I am able to check if the interval contains 0 which I can find out by just plugging numbers into the above formula but I am curious as to why this interval containing zero means that the two CI's of $\theta_1$ and $\theta_2$ overlap.

  • 3
    $\begingroup$ Because the answer comes down to the meaning of subtraction, one wonders what you are really trying to ask. Could you elaborate a little? $\endgroup$
    – whuber
    Mar 15, 2021 at 21:32

1 Answer 1


If they overlap, there is a number $\color{blue}x$ from their intersection, i. e.

$$\color{blue}x \in \theta_1 \pm z(\text{SE}_1)\\ \color{blue}x \in \theta_2 \pm z(\text{SE}_2)$$

By subtracting them, we obtain

$$\color{red}0 \in (\theta_1 - \theta_2)\pm z(\text{SE}_1 + \text{SE}_2)$$

(since $\color{blue}x - \color{blue}x = \color{red}0$).


I used the shortened notation; the expanded one is

$$ \begin{aligned} \ \theta_1 - z(\text{SE}_1) \le\, &\color{blue}x \le \theta_1 + z(\text{SE}_1)\\ \theta_2 - z(\text{SE}_2) \le\, &\color{blue}x \le \theta_2 + z(\text{SE}_2)\\ \hline (\theta_1 - \theta_2) - z(\text{SE}_1 + \text{SE}_2) \le\, &\color{red}0 \le (\theta_1 - \theta_2) + z(\text{SE}_1 + \text{SE}_2) \end{aligned} $$


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.