Question 7.4: (Quantiles of the standard normal distribution) We have just......

Since for any real z
P(Z \gt z) = 1 − P(Z \leq z) = 1 − \Phi (z),

we see that, if z_\alpha is the upper 𝛼-quantile of Φ, we have 1 − \Phi(z_\alpha) = 𝛼, or
\Phi(z_\alpha) = 1 − 𝛼.

We thus look at the tables of the standard normal distribution and we seek, for different values of 𝛼, the closest z-value (up to two decimal places) for which the value of the distribution function is 1 − 𝛼.

(i) For 𝛼 = 0.01, the last equation becomes \Phi(z_{0.01}) = 0.99, and we see from the table in Appendix B that the corresponding quantile is

z_{0.01} ≅ 2.33.

(ii) Similarly for 𝛼 = 0.05, we get that \Phi(z_{0.05}) = 0.95, and from the tables we find the upper 0.05-quantile to be

z_{0.05} ≅ 1.645

(since the value appears to be halfway between 1.64 and 1.65).

(iii) Now we have \Phi(z_{0.10}) = 0.10 and this gives, by using the table in Appendix B again, that

z_{0.10} ≅ 1.28.