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Chapter 6: Problem 89

a. Find a \(z_{0}\) such that \(P\left(z>z_{0}\right)=.9750 .\) b. Find a \(z_{0}\) such that \(P\left(z>z_{0}\right)=.3594\).

Short Answer

Expert verified
Question: Determine the value of \(z_{0}\) for each of the following cases: a) \(P\left(z>z_{0}\right)=.9750\) b) \(P\left(z>z_{0}\right)=.3594\) Answer: a) \(z_{0} \approx -1.96\) b) \(z_{0} \approx 0.37\)

Step by step solution

01

a. Finding \(z_{0}\) for \(P\left(z>z_{0}\right)=.9750\)

Since the standard normal table typically provides the area/probability under the curve to the left of \(z\), we need to convert the given probability to its complement: $$P\left(zz_{0}\right)=1-0.9750=0.0250$$ Now, look up the z-score corresponding to the probability \(0.0250\) in a standard normal table (or use an online calculator). You will find that the \(z\)-score is approximately \(-1.96\). Thus, the \(z_{0}\) you are looking for is: $$z_{0} \approx -1.96$$
02

b. Finding \(z_{0}\) for \(P\left(z>z_{0}\right)=.3594\)

Again, we need to convert the given probability to its complement: $$P\left(zz_{0}\right)=1-0.3594=0.6406$$ Now, look up the z-score corresponding to the probability \(0.6406\) in a standard normal table (or use an online calculator). You will find that the \(z\)-score is approximately \(0.37\). Thus, the \(z_{0}\) you are looking for is: $$z_{0} \approx 0.37$$

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