## Confidence Interval for paired t-test

Let $X_1, X_2, \cdots, X_n$ and $Y_1, Y_2, \cdots, Y_n$ be two dependent samples each of size $n$ with respective means $\mu_1$ and $\mu_2$. Define $d_i = X_i - Y_i$, $i=1,2,\cdots, n$.

## Formula

$100(1-\alpha)$% confidence interval for the mean of the difference is

### $\overline{d} - E \leq \mu \leq \overline{d} + E$

where,

• $1-\alpha$ is the confidence coefficient,
• $E = t_{(\alpha/2,n-1)} \dfrac{s_d}{\sqrt{n}}$ is the margin of error,
• $\overline{d}=\frac{1}{n} \sum_{i=1}^n d_i$ is the mean of the difference,
• $t_{(\alpha/2,n-1)}$ is the critical value of $t$
• $s_d=\sqrt{\dfrac{1}{n-1}\sum (d_i - \overline{d})^2}$ is the sample standard deviation of the difference