(Quantiles of the standard normal distribution) We have just seen how we can use the table of values for the standard normal distribution to find, for a given real z, the corresponding probability Φ(z) = P(Z ≤ z). In many cases, interest lies in the opposite direction, so that for a given value 𝛼

Question

(Quantiles of the standard normal distribution) We have just seen how we can use the table of values for the standard normal distribution to find, for a given real z, the corresponding probability Φ(z) = P(Z ≤ z). In many cases, interest lies in the opposite direction, so that for a given value 𝛼 in the interval (0, 1), we want to find the value of z, denoted by $\pmb{z_\alpha}$ , such that

\pmb{P(Z\gt z_{\alpha})=\alpha.}

In such a case, the value $\pmb{z_\alpha}$ is called the (upper) 𝛂-quantile of the distribution Φ (Figure 7.4).
Find the 𝛼-quantiles of Φ for the cases when

(i) 𝛼 = 0.01, (ii) 𝛼 = 0.05, (iii) 𝛼 = 0.10.

[latex]\pmb{P(Z\gt z_{\alpha})=\alpha.}[/latex]

In such a case, the value [latex]\pmb{z_\alpha}[/latex] is called the (upper) 𝛂-quantile of the distribution Φ (Figure 7.4).
Find the 𝛼-quantiles of Φ for the cases when

(i) 𝛼 = 0.01, (ii) 𝛼 = 0.05, (iii) 𝛼 = 0.10.

Accepted Answer

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

we see that, if [latex]z_\alpha[/latex] is the upper 𝛼-quantile of Φ, we have 1 − [latex]\Phi(z_\alpha)[/latex] = 𝛼, or
[latex]\Phi(z_\alpha)[/latex] = 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 [latex]\Phi(z_{0.01})[/latex] = 0.99, and we see from the table in Appendix B that the corresponding quantile is

[latex]z_{0.01}[/latex] ≅ 2.33.

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

[latex]z_{0.05}[/latex] ≅ 1.645

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

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

[latex]z_{0.10}[/latex] ≅ 1.28.

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

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