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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.

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    $\begingroup$ 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

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


Note:

I used the shortened notation; the expanded one is

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

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