Statistics for Engineers and Scientists
4th Edition
ISBN: 9780073401331
Author: William Navidi Prof.
Publisher: McGraw-Hill Education
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Textbook Question
Chapter 2, Problem 4SE
Two production lines are used to pack sugar into 5 kg bags. Line 1 produces twice as many bags as does line 2. One percent of the bags from line 1 are defective in that they fail to meet a purity specification, while 3% of the bags from line 2 are defective. A bag is randomly chosen for inspection.
- a. What is the
probability that it came from line 1? - b. What is the probability that it is defective?
- c. If the bag is defective, what is the probability that it came from line 1?
- d. If the bag is not defective, what is the probability that it came from line 1?
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Throughout, A, B, (An, n≥ 1), and (Bn, n≥ 1) are subsets of 2.
1. Show that
AAB (ANB) U (BA) = (AUB) (AB),
Α' Δ Β = Α Δ Β,
{A₁ U A2} A {B₁ U B2) C (A1 A B₁}U{A2 A B2).
16. Show that, if X and Y are independent random variables, such that E|X|< ∞,
and B is an arbitrary Borel set, then
EXI{Y B} = EX P(YE B).
Proposition 1.1 Suppose that X1, X2,... are random variables. The following
quantities are random variables:
(a) max{X1, X2) and min(X1, X2);
(b) sup, Xn and inf, Xn;
(c) lim sup∞ X
and lim inf∞ Xn-
(d) If Xn(w) converges for (almost) every w as n→ ∞, then lim-
random variable.
→ Xn is a
Chapter 2 Solutions
Statistics for Engineers and Scientists
Ch. 2.1 - The probability that a bearing fails during the...Ch. 2.1 - A die (six faces) has the number 1 painted on...Ch. 2.1 - A section of an exam contains four True-False...Ch. 2.1 - Three times each day, a quality engineer samples a...Ch. 2.1 - Four candidates are to be interviewed for a job....Ch. 2.1 - Refer to Exercise 5. Two candidates are randomly...Ch. 2.1 - In a survey of households with television sets,...Ch. 2.1 - An automobile insurance company divides customers...Ch. 2.1 - Among the cast aluminum parts manufactured on a...Ch. 2.1 - The article High Cumulative Risk of Lung Cancer...
Ch. 2.1 - A quality-control engineer samples 100 items...Ch. 2.1 - Let V be the event that a computer contains a...Ch. 2.1 - Let S be the event that a randomly selected...Ch. 2.1 - Six hundred paving stones were examined for...Ch. 2.1 - All the fourth-graders in a certain elementary...Ch. 2.1 - A system contains two components, A and B. The...Ch. 2.1 - A system contains two components, A and B. The...Ch. 2.1 - Human blood may contain either or both of two...Ch. 2.1 - True or false: If A and B are mutually exclusive,...Ch. 2.1 - A flywheel is attached to a crankshaft by 12...Ch. 2.2 - DNA molecules consist of chemically linked...Ch. 2.2 - A metallurgist is designing an experiment to...Ch. 2.2 - The article Improved Bioequivalence Assessment of...Ch. 2.2 - A group of 18 people have gotten together to play...Ch. 2.2 - In horse racing, one can make a trifecta bet by...Ch. 2.2 - A college math department consisting of 10 faculty...Ch. 2.2 - A test consists of 15 questions. Ten are...Ch. 2.2 - In a certain state, license plates consist of...Ch. 2.2 - A computer password consists of eight characters....Ch. 2.2 - A company has hired 15 new employees, and must...Ch. 2.2 - Let A and B be events with P(A) = 0.8 and P(A B)...Ch. 2.2 - A drawer contains 6 red socks, 4 green socks, and...Ch. 2.3 - Let A and B be events with P(A) = 0.8 and P(A B)...Ch. 2.3 - Let A and B be events with P(A) = 0.5 and P(A Bc)...Ch. 2.3 - A box contains 15 resistors. Ten of them are...Ch. 2.3 - Prob. 4ECh. 2.3 - On graduation day at a large university, one...Ch. 2.3 - The article Integrating Risk Assessment and Life...Ch. 2.3 - Suppose that start-up companies in the area of...Ch. 2.3 - A drag racer has two parachutes, a main and a...Ch. 2.3 - Of people in a certain city who bought a new...Ch. 2.3 - Of all failures of a certain type of computer hard...Ch. 2.3 - In the process of producing engine valves, the...Ch. 2.3 - Sarah and Thomas are going bowling. The...Ch. 2.3 - A particular automatic sprinkler system has two...Ch. 2.3 - Laura and Philip each fire one shot at a target....Ch. 2.3 - A population of 