## 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,