The two events are not independent For small samples ($n < 30$), we use $t$-distribution If “0” is inside the C.I., then the event is believable
Let $\bar d$ and $s_d$ be the mean and standard deviation, respectively, of the normally distributed differences of a small random pairs ($n\lt 30$) of measurements. A $100(1-\alpha)\%$ confidence interval for $\mu_D =\mu_1-\mu_2$ is
$$ \bar d - t_{\alpha / 2}\frac{s_d}{\sqrt n } \lt \mu_D \lt \bar d + t_{\alpha / 2}\frac{s_d}{\sqrt n } $$
where $t_{\alpha / 2}$ is the $t$-value with $n-1$ degrees of freedom, leaving an area of $\alpha / 2$ to the right.
Note: $\bar d = \frac{\sum^n_{i=1}d_i}{n}$, where $d_i = x_{i_1} - x_{i_2}$ $s_d = \sqrt{\frac{1}{n-1}\sum^n_{i=1}(d_i - \bar d)}$