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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

Let \(x\) denote the duration of a randomly selected pregnancy (the time elapsed between conception and birth). Accepted values for the mean value and standard deviation of \(x\) are 266 days and 16 days, respectively. Suppose that a normal distribution is an appropriate model for the probability distribution of \(x\). a. What is the probability that the duration of pregnancy is between 250 and 300 days? b. What is the probability that the duration of pregnancy is at most 240 days? c. What is the probability that the duration of pregnancy is within 16 days of the mean duration? d. A Dear Abby column dated January 20,1973 , contained a letter from a woman who stated that the duration of her pregnancy was exactly 310 days. (She wrote that the last visit with her husband, who was in the navy, occurred 310 days before the birth.) What is the probability that the duration of a pregnancy is at least 310 days? Does this probability make you a bit skeptical of the claim? e. Some insurance companies will pay the medical expenses associated with childbirth only if the insurance has been in effect for more than 9 months ( 275 days). This restriction is designed to ensure that the insurance company has to pay benefits only for those pregnancies for which conception occurred during coverage. Suppose that conception occurred 2 weeks after coverage began. What is the probability that the insurance company will refuse to pay benefits because of the 275 -day insurance requirement?

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.

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?

The time that it takes a randomly selected job applicant to perform a certain task has a distribution that can be approximated by a normal distribution with a mean value of \(120 \mathrm{sec}\) and a standard deviation of \(20 \mathrm{sec}\). The fastest \(10 \%\) are to be given advanced training. What task times qualify individuals for such training?

Consider babies bom in the "normal" range of 37-43 weeks gestational age. Extensive data support the assumption that for such babies born in the United States, birth weight is normally distributed with mean \(3432 \mathrm{~g}\) and standard deviation \(482 \mathrm{~g}\) ("Are Babies Normal," The \(-302\) ). (The investigad data from a particular s intervals, the hisbut after further investi1 that this was due ht in grams and others measuring to the nearest ounce and then converting to grams. A modified choice of class intervals that allowed for this measurement difference gave a histogram that was well described by a normal distribution.) a. What is the probability that the birth weight of a randomly selected baby of this type exceeds \(4000 \mathrm{~g}\) ? is between 3000 and \(4000 \mathrm{~g}\) ? b. What is the probability that the birth weight of a randomly selected baby of this type is either less than \(2000 \mathrm{~g}\) or greater than \(5000 \mathrm{~g}\) ? c. What is the probability that the birth weight of a randomly selected baby of this type exceeds \(7 \mathrm{lb}\) ? (Hint: \(1 \mathrm{lb}=453.59 \mathrm{~g} .\) ) d. How would you characterize the most extreme \(0.1 \%\) of all birth weights?
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