## CI for difference between two population means (variances are unknown and unequal)

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,

`$1-\alpha$`

is the confidence coefficient,`$\overline{X} = \dfrac{1}{n_1}\sum X_i$`

is the sample mean of first sample,`$\overline{Y} = \dfrac{1}{n_2}\sum Y_i$`

is the sample mean second sample,`$E = t_{\alpha/2,n_1+n_2-2} \sqrt{\dfrac{s_1^2}{n_1}+\dfrac{s_2^2}{n_2}}$`

is the margin of error,`$t_{\alpha/2,n_1+n_2-2}$`

is the critical value of $t$,`$s_1^2 = \dfrac{1}{n_1-1}\sum (X_i -\overline{X})^2$`

is the sample variance of first sample,`$s_2^2 = \dfrac{1}{n_2-1}\sum (Y_i -\overline{Y})^2$`

is the the sample variance of second sample,