600 semiconductor wafers contains...Ch. 2.3 - Refer to Exercise 15. Let E1 be the event that the...Ch. 2.3 - A geneticist is studying two genes. Each gene can...Ch. 2.3 - A car dealer sold 750 automobiles last year. The...Ch. 2.3 - The following table presents the 100 senators of...Ch. 2.3 - An automobile insurance company divides customers...Ch. 2.3 - Nuclear power plants have redundant components in...Ch. 2.3 - Prob. 22ECh. 2.3 - A lot of 10 components contains 3 that are...Ch. 2.3 - A lot of 1000 components contains 300 that are...Ch. 2.3 - In a lot of n components, 30% are defective. Two...Ch. 2.3 - Prob. 26ECh. 2.3 - Each day, a weather forecaster predicts whether or...Ch. 2.3 - Items are inspected for flaws by two quality...Ch. 2.3 - Refer to Exercise 28. Assume that both inspectors...Ch. 2.3 - Refer to Example 2.26. Assume that the proportion...Ch. 2.3 - Sickle-cell anemia is an inherited disease in...Ch. 2.3 - A quality-control program at a plastic bottle...Ch. 2.3 - Refer to Example 2.26. a. If a man tests negative,...Ch. 2.3 - A system consists of four components connected as...Ch. 2.3 - A system consists of four components, connected as...Ch. 2.3 - A system contains two components, A and B,...Ch. 2.3 - A system contains two components, C and D,...Ch. 2.3 - If A and B are independent events, prove that the...Ch. 2.4 - Determine whether each of the following random...Ch. 2.4 - Computer chips often contain surface...Ch. 2.4 - A chemical supply company ships a certain solvent...Ch. 2.4 - Let X represent the number of tires with low air...Ch. 2.4 - A survey of cars on a certain stretch of highway...Ch. 2.4 - The element titanium has five stable occurring...Ch. 2.4 - A computer sends a packet of information along a...Ch. 2.4 - After manufacture, computer disks are tested for...Ch. 2.4 - On 100 different days, a traffic engineer counts...Ch. 2.4 - Microprocessing chips are randomly sampled one by...Ch. 2.4 - Refer to Exercise 10. Let Y be the number of chips...Ch. 2.4 - Three components are randomly sampled, one at a...Ch. 2.4 - Resistors labeled 100 have true resistances that...Ch. 2.4 - Elongation (in percent) of steel plates treated...Ch. 2.4 - The lifetime in months of a transistor in a...Ch. 2.4 - A process that manufactures piston rings produces...Ch. 2.4 - Refer to Exercise 16. A competing process produces...Ch. 2.4 - The lifetime, in years, of a certain type of pump...Ch. 2.4 - The level of impurity (in percent) in the product...Ch. 2.4 - The main bearing clearance (in mm) in a certain...Ch. 2.4 - The error in the length of a part (absolute value...Ch. 2.4 - Prob. 22ECh. 2.4 - The thickness of a washer (in mm) is a random...Ch. 2.4 - Particles are a major component of air pollution...Ch. 2.4 - The repair time (in hours) for a certain machine...Ch. 2.4 - The diameter of a rivet (in mm) is a random...Ch. 2.5 - Prob. 1ECh. 2.5 - The bottom of a cylindrical container has an area...Ch. 2.5 - The lifetime of a certain transistor in a certain...Ch. 2.5 - Two batteries, with voltages V1 and V2, are...Ch. 2.5 - A laminated item is composed of five layers. The...Ch. 2.5 - Two independent measurements are made of the...Ch. 2.5 - The molarity of a solute in solution is defined to...Ch. 2.5 - A machine that fills bottles with a beverage has a...Ch. 2.5 - The four sides of a picture frame consist of two...Ch. 2.5 - A gas station earns 2.60 in revenue for each...Ch. 2.5 - A certain commercial jet plane uses a mean of 0.15...Ch. 2.5 - Prob. 12ECh. 2.5 - In the article An Investigation of the...Ch. 2.5 - Prob. 14ECh. 2.5 - Prob. 15ECh. 2.5 - The thickness X of a wooden shim (in mm) has...Ch. 2.5 - Prob. 17ECh. 2.5 - Prob. 18ECh. 2.6 - In a certain community, levels of air pollution...Ch. 2.6 - Prob. 2ECh. 2.6 - Refer to Exercise 1. a. Find the conditional...Ch. 2.6 - Prob. 4ECh. 2.6 - Refer to Exercise 4. The total number of...Ch. 2.6 - Refer to Exercise 4. a. Find the conditional...Ch. 2.6 - Refer to Exercise 4. Assume that the cost of...Ch. 2.6 - The number of customers in line at a supermarket...Ch. 2.6 - Prob. 9ECh. 2.6 - Refer to Exercise 9. a. Find the mean of the total...Ch. 2.6 - Refer to Exercise 9. a. Find the conditional...Ch. 2.6 - Prob. 12ECh. 2.6 - Refer to Exercise 12. Let Z = X + Y represent the...Ch. 2.6 - Refer to Exercise 12. Assume that the cost of an...Ch. 2.6 - Automobile engines and transmissions are produced...Ch. 2.6 - Prob. 16ECh. 2.6 - Prob. 17ECh. 2.6 - A production facility contains two machines that...Ch. 2.6 - Prob. 19ECh. 2.6 - Prob. 20ECh. 2.6 - The lifetime of a certain component, in years, has...Ch. 2.6 - Prob. 22ECh. 2.6 - An investor has 100 to invest, and two investments...Ch. 2.6 - Prob. 24ECh. 2.6 - Let R denote the resistance of a resistor that is...Ch. 2.6 - Prob. 26ECh. 2.6 - Prob. 27ECh. 2.6 - Prob. 28ECh. 2.6 - Let X and Y be jointly distributed random...Ch. 2.6 - Prob. 30ECh. 2.6 - Prob. 31ECh. 2.6 - Prob. 32ECh. 2.6 - Prob. 33ECh. 2 - A system consists of four components connected as...Ch. 2 - A fair coin is tossed until a head appears. What...Ch. 2 - Silicon wafers are used in the manufacture of...Ch. 2 - Two production lines are used to pack sugar into 5...Ch. 2 - Prob. 5SECh. 2 - In a certain type of automobile engine, the...Ch. 2 - An electronic message consists of a string of bits...Ch. 2 - The reading given by a thermometer calibrated in...Ch. 2 - Two dice are rolled. Given that two different...Ch. 2 - In a lot of 10 components, 2 are sampled at random...Ch. 2 - Prob. 11SECh. 2 - Prob. 12SECh. 2 - A snowboard manufacturer has three plants, one in...Ch. 2 - The article Traps in Mineral ValuationsProceed...Ch. 2 - Six new graduates are hired by an engineering...Ch. 2 - Prob. 16SECh. 2 - Let X and Y be independent random variables with x...Ch. 2 - Prob. 18SECh. 2 - Prob. 19SECh. 2 - Prob. 20SECh. 2 - Prob. 21SECh. 2 - A certain plant runs three shifts per day. Of all...Ch. 2 - Prob. 23SECh. 2 - Prob. 24SECh. 2 - Prob. 25SECh. 2 - A stock solution of hydrochloric acid (HC1)...Ch. 2 - Prob. 27SECh. 2 - Prob. 28SECh. 2 - A penny and a nickel are tossed. The penny has...Ch. 2 - Prob. 30SECh. 2 - Prob. 31SECh. 2 - Prob. 32SECh. 2 - Prob. 33SECh. 2 - Prob. 34SECh. 2 - Blood is taken from each of n individuals to be...
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- Exercise 4.2 Prove that, if A and B are independent, then so are A and B, Ac and B, and A and B.arrow_forward8. Show that, if {Xn, n ≥ 1) are independent random variables, then sup X A) < ∞ for some A.arrow_forward8- 6. Show that, for any random variable, X, and a > 0, 8 心 P(xarrow_forward15. This problem extends Problem 20.6. Let X, Y be random variables with finite mean. Show that 00 (P(X ≤ x ≤ Y) - P(X ≤ x ≤ X))dx = E Y — E X.arrow_forward(b) Define a simple random variable. Provide an example.arrow_forward17. (a) Define the distribution of a random variable X. (b) Define the distribution function of a random variable X. (c) State the properties of a distribution function. (d) Explain the difference between the distribution and the distribution function of X.arrow_forward16. (a) Show that IA(w) is a random variable if and only if A E Farrow_forward15. Let 2 {1, 2,..., 6} and Fo({1, 2, 3, 4), (3, 4, 5, 6}). (a) Is the function X (w) = 21(3, 4) (w)+711.2,5,6) (w) a random variable? Explain. (b) Provide a function from 2 to R that is not a random variable with respect to (N, F). (c) Write the distribution of X. (d) Write and plot the distribution function of X.arrow_forward20. Define the o-field R2. Explain its relation to the o-field R.arrow_forward7. Show that An → A as n→∞ I{An} - → I{A} as n→ ∞.arrow_forward7. (a) Show that if A,, is an increasing sequence of measurable sets with limit A = Un An, then P(A) is an increasing sequence converging to P(A). (b) Repeat the same for a decreasing sequence. (c) Show that the following inequalities hold: P (lim inf An) lim inf P(A) ≤ lim sup P(A) ≤ P(lim sup A). (d) Using the above inequalities, show that if A, A, then P(A) + P(A).arrow_forward19. (a) Define the joint distribution and joint distribution function of a bivariate ran- dom variable. (b) Define its marginal distributions and marginal distribution functions. (c) Explain how to compute the marginal distribution functions from the joint distribution function.arrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
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