Question

Determine the value of \(z^{*}\) such that a. \(-z^{*}\) and \(z^{*}\) separate the middle \(95 \%\) of all \(z\) values from the most extreme \(5 \%\) b. \(-z^{*}\) and \(z^{*}\) separate the middle \(90 \%\) of all \(z\) values from the most extreme \(10 \%\) c. \(-z^{*}\) and \(z^{*}\) separate the middle \(98 \%\) of all \(z\) values from the most extreme \(2 \%\) d. \(-z^{*}\) and \(z^{*}\) separate the middle \(92 \%\) of all \(z\) values from the most extreme \(8 \%\)

Short answer

The values of \(z^{*}\) that separate the given percentages of \(z\) values are: a) \(\pm 1.96\), b) \(\pm 1.645\), c) \(\pm 2.33\), d) \(\pm 1.75\)

Step-by-step solution

Understand the Problem

In each part of the problem, a percentage of the total area under the standard normal curve is given. We need to find the z-scores \(z^{*}\) and \(-z^{*}\) that separate this area from the remaining area.

Locate the Area on the Curve

The area under the standard normal curve that lies between \(-z^{*}\) and \(z^{*}\) corresponds to the given percentage. Since the curve is symmetrical, this area is divided evenly between \(-z^{*}\) and \(z^{*}\). Thus, if we want to find the z-score that separates the middle 90% of the data, we look for the z-score that has 5% of the data to its right (because the 90% in the middle is split evenly between the two tails).

Use Standard Normal Table or Calculator

Now that we understand the distribution of the standard normal curve for the given percentages, we can calculate the critical z-scores \(z^{*}\) using a standard normal table or a calculator with a built-in standard normal distribution function.

Calculate z-scores

For part a (95%), the z-score is approximately ±1.96 (because 2.5% is to the left of -1.96 and 2.5% is to the right of 1.96). For part b (90%), the z-score is ±1.645. For part c, which represents 98% of the data, the z-score is ±2.33. For part d (92%), the z-score is approximately ±1.75.