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Principles Of Statistics For Engineers And Scientists 2nd International Edition William Navidi - Solutions

As part of a quality-control study aimed at improving a production line, the weights (in ounces) of 50 bars of soap are measured. The results are as follows, sorted from smallest to largest.11.6 12.6 12.7 12.8 13.1 13.3 13.6 13.7 13.8 14.1 14.3 14.3 14.6 14.8 15.1 15.2 15.6 15.6 15.7 15.8 15.8 15.9
Each of three samples has been plotted on a normal probability plot. For each, say whether the sample appears to have come from an approximately normal population. 0.999 0.99 0.95 0.999 0.99 0.95 0.9 0.75 0.5 0.25 0.1 0.9 0.75 0.5 0.25 0.1 0.05 0.05 0.01 0.001 (a) 0.01 0.001 0.999 0.99 0.95 0.9
Someone suggests that the lifetime T (in days) of a certain component can be modeled with the Weibull distribution with parameters ???? = 3 and ???? = 0.01.a. If this model is correct, what is P(T ≤ 1)?b. Based on the answer to part (a), if the model is correct, would one day be an unusually
The lifetime of a cooling fan, in hours, that is used in a computer system has the Weibull distribution with???? = 1.5 and ???? = 0.0001.a. What is the probability that a fan lasts more than 10,000 hours?b. What is the probability that a fan lasts less than 5000 hours?c. What is the probability
The lifetime of a certain battery is modeled with the Weibull distribution with ???? = 2 and ???? = 0.1.a. What proportion of batteries will last longer than 10 hours?b. What proportion of batteries will last less than 5 hours?c. What proportion of batteries will last longer than 20 hours?d. The
What is the hazard at t = 2000 hours?
In the article “Parameter Estimation with Only One Complete Failure Observation” (W. Pang, P. Leung, et al., International Journal of Reliability, Quality, and Safety Engineering, 2001:109–122), the lifetime, in hours, of a certain type of bearing is modeled with the Weibull distribution with
If T is a continuous random variable that is always positive (such as a waiting time), with probability density function f(t) and cumulative distribution function F(t), then the hazard function is defined to be the function h(t) = f(t)1 − F(t)The hazard function is the rate of failure per unit
Let T ∼ Γ(6, 2).a. Find ????T .b. Find ????T .
Resistors are labeled 100 Ω. In fact, the actual resistances are uniformly distributed on the interval(95, 103).a. Find the mean resistance.b. Find the standard deviation of the resistances.
The distance advanced in a day by a tunnel boring machine, in meters, is uniformly distributed on the interval (30, 50).a. Find the mean distance.b. Find the standard deviation of the distances.c. Find the probability that the distance is between 35 and 42 meters.d. Suppose the distances advanced
Refer to Exercise 2.a. Find the probability that there will be exactly 5 requests in a 2-second time interval.b. Find the probability that there will be more than 1 request in a 1.5-second time interval.c. Find the probability that there will be no requests in a 1-second time interval.d. Find the
Refer to Exercise 4.a. Find the probability that there are exactly 5 flaws in a 50 m length of cable.b. Find the probability that there are more than two flaws in a 20 m length of cable.c. Find the probability that there are no flaws in a 15 m length of cable.d. Find the probability that the
A radioactive mass emits particles according to a Poisson process at a mean rate of 2 per second. Let T be the waiting time, in seconds, between emissions.a. What is the mean waiting time?b. What is the median waiting time?c. Find P(T > 2).d. Find P(T < 0.1).e. Find P(0.3 < T < 1.5).f. If 3 seconds
The distance between flaws on a long cable is exponentially distributed with mean 12 m.a. What is the value of the parameter ?????b. Find the median distance.c. Find the standard deviation of the distances.d. Find the 65th percentile of the distances.
A catalyst researcher states that the diameters, in microns, of the pores in a new product she has made follow the exponential distribution with parameter???? = 0.25.a. What is the mean pore diameter?b. What is the standard deviation of the pore diameters?c. What proportion of the pores are less
The time between requests to a web server is exponentially distributed with mean 0.5 seconds.a. What is the value of the parameter ?????b. What is the median time between requests?c. What is the standard deviation?d. What is the 80th percentile?
