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Chapter 7: Problem 22

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

Expert verified
The z-values for (a) 95% middle is 卤1.96, (b) 90% middle is 卤1.645, (c) 98% middle is 卤2.33, and (d) 92% middle is 卤1.75.

Step by step solution

01

Understanding the concept

A z-table, also called the standard normal table, is a mathematical table that allows us to know the percentage of values below (to the left of) a z-score in a standard normal distribution. For this problem, each case is a two-tail problem where you have some percentage in the middle and the rest is equally distributed in both tails. In each case, you need to first identify the total tail percentage and divide by two as the Z-table works for the area to the left of the z-score.
02

Locate z-score for part (a)

For part (a), the middle 95% means you have 5% in the tails. Since this is a two-tailed test, you have 2.5% in each tail. In this case, look for closest to 0.9750 (0.5000 + 0.4750) in the Z-table which gives a z-score of 1.96.
03

Locate z-score for part (b)

For part (b), the middle 90% means you have 10% in the tails. Therefore 5% in each tail. Look for closest to 0.9500 (0.5000 + 0.4500) in the Z-table which gives a z-score of 1.645.
04

Locate z-score for part (c)

For part (c), the middle 98% means you have 2% in the tails. Therefore 1% in each tail. Look for closest to 0.9900 (0.5000 + 0.4900) in the Z-table which gives a z-score of 2.33.
05

Locate z-score for part (d)

For part (d), the middle 92% means you have 8% in the tails. Therefore, 4% in each tail. Look for closest to 0.9600 (0.5000 + 0.4600) in the Z-table which gives a z-score of 1.75.

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Most popular questions from this chapter

Consider the variable \(x=\) time required for a college student to complete a standardized exam. Suppose that for the population of students at a particular university, the distribution of \(x\) is well approximated by a normal curve with mean \(45 \mathrm{~min}\) and standard deviation \(5 \mathrm{~min}\). a. If \(50 \mathrm{~min}\) is allowed for the exam, what proportion of students at this university would be unable to finish in the allotted time? b. How much time should be allowed for the exam if we wanted \(90 \%\) of the students taking the test to be able to finish in the allotted time? c. How much time is required for the fastest \(25 \%\) of all students to complete the exam?

Determine the following standard normal ( \(z\) ) curve areas: a. The area under the \(z\) curve to the left of \(1.75\) b. The area under the \(z\) curve to the left of \(-0.68\) c. The area under the \(z\) curve to the right of \(1.20\) d. The area under the \(z\) curve to the right of \(-2.82\) e. The area under the \(z\) curve between \(-2.22\) and \(0.53\) f. The area under the \(z\) curve between \(-1\) and 1 g. The area under the \(z\) curve between \(-4\) and 4

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 \(\left.z<-z^{*}\right)=.20\)

Consider the population of all 1 -gal cans of dusty rose paint manufactured by a particular paint company. Suppose that a normal distribution with mean \(\mu=5 \mathrm{ml}\) and standard deviation \(\sigma=0.2 \mathrm{ml}\) is a reasonable model for the distribution of the variable \(x=\) amount of red dye in the paint mixture. Use the normal distribution model to calculate the following probabilities. a. \(P(x<5.0)\) b. \(P(x<5.4)\) c. \(P(x \leq 5.4)\) d. \(P(4.64.5)\) f. \(P(x>4.0)\)

Consider the following sample of 25 observations on the diameter \(x\) (in centimeters) of a disk used in a certain system. \(\begin{array}{lllllll}16.01 & 16.08 & 16.13 & 15.94 & 16.05 & 16.27 & 15.89 \\ 15.84 & 15.95 & 16.10 & 15.92 & 16.04 & 15.82 & 16.15 \\ 16.06 & 15.66 & 15.78 & 15.99 & 16.29 & 16.15 & 16.19 \\ 16.22 & 16.07 & 16.13 & 16.11 & & & \end{array}\) The 13 largest normal scores for a sample of size 25 are \(1.965,1.524,1.263,1.067,0.905,0.764,0.637,0.519\) \(0.409,0.303,0.200,0.100\), and \(0 .\) The 12 smallest scores result from placing a negative sign in front of each of the given nonzero scores. Construct a normal probability plot. Does it appear plausible that disk diameter is normally distributed? Explain.
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