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I want to know for sure when we compute the 95% confidence interval for mean/prop, we get z = qnorm(0.95) or z = qnorm(0.975)?

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As I understand it, you want to know when to use a certain quantile (qnorm). This will depend on the level of significance set for your test. For example, consider a hypothesis test on which you want to test, $H_0: \mu = \mu_0 \: \times \: \mu \neq \mu_0$. It is known that $$Z_{test} = \frac {X- \mu_0}{\sigma} \sim N(0,1)$$ As it is a bilateral test (see the hypotheses), the level of significance set is associated with the quantile of your test statistic ($ Z_ {test} $). For, $\alpha = 1- \lambda/2$, because the test is bilateral ($\alpha$ being the level of significance and $\lambda$ being the associated trust).

If you set a significance level of say $ \alpha = 0.05 $ you will have the equivalent quantile will be the $\lambda = 1-\alpha/2 = 0.975$ quantile of an $N(0.1)$, that is, $qnorm(.975) \approx 1.959964$, if $\alpha = 0.1$ the quantile will be $qnorm(.95) \approx 1.644854$.

Hope this helps!

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  • $\begingroup$ despite having interpreted hypothesis tests, the connection is direct to confidence intervals. $\endgroup$
    – jassis
    Jan 26, 2021 at 18:01

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