Let T ∼ Exp(0.5). Finda. ????Tb. ????2 Tc. P(T > 5)d. The median of T
A manufacturer claims that the tensile strength of a certain composite (in MPa) has the lognormal distribution with ???? = 5 and ???? = 0.5. Let X be the strength of a randomly sampled specimen of this composite.a. If the claim is true, what is P(X < 20)?b. Based on the answer to part (a), if the
The prices of stocks or other financial instruments are often modeled with a lognormal distribution. An investor is considering purchasing stock in one of two companies, A or B. The price of a share of stock today is $1 for both companies. For company A, the value of the stock one year from now is
The article “Stochastic Estimates of Exposure and Cancer Risk from Carbon Tetrachloride Released to the Air from the Rocky Flats Plant” (A. Rood, P. McGavran, et al., Risk Analysis, 2001:675–695) models the increase in the risk of cancer due to exposure to carbon tetrachloride as lognormal
The body mass index (BMI) of a person is defined to be the person’s body mass divided by the square of the person’s height. The article “Influences of Parameter Uncertainties within the ICRP 66 Respiratory Tract Model: Particle Deposition” (W. Bolch, E. Farfan, et al., Health Physics,
The article “Assessment of Dermopharmacokinetic Approach in the Bioequivalence Determination of Topical Tretinoin Gel Products” (L. Pershing, J. Nelson, et al., Journal of the American Academy of Dermatology, 2003:740–751) reports that the amount of a certain antifungal ointment that is
The molarity of a solute in solution is defined to be the number of moles of solute per liter of solution(1 mole = 6.02 × 1023 molecules). If X is the molarity of a solution of sodium chloride (NaCl), and Y is the molarity of a solution of sodium carbonate (Na2CO3), the molarity of sodium ion
The quality assurance program for a certain adhesive formulation process involves measuring how well the adhesive sticks a piece of plastic to a glass surface. When the process is functioning correctly, the adhesive strength X is normally distributed with a mean of 200 N and a standard deviation of
Penicillin is produced by the Penicillium fungus, which is grown in a broth whose sugar content must be carefully controlled. The optimum sugar concentration is 4.9 mg/mL. If the concentration exceeds 6.0 mg/mL, the fungus dies and the process must be shut down for the day.a. If sugar concentration
The area covered by 1 L of a certain stain is normally distributed with mean 10 m2 and standard deviation 0.2 m2.a. What is the probability that 1 L of stain will be enough to cover 10.3 m2?b. What is the probability that 2 L of stain will be enough to cover 19.9 m2?
The strength of an aluminum alloy is normally distributed with mean 10 gigapascals (GPa) and standard deviation 1.4 GPa.a. What is the probability that a specimen of this alloy will have a strength greater than 12 GPa?b. Find the first quartile of the strengths of this alloy.c. Find the 95th
At a certain university, math SAT scores for the entering freshman class averaged 650 and had a standard deviation of 100. The maximum possible score is 800.Is it possible that the scores of these freshmen are normally distributed? Explain.
In a recent study, the Centers for Disease Control reported that diastolic blood pressures (in mmHg) of adult women in the United States are approximately normally distributed with mean 80.5 and standard deviation 9.9.a. What proportion of women have blood pressures lower than 70?b. What is the
Depths of pits on a corroded steel surface are normally distributed with mean 822 ????m and standard deviation 29 ????m.a. Find the 10th percentile of pit depths.b. A certain pit is 780 ????m deep. What percentile is it on?c. What proportion of pits have depths between 800 and 830 ????m?
Scores on a standardized test are approximately normally distributed with a mean of 460 and a standard deviation of 80.a. What proportion of the scores are above 550?b. What is the 35th percentile of the scores?c. If someone’s score is 600, what percentile is she on?d. What proportion of the
Find the area under the normal curvea. To the left of z = 0.56.b. Between z = −2.93 and z = −2.06.c. Between z = −1.08 and z = 0.70.d. Outside z = 0.96 to z = 1.62.
