Deriving the confidence interval $P(-\Phi^{-1}_\alpha < X < \Phi^{-1}_\alpha) = 1-2\alpha$

Question

I'm trying to derive the confidence interval for the standard normal distribution.

Let $P\sim\mathcal N(0,1)$

\begin{align} &P(-\Phi^{-1}_\alpha < X < \Phi^{-1}_\alpha)&&&(1)\\ &P(X < \Phi^{-1}_\alpha) - P(X<-\Phi^{-1}_\alpha) &&P(A<X<B) = P(X<B)-P(X<A) &(2) \\ &\Phi(\Phi^{-1}_\alpha)) - \Phi(-\Phi^{-1}_\alpha)) &&P(X<A) = \Phi(A) &(3)\\ &\Phi(\Phi^{-1}_\alpha)) - (1-\Phi(\Phi^{-1}_\alpha))) &&\Phi(-x) = 1-\Phi(x) &(4)\\ &\alpha-1+\alpha &&f(f^{-1}) = id &(5)\\ &\color{red}{-1}\times(1-2\alpha) &&&(6) \end{align}

As you can see, my method is correct up to a factor of $-1$, but I'm not sure where I went wrong... Any help is appreciated!

Answer 1

Note that for $\alpha <0.5$, you get $\phi^{-1}(a) < 0$.

Hence,

$1-2\alpha \ge 0$ so for sure $\alpha \le 0.5$. Then, $$ P(\phi^{-1}(\alpha) < X < -\phi^{-1}(\alpha)) = 1- \phi(\phi^{-1}(\alpha)) - \phi(\phi^{-1}(\alpha)) = 1-2\alpha $$.