Areas of Tails of Distributions

Key Takeaways

  • The problem of finding the number \mathrm{z}^{*} so that the probability P\left(Z < \mathrm{z}^{*}\right) is a specified value c is solved by looking for the number c in the interior of Figure 12.2 "Cumulative Normal Probability" and reading \mathrm{z}^{*} from the margins.
  • The problem of finding the number \mathrm{z}^{*} so that the probability P\left(Z > \mathrm{z}^{*}\right) is a specified value c is solved by looking for the complementary probability 1-c in the interior of Figure 12.2 "Cumulative Normal Probability" and reading z^* from the margins.
  • For a normal random variable X with mean \mu and standard deviation \sigma, the problem of finding the number \mathrm{x}^{*} so that P\left(X < \mathrm{x}^{*}\right) is a specified value c (or so that P\left(X > \mathrm{x}^{*}\right) is a specified value c) is solved in two steps: (1) solve the corresponding problem for Z with the same value of c, thereby obtaining the z-score, \mathrm{z}^{*}, of \mathrm{x}^{*}; (2) find \mathrm{x}^{*} using \mathrm{x}^{*}=\mu+\mathrm{z}^{*} \cdot \sigma.
  • The value of Z that cuts off a right tail of area c in the standard normal distribution is denoted z_{c}.