Someone claims that a certain suspension contains at least seven particles per mL. You sample 1 mL of solution. Let X be the number of particles in the sample.a. If the mean number of particles is exactly seven per mL (so that the claim is true, but just barely), what is P(X ≤ 1)?b. Based on the
You have received a radioactive mass that is claimed to have a mean decay rate of at least 1 particle per second. If the mean decay rate is less than 1 per second, you may return the product for a refund. Let X be the number of decay events counted in 10 seconds.a. If the mean decay rate is exactly
The number of flaws in a certain type of lumber follows a Poisson distribution with a rate of 0.45 per linear meter.a. What is the probability that a board 3 meters in length has no flaws?b. How long must a board be so that the probability it has no flaw is 0.5?
The number of cars arriving at a given intersection follows a Poisson distribution with a mean rate of 4 per second.a. What is the probability that 3 cars arrive in a given second?b. What is the probability that 8 cars arrive in three seconds?c. What is the probability that more than 3 cars arrive
Is it possible to determine which random variable has the larger variance?Choose one of the following answers:i. Yes, X has the larger variance.ii. Yes, Y has the larger variance.iii. No, we need to know the number of trials, n, for X.iv. No, we need to know the success probability, p, for X.v. No,
A random variable X has a binomial distribution, and a random variable Y has a Poisson distribution. Both X and Y have means equal to
Geologists estimate the time since the most recent cooling of a mineral by counting the number of uranium fission tracks on the surface of the mineral. A certain mineral specimen is of such an age that there should be an average of 6 tracks per cm2 of surface area. Assume the number of tracks in an
The number of hits on a certain website follows a Poisson distribution with a mean rate of 4 per minute.a. What is the probability that five messages are received in a given minute?b. What is the probability that 9 messages are received in 1.5 minutes?c. What is the probability that fewer than
The number of flaws in a given area of aluminum foil follows a Poisson distribution with a mean of 3 per m2.Let X represent the number of flaws in a 1 m2 sample of foil.a. P(X = 5)b. P(X = 0)c. P(X < 2)d. P(X > 1)e. ????Xf. ????X
The number of large packages delivered by a courier service follows a Poisson distribution with a rate of 5 per day. Let X be the number of large packages delivered on a given day. Finda. P(X = 6)b. P(X ≤ 2)c. P(5 < X < 8)d. ????Xe. ????X
The number of pits in a corroded steel coupon follows a Poisson distribution with a mean of 6 pits per cm2.Let X represent the number of pits in a 1 cm2 area.Finda. P(X = 8)b. P(X = 2)c. P(X < 3)d. P(X > 1)e. ????Xf. ????X
One design for a system requires the installation of two identical components. The system will work if at least one of the components works. An alternative design requires four of these components, and the system will work if at least two of the four components work. If the probability that a
A message consists of a string of bits (0s and 1s).Due to noise in the communications channel, each bit has probability 0.3 of being reversed (i.e., a 1 will be changed to a 0 or a 0 to a 1). To improve the accuracy of the communication, each bit is sent five times, so, for example, 0 is sent as
An insurance company offers a discount to homeowners who install smoke detectors in their homes. A company representative claims that 80% or more of policyholders have smoke detectors. You draw a random sample of eight policyholders. Let X be the number of policyholders in the sample who have smoke
A certain large shipment comes with a guarantee that it contains no more than 15% defective items. If the proportion of defective items in the shipment is greater than 15%, the shipment may be returned. You draw a random sample of 10 items. Let X be the number of defective items in the sample.a. If
Refer to Exercise 15 for the definition of a k out of n system. For a certain 4 out of 6 system, assume that on a rainy day each component has probability 0.7 of functioning and that on a nonrainy day each component has probability 0.9 of functioning.a. What is the probability that the system
A k out of n system is one in which there is a group of n components, and the system will function if at least k of the components function. Assume the components function independently of one another.a. In a 3 out of 5 system, each component has probability 0.9 of functioning. What is the
A distributor receives a large shipment of components. The distributor would like to accept the shipment if 10% or fewer of the components are defective and to return it if more than 10% of the components are defective. She decides to sample 10 components and to return the shipment if more than 1
A commuter must pass through three traffic lights on her way to work. For each light, the probability that it is green when she arrives is 0.6. The lights are independent.a. What is the probability that all three lights are green?b. The commuter goes to work five days per week.Let X be the number
Gears produced by a grinding process are categorized as either conforming (suitable for their intended purpose), downgraded (unsuitable for the intended purpose but usable for another purpose), or scrap (not usable). Suppose that 80% of the gears produced are conforming, 15% are downgraded, and 5%
Of the bolts manufactured for a certain application, 90% meet the length specification and can be used immediately, 6% are too long and can be used after being cut, and 4% are too short and must be scrapped.a. Find the probability that a randomly selected bolt can be used (either immediately or
A general contracting firm experiences cost overruns on 20% of its contracts. In a company audit, 20 contracts are sampled at random.a. What is the probability that exactly four of them experience cost overruns?b. What is the probability that fewer than three of them experience cost overruns?c.
