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Chapter 7: Problem 68

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

Expert verified
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

01

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.
02

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.
03

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).

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Most popular questions from this chapter

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