Deriving the confidence intervalP(−Φα−1&lt;X&lt;Φα−1)=1−2αLet&nbsp;P∼N(0,1)&nbsp;P(−Φα−1&lt;X&lt;Φα−1)&nbsp;&nbsp;(1)&nbsp;P(X&lt;Φα−1)−P(X&lt;−Φα−1)&nbsp;P(A&lt;X&lt;B)=P(X&lt;B)−P(X&lt;A)(2)&nbsp;Φ(Φα−1))−Φ(−Φα−1))&nbsp;P(X&lt;A)=Φ(A)(3)&nbsp;Φ(Φα−1))−(1−Φ(Φα−1)))&nbsp;Φ(−x)=1−Φ(x)(4)&nbsp;α−1+α&nbsp;f(f−1)=id(5)&nbsp;−1×(1−2α)&nbsp;&nbsp;(6)

Question

Deriving the confidence intervalP(−Φα−1&amp;lt;X&amp;lt;Φα−1)=1−2αLet&amp;nbsp;P∼N(0,1)&amp;nbsp;P(−Φα−1&amp;lt;X&amp;lt;Φα−1)&amp;nbsp;&amp;nbsp;(1)&amp;nbsp;P(X&amp;lt;Φα−1)−P(X&amp;lt;−Φα−1)&amp;nbsp;P(A&amp;lt;X&amp;lt;B)=P(X&amp;lt;B)−P(X&amp;lt;A)(2)&amp;nbsp;Φ(Φα−1))−Φ(−Φα−1))&amp;nbsp;P(X&amp;lt;A)=Φ(A)(3)&amp;nbsp;Φ(Φα−1))−(1−Φ(Φα−1)))&amp;nbsp;Φ(−x)=1−Φ(x)(4)&amp;nbsp;α−1+α&amp;nbsp;f(f−1)=id(5)&amp;nbsp;−1×(1−2α)&amp;nbsp;&amp;nbsp;(6)

Pubephenedsjq · Accepted Answer

Note that for α&amp;lt;0.5, you get ϕ−1(a)&amp;lt;0.Hence, 1−2α≥0 so for sure α≤0.5. Then,P(ϕ−1(α)&amp;lt;X&amp;lt;−ϕ−1(α))=1−ϕ(ϕ−1(α))−ϕ(ϕ−1(α))=1−2α

Deriving the confidence interval \(\displaystyle{P}{\left(-\Phi^{{-{1}}}_\alpha{ < }{X}{

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