A fair die is rolled eight times.a. What is the probability that the die comes up 6 exactly twice?b. What is the probability that the die comes up an odd number exactly five times?c. Find the mean number of times a 6 comes up.d. Find the mean number of times an odd number comes up.e. Find the
A fair coin is tossed eight times.a. What is the probability of obtaining exactly five heads?b. Find the mean number of heads obtained.c. Find the variance of the number of heads obtained.d. Find the standard deviation of the number of heads obtained.
A large industrial firm allows a discount on any invoice that is paid within 30 days. Of all invoices, 10%receive the discount. In a company audit, 12 invoices are sampled at random.a. What is the probability that exactly four of them receive the discount?b. What is the probability that fewer than
Of all the weld failures in a certain assembly, 85% of them occur in the weld metal itself, and the remaining 15% occur in the base metal. A sample of 20 weld failures is examined.a. What is the probability that exactly five of them are base metal failures?b. What is the probability that fewer than
The article “Uncertainty and Climate Change”(G. Heal and B. Kristrom, ¨ Environmental and Resource Economics, 2002:3–39) considers three scenarios, labeled A, B, and C, for the impact of global warming on income. For each scenario, a probability mass function for the loss of income is
The oxygen equivalence number of a weld is a number that can be used to predict properties such as hardness, strength, and ductility. The article “Advances in Oxygen Equivalence Equations for Predicting the Properties of Titanium Welds” (D. Harwig, W.Ittiwattana, and H. Castner, The Welding
In the article “An Investigation of the Ca–CO3–CaF2–K2SiO3–SiO2–Fe Flux System Using the Submerged Arc Welding Process on HSLA-100 and AISI-1018 Steels” (G. Fredrickson, M.S. thesis, Colorado School of Mines, 1992), the carbon equivalent P of a weld metal is defined to be a linear
In a lot of 10 components, 2 are sampled at random for inspection. Assume that in fact exactly 2 of the 10 components in the lot are defective. Let X be the number of sampled components that are defective.a. Find P(X = 0).b. Find P(X = 1).c. Find P(X = 2).d. Find the probability mass function of
Refer to Exercise 12. A competing process produces rings whose diameters (in centimeters) vary according to the probability density function f(x) ={15[1 − 25(x − 10.05)2]∕4 9.85 < x < 10.25 0 otherwise Specifications call for the diameter to be 10.0 ±0.1 cm. Which process is better, this one
A process that manufactures piston rings produces rings whose diameters (in centimeters) vary according to the probability density function f(x) ={3[1 − 16(x − 10)2] 9.75
The repair time (in hours) for a certain machine is a random variable with probability density function f(x) ={xe−x x > 0 0 x ≤ 0a. What is the probability that the repair time is less than 2 hours?b. What is the probability that the repair time is between 1.5 and 3 hours?c. Find the mean
The pressure P, temperature T, and volume V of one mole of an ideal gas are related by the equation T = 0.1203PV, when P is measured in kilopascals, T is measured in kelvins, and V is measured in liters.a. Assume that P is measured to be 242.52 kPa with a standard deviation of 0.03 kPa and V is
The period T of a simple pendulum is given by T = 2????√L∕g, where L is the length of the pendulum and g is the acceleration due to gravity. Assume g = 9.80 m/s2 exactly, and L is measured to be 0.742 m with a standard deviation of 0.005 m.Estimate T, and find the standard deviation of the
The article “Traps in Mineral Valuations—Proceed With Care” (W. Lonegan, Journal of the Australasian Institute of Mining and Metallurgy, 2001:18–22)models the value (in millions of dollars) of a mineral deposit yet to be mined as a random variable X with probability mass function p(x) given
Let X and Y be independent random variables with????X =2, ????X =1, ????Y =2, and ????Y =3. Find the means and variances of the following quantities.a. 3Xb. X + Yc. X − Yd. 2X + 6Y
Let A and B be events with P(A) = 0.4 and P(A ∩ B) = 0.2. For what value of P(B) will A and B be independent?
Four bricks are sampled from a large load, and the crushing strength of each is measured to determine whether it meets a specification. If any of the four fail to meet the specification, the load will be returned. If in fact 10% of the bricks in the load fail to meet the specification, what is the
The lifetime, in years, of a certain type of fuel cell is a random variable with probability density function f(x) =⎧⎪⎨⎪⎩81(x + 3)4 x > 0 0 x ≤ 0a. What is the probability that a fuel cell lasts more than 3 years?b. What is the probability that a fuel cell lasts between 1 and 3 years?c.
Nuclear power plants have redundant components in important systems to reduce the chance of catastrophic failure. Assume that a plant has two gauges to measure the level of coolant in the reactor core and that each gauge has probability 0.01 of failing. Assume that one potential cause of gauge
In a certain type of automobile engine, the cylinder head is fastened to the block by 10 bolts, each of which should be torqued to 60 N⋅m. Assume that the torques of the bolts are independent. If each bolt is torqued correctly with probability 0.99, what is the probability that all the bolts on a
Among the cast aluminum parts manufactured on a certain day, 80% were flawless, 15% had only minor flaws, and 5% had major flaws. Find the probability that a randomly chosen parta. has a flaw (major or minor).b. has no major flaw.
One way to measure the water content of a soil is to weigh the soil both before and after drying it in an oven. The water content is W = (M1 − M2)∕M1, where M1 is the mass before drying and M2 is the mass after drying. Assume that M1 = 1.32 ± 0.01 kg and M2 = 1.04 ± 0.01 kg.a. Estimate W, and
The acceleration g due to gravity is estimated by dropping an object and measuring the time it takes to travel a certain distance. Assume the distance s is known to be exactly 5 m, and the time is measured to be t = 1.01 ± 0.02 s. Estimate g, and find the standard deviation of the estimate. (Note
The Needleman–Wunsch method for aligning DNA sequences assigns 1 point whenever a mismatch occurs, and 3 points whenever a gap (insertion or deletion) appears in a sequence. Assume that under certain conditions, the number of mismatches has mean 5 and standard deviation 2, and the number of gaps
A certain commercial jet plane uses a mean of 0.15 gallons of fuel per passenger-mile, with a standard deviation of 0.01 gallons.a. Find the mean number of gallons the plane uses to fly 8000 miles if it carries 210 passengers.b. Assume the amounts of fuel used are independent for each
A gas station earns $2.60 in revenue for each gallon of regular gas it sells, $2.75 for each gallon of midgrade gas, and $2.90 for each gallon of premium gas. Let X1, X2, and X3 denote the numbers of gallons of regular, midgrade, and premium gasoline sold in a day.Assume that X1, X2, and X3 have
The four sides of a picture frame consist of two pieces selected from a population whose mean length is 30 cm with standard deviation 0.1 cm, and two pieces selected from a population whose mean length is 45 cm with standard deviation 0.3 cm.a. Find the mean perimeter.b. Assuming the four pieces
Two independent measurements are made of the lifetime of a charmed strange meson. Each measurement has a standard deviation of 7×10−15 seconds. The lifetime of the meson is estimated by averaging the two measurements. What is the standard deviation of this estimate?
The molarity of a solute in solution is defined to be the number of moles of solute per liter of solution(1 mole = 6.02×1023 molecules). If X is the molarity of a solution of magnesium chloride (MgCl2) and Y is the molarity of a solution of ferric chloride (FeCl3), the molarity of chloride ion
The period of a pendulum is estimated by measuring the starting and stopping times and taking their difference. If the starting and stopping times are measured independently, each with standard deviation 0.2 s, what is the standard deviation of the estimated period?
A laminated item is composed of five layers. The layers are a simple random sample from a population whose thickness has mean 