Question

Let \(z\) denote a variable that has a standard normal distribution. Determine the value \(z^{*}\) to satisfy the following conditions: a. \(P\left(zz^{*}\right)=.02\) e. \(P\left(z>z^{*}\right)=.01\) f. \(P\left(z>z^{*}\right.\) or \(\left.z<-z^{*}\right)=.20\)

Short answer

The values of \(z^{*}\) can be determined via quantile function Q, using a standard normal distribution table or software capable of providing these calculations. It's important to take the symmetry of normal distribution into consideration.

Step-by-step solution

Identify the Standard Normal Distribution Properties

When a variable \(z\) has a standard normal distribution, its mean is 0 and standard deviation is 1. The cumulative distribution function (CDF) gives the probability that the variable is less than or equal to a certain value.

Use the CDF to find \(z^{*}\)

We have to refer to the inverse of the CDF (also called the quantile function) to find the values of \(z^{*}\). For instance, for problem A; \(P(z<z^{*})=0.025\), implies \(z^{*}=Q(0.025)\) where Q is the quantile function. Use this Q function to solve the problems separately.

Solve for \(z^{*}\)

The solution for the six tasks will be found by referring to the standard normal distribution table or by using a calculator or software that can calculate the quantile function. a. \(z^{*}=Q(0.025)\), b. \(z^{*}=Q(0.01)\), c. \(z^{*}=Q(0.05)\), d. \(z^{*}=Q(1-0.02)\) (because \(P(z>z^{*})=1-P(zz^{*}\}\) or \(\{z<-z^{*}\}\) is equivalent to the event \(\{|z|>z^{*}\}\). Therefore \(P(|z|>z^{*})=0.20\), implies \(z^{*}=Q(1-0.20/2)\) (this comes from the symmetry of the standard normal distribution and the definition of absolute value).