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Chapter 7: Problem 69

Determine the value \(z^{*}\) that a. Separates the largest \(3 \%\) of all \(z\) values from the others b. Separates the largest \(1 \%\) of all \(z\) values from the others c. Separates the smallest \(4 \%\) of all \(z\) values from the others d. Separates the smallest \(10 \%\) of all \(z\) values from the others

Short Answer

Expert verified
The z-scores that separate the specified percentages from the rest of z values are \(z^{*} = 1.88\) for 3%, \(z^{*} = 2.33\) for 1%, \(z^{*} = -1.75\) for 4%, and \(z^{*} = -1.28\) for 10%.

Step by step solution

01

Understand the Percentages

Identify the percentage from the problem. For example, for a. it is \(3\%\) which can be used as the area to the right under the standard normal distribution. In the case of b. this is \(1\%\) to the right, for c. \(4\%\) to the left and for d. \(10\%\) to the left of the distribution.
02

Use the Standard Normal Table

The standard normal table gives the area under the curve from z = 0 up to the z-score. For percentages/areas to the right of the distribution as in steps a. and b., one can subtract the given percentage from \(100\%\) and then look up the z score in the table. For example, in step a, \(100\% - 3\% = 97\%\) is looked up and the corresponding z-score is found to be \(z^{*} = 1.88\). The same method is used for part b. where \(z^{*} = 2.33\). In steps c. and d. where the percentages to the left are given, it's a straight forward look up in the standard normal table to find the z score. In the case of c, \(z^{*} = -1.75\) and for d, \(z^{*} = -1.28\) are corresponding z-scores.
03

Recap Findings

After using the standard normal table to find the corresponding z-scores that separate the specified percentages, make sure to recap and interpret them correctly. In the context of this problem, the z-scores found are the values that separate the largest/smallest specified percentage of all z values from the others.

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Most popular questions from this chapter

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