## 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 $Z$.