Explanation Given info: Checbyshev’s inequality: For any random variable X having mean μ and variance σ 2 , P ( | X − μ | ≥ k σ ) ≤ 1 k 2 Where k is any positive number. Assume that the random variable X follows normal with mean Calculation: The formula to convert X values into z score is, z = x − μ σ The required probability is, P ( | X − μ | ≥ k σ ) = P ( X − μ ≥ k σ ) + P ( X − μ ≤ − k σ ) Then, P ( | X − μ | ≥ k σ ) = P ( X − μ ≥ k σ ) + P ( X − μ ≤ − k σ ) = P ( X − μ σ ≥ k ) + P ( X − μ σ ≤ − k ) = P ( Z ≥ k ) + P ( Z ≤ − k ) = 2 P ( Z ≥ k ) = 2 ( 1 − P ( Z ≤ k ) ) For k = 1 , P ( | X − μ | ≥ σ ) = 2 ( P ( Z ≥ 1 ) ) = 2 ( 1 − P ( Z ≤ 1 ) ) The value of P ( | X − μ | ≥ σ ) is obtained by finding the area to the left of –1 and right of 1. The shaded region represents of the area to the right of 1 and left of –1 of a random variable is shown below: Use Table A.2: Cumulative Normal Distribution to find the areas. Procedure: For z at 1.00, Locate 1.0 in the left column of the Table A.2. Obtain the value in the corresponding row below 0.00. That is, P ( Z ≤ 1 ) = 0.8413 Then, P ( | X − μ | ≥ σ ) = 2 ( 1 − 0.8413 ) = 2 ( 0.1587 ) = 0.3174 Thus, for k = 1 , the required probability is P ( | X − μ | ≥ σ ) = 0.3174 . The Chebyshev’s bound for k = 1 is, 1 k 2 = 1 . Here, the actual probability is less than the Chebyshev bound. For k = 2 , P ( | X − μ | ≥ 2 σ ) = 2 ( P ( Z ≥ 2 ) ) = 2 ( 1 − P ( Z ≤ 2 ) ) The value of P ( | X − μ | ≥ 2 σ ) is obtained by finding the area to the left of –2 and right of 2

4th Edition

ISBN: 9780073401331

Author: William Navidi Prof.

Publisher: McGraw-Hill Education

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Question

Chapter 4.5, Problem 26E

To determine

Find the value of P(|X−μ|≥kσ) for the values k=1,2,and 3.

Check whether the actual probabilities close to the Chebyshev bound or smaller than Chebyshev bound.

Expert Solution & Answer

Chapter 4 Solutions

Statistics for Engineers and Scientists

Ch. 4.1 - After scoring a touchdown, a football team may...Ch. 4.1 - A certain brand of dinnerware set comes in three...Ch. 4.1 - When a certain glaze is applied to a ceramic...Ch. 4.1 - Let X and Y be Bernoulli random variables. Let Z =...Ch. 4.1 - A penny and a nickel are tossed. Both are fair...Ch. 4.1 - Prob. 6E Ch. 4.1 - Prob. 7E Ch. 4.2 - Let X Bin(7, 0.3). Find a. P(X = 1) b. P(X = 2)...Ch. 4.2 - Let X Bin(9, 0.4). Find a. P(X 6) b. P(X 2) c....Ch. 4.2 - Prob. 3E

Ch. 4.2 - At a certain airport, 75% of the flights arrive on...Ch. 4.2 - Of all the registered automobiles in a certain...Ch. 4.2 - A fair die is rolled 8 times. a. What is the...Ch. 4.2 - Of all the weld failures in a certain assembly,...Ch. 4.2 - A general contracting firm experiences cost...Ch. 4.2 - Several million lottery tickets are sold, and 60%...Ch. 4.2 - A quality engineer takes a random sample of 100...Ch. 4.2 - In a random sample of 100 parts ordered from...Ch. 4.2 - Of the items manufactured by a certain process,...Ch. 4.2 - Of the bolts manufactured for a certain...Ch. 4.2 - Gears produced by a grinding process are...Ch. 4.2 - A commuter must pass through three traffic lights...Ch. 4.2 - A distributor receives a large shipment of...Ch. 4.2 - A k out of n system is one in which there is a...Ch. 4.2 - Refer to Exercise 17 for the definition of a k out...Ch. 4.2 - A certain large shipment comes with a guarantee...Ch. 4.2 - An insurance company offers a discount to...Ch. 4.2 - A message consists of a string of bits (0s and...Ch. 4.2 - Let X Bin(n, p), and let Y = n X. Show that Y ...Ch. 4.2 - Porcelain figurines are sold for 10 if flawless,...Ch. 4.2 - Prob. 24E Ch. 4.2 - (Requires material from Section 3.3.) Refer to...Ch. 4.2 - If p is a success probability, the quantity p/(1 ...Ch. 4.3 - Let X Poisson(4). Find a. P(X = 1) b. P(X = 0) c....Ch. 4.3 - The number of flaws in a given area of aluminum...Ch. 4.3 - In a certain city, the number of potholes on a...Ch. 4.3 - Geologists estimate the time since the most recent...Ch. 4.3 - Prob. 5E Ch. 4.3 - One out of every 5000 individuals in a population...Ch. 4.3 - The number of hits on a certain website follows a...Ch. 4.3 - The number of cars arriving at a given...Ch. 4.3 - Prob. 9E Ch. 4.3 - Prob. 10E Ch. 4.3 - A microbiologist wants to estimate the...Ch. 4.3 - Two-dimensional Poisson process. The number of...Ch. 4.3 - The number of defective components produced by a...Ch. 4.3 - The probability that a certain radioactive mass...Ch. 4.3 - The number of flaws in a certain type of lumber...Ch. 4.3 - Grandma is trying out a new recipe for raisin...Ch. 4.3 - Mom and Grandma are each baking chocolate chip...Ch. 4.3 - Prob. 18E Ch. 4.3 - Someone claims that a certain suspension contains...Ch. 4.3 - A physicist wants to estimate the rate of...Ch. 4.3 - Prob. 21E Ch. 4.4 - Twenty air-conditioning units have been brought in...Ch. 4.4 - There are 30 restaurants in a certain town. Assume...Ch. 4.4 - The probability that a computer running a certain...Ch. 4.4 - A traffic light at a certain intersection is green...Ch. 4.4 - Refer to Exercise 4. Let Y denote the number of...Ch. 4.4 - Refer to Exercise 4. What is the probability that...Ch. 4.4 - If X Geom(p), what is the most probable value of...Ch. 4.4 - A process that fills packages is stopped whenever...Ch. 4.4 - A system is tested for faults once per hour. If...Ch. 4.4 - Prob. 10E Ch. 4.4 - In a lot of 10 microcircuits, 3 are defective....Ch. 4.4 - A lot of parts contains 500 items, 100 of which...Ch. 4.4 - Ten items are to be sampled from a lot of 60. If...Ch. 4.4 - Prob. 14E Ch. 4.4 - Prob. 15E Ch. 4.4 - A thermocouple placed in a certain medium produces...Ch. 4.4 - Prob. 17E Ch. 4.4 - Prob. 18E Ch. 4.5 - Find the area under the normal curve a. To the...Ch. 4.5 - Find the area under the normal curve a. To the...Ch. 4.5 - Let Z N(0, 1). Find a constant c for which a. P(Z...Ch. 4.5 - If X N(2, 9), compute a. P(X 2) b. P(1 X 7) c....Ch. 4.5 - The lifetime of a battery in a certain application...Ch. 4.5 - The temperature recorded by a certain thermometer...Ch. 4.5 - Scores on a standardized test are approximately...Ch. 4.5 - Weights of female cats of a certain breed are...Ch. 4.5 - The lifetime of a lightbulb in a certain...Ch. 4.5 - In a certain university, math SAT scores for the...Ch. 4.5 - Penicillin is produced by the Penicillium fungus,...Ch. 4.5 - Specifications for an aircraft bolt require that...Ch. 4.5 - A cylindrical hole is drilled in a block, and a...Ch. 4.5 - Shafts manufactured for use in optical storage...Ch. 4.5 - The fill volume of cans filled by a certain...Ch. 4.5 - The amount of paint required to paint a surface...Ch. 4.5 - A fiber-spinning process currently produces a...Ch. 4.5 - The area covered by 1 L of a certain stain is...Ch. 4.5 - Prob. 19E Ch. 4.5 - The quality-assurance program for a certain...Ch. 4.5 - Two resistors, with resistances R1 and R2, are...Ch. 4.5 - The molarity of a solute in solution is defined to...Ch. 4.5 - A binary message m, where m is equal either to 0...Ch. 4.5 - Refer to Exercise 23. Assume that if m = 0, the...Ch. 4.5 - A company receives a large shipment of bolts. The...Ch. 4.5 - Prob. 26E Ch. 4.6 - The lifetime (in days) of a certain electronic...Ch. 4.6 - The article Assessment of Dermopharmacokinetic...Ch. 4.6 - The body mass index (BMI) of a person is defined...Ch. 4.6 - The article Stochastic Estimates of Exposure and...Ch. 4.6 - If a resistor with resistance R ohms carries a...Ch. 4.6 - Refer to Exercise 5. Suppose 10 circuits are...Ch. 4.6 - The article Withdrawal Strength of Threaded Nails...Ch. 4.6 - Choose the best answer, and explain. If X is a...Ch. 4.6 - Prob. 9E Ch. 4.6 - A manufacturer claims that the tensile strength of...Ch. 4.6 - Let X1,...., Xn be independent lognormal random...Ch. 4.7 - Let T Exp(0.45). Find a. T b. T2 c. P(T 3) d....Ch. 4.7 - The time between requests to a web server is...Ch. 4.7 - A catalyst researcher states that the diameters,...Ch. 4.7 - The distance between flaws on a long cable is...Ch. 4.7 - Refer to Exercise 3. Suppose that the diameters of...Ch. 4.7 - Someone claims that the waiting time, in minutes,...Ch. 4.7 - A certain type of component can be purchased new...Ch. 4.7 - A radioactive mass emits particles according to a...Ch. 4.7 - The number of traffic accidents at a certain...Ch. 4.7 - The distance between consecutive flaws on a roll...Ch. 4.7 - A light fixture contains five lightbulbs. The...Ch. 4.8 - Prob. 1E Ch. 4.8 - Resistors are labeled 100 . In fact, the actual...Ch. 4.8 - Let T (4,0.5). a.Find T. b.Find T. c.Find P(T ...Ch. 4.8 - Prob. 4E Ch. 4.8 - Prob. 5E Ch. 4.8 - The lifetime, in years, of a type of small...Ch. 4.8 - Let T Weibull (0.5, 3). a.Find T. b.Find T....Ch. 4.8 - Prob. 8E Ch. 4.8 - Prob. 9E Ch. 4.8 - The lifetime of a certain battery is modeled with...Ch. 4.8 - The lifetime of a cooling fan, in hours, that is...Ch. 4.8 - Someone suggests that the lifetime T (in days) of...Ch. 4.8 - A system consists of two components connected in...Ch. 4.8 - Prob. 14E Ch. 4.8 - Prob. 15E Ch. 4.8 - Prob. 16E Ch. 4.8 - Prob. 17E Ch. 4.9 - Choose the best answer to fill in the blank. If an...Ch. 4.9 - Choose the best answer to fill in the blank. The...Ch. 4.9 - Let X1 and X2 be independent, each with unknown...Ch. 4.9 - Let X1,, Xn be a simple random sample from a N(,2)...Ch. 4.9 - Let X Geom (p). Find the MLE of p.Ch. 4.9 - Let X1, Xn be a random sample from a population...Ch. 4.9 - Maximum likelihood estimates possess the property...Ch. 4.9 - Let X1,, Xn be a random sample from a N(, 1)...Ch. 4.9 - Let X1, Xn be a random sample from a N(0,2)...Ch. 4.9 - Let X1,, Xn be a random sample from a N(,2)...Ch. 4.10 - Prob. 1E Ch. 4.10 - As part of a quality-control study aimed at...Ch. 4.10 - Below are the durations (in minutes) of 40...Ch. 4.10 - Prob. 4E Ch. 4.10 - Construct a normal probability plot for the PM...Ch. 4.10 - Construct a normal probability plot for the logs...Ch. 4.10 - Can the plot in Exercise 6 be used to determine...Ch. 4.10 - In the article Assessment of...Ch. 4.11 - Bottles filled by a certain machine are supposed...Ch. 4.11 - A 500-page book contains 250 sheets of paper. The...Ch. 4.11 - A commuter encounters four traffic lights each day...Ch. 4.11 - Among all the income-tax forms filed in a certain...Ch. 4.11 - Bags checked for a certain airline flight have a...Ch. 4.11 - The amount of warpage in a type of wafer used in...Ch. 4.11 - The time spent by a customer at a checkout counter...Ch. 4.11 - Drums labeled 30 L are filled with a solution from...Ch. 4.11 - The temperature of a solution will be estimated by...Ch. 4.11 - Among the adults in a large city, 30% have a...Ch. 4.11 - In a process that manufactures bearings, 90% of...Ch. 4.11 - A machine produces 1000 steel O-rings per day....Ch. 4.11 - Radioactive mass A emits particles at a mean rate...Ch. 4.11 - The concentration of particles in a suspension is...Ch. 4.11 - The concentration of particles in a suspension is...Ch. 4.11 - A battery manufacturer claims that the lifetime of...Ch. 4.11 - A new process has been designed to make ceramic...Ch. 4.11 - The manufacture of a certain part requires two...Ch. 4.11 - Seventy percent of rivets from vendor A meet a...Ch. 4.11 - Radiocarbon dating: Carbon-14 is a radioactive...Ch. 4 - An airplane has 100 seats for passengers. Assume...Ch. 4 - The number of large cracks in a length of pavement...Ch. 4 - Pea plants contain two genes for seed color, each...Ch. 4 - A simple random sample X1,,Xn is drawn from a...Ch. 4 - The Environmental Protection Agency (EPA) has...Ch. 4 - In the article Occurrence and Distribution of...Ch. 4 - Medication used to treat a certain condition is...Ch. 4 - A certain type of plywood consists of five layers....Ch. 4 - Prob. 9SE Ch. 4 - Prob. 10SE Ch. 4 - In a certain process, the probability of producing...Ch. 4 - A process that polishes a mirrored surface leaves...Ch. 4 - Prob. 13SE Ch. 4 - Prob. 14SE Ch. 4 - Thicknesses of shims are normally distributed with...Ch. 4 - The lifetime of a microprocessor is exponentially...Ch. 4 - Prob. 17SE Ch. 4 - Prob. 18SE Ch. 4 - A cereal manufacturer claims that the gross weight...Ch. 4 - Prob. 20SE Ch. 4 - Prob. 21SE Ch. 4 - Prob. 22SE Ch. 4 - Prob. 23SE Ch. 4 - Prob. 24SE Ch. 4 - Let X Geom(p). Let s 0 be an integer. a.Show...Ch. 4 - Prob. 26SE Ch. 4 - Prob. 27SE Ch. 4 - Prob. 28SE Ch. 4 - Prob. 29SE Ch. 4 - Prob. 30SE

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Find the value of P ( | X − μ | ≥ k σ ) for the values k = 1 , 2 , and 3 . Check whether the actual probabilities close to the Chebyshev bound or smaller than Chebyshev bound.

Solution Summary: The author explains how to find the value of Pleft for the values k=1,2,and3. Check whether the actual probabilities are close to the Chebyshev bound or smaller

Find the value of P(|X−μ|≥kσ) for the values k=1,2,and 3.

Check whether the actual probabilities close to the Chebyshev bound or smaller than Chebyshev bound.

Chapter 4 Solutions

Find the value of P ( | X − μ | ≥ k σ ) for the values k = 1 , 2 , and 3 . Check whether the actual probabilities close to the Chebyshev bound or smaller than Chebyshev bound.

Solution Summary: The author explains how to find the value of Pleft for the values k=1,2,and3. Check whether the actual probabilities are close to the Chebyshev bound or smaller

Find the value of P(|X−μ|≥kσ) for the values k=1,2,and 3. Check whether the actual probabilities close to the Chebyshev bound or smaller than Chebyshev bound.

Chapter 4 Solutions

Find the value of P(|X−μ|≥kσ) for the values k=1,2,and 3.

Check whether the actual probabilities close to the Chebyshev bound or smaller than Chebyshev bound.