with mean $\mu_1$ and standard deviation $\sigma_1$.

Let $Y_1, Y_2, \cdots, Y_{n_2}$ be a random sample of size $n_2$ from a population with mean $\mu_2$ and standard deviation $\sigma_2$. And the two sample are independent.

## Formula

$100(1-\alpha)$% confidence interval estimate for the difference $(\mu_1-\mu_2)$ is

### $(\overline{X} -\overline{Y})- E \leq (\mu_1-\mu_2) \leq (\overline{X} -\overline{Y}) + E$

where,

• $\overline{X} = \dfrac{1}{n_1}\sum X_i$ and $\overline{Y} =\dfrac{1}{n_2}\sum Y_i$ are the sample means of first and second sample respectively,
• $1-\alpha$ is the confidence coefficient,
• $E = Z_{\alpha/2} \sqrt{\dfrac{\sigma_1^2}{n_1}+\dfrac{\sigma_2^2}{n_2}}$ is the margin of error,
• $Z_{\alpha/2}$ is the critical value of $Z$.