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

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 `$\sigma_2$`

. The two sample are independent. Assume that the standard deviations are unknown but equal.

## 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,`$E = t_{\alpha/2,n_1+n_2-2} s_p\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}$`

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,`$s_p=\sqrt{\dfrac{(n_1-1)s_1^2+(n_2-1)s_2^2}{n_1+n_2-2}}$`

be the pooled standard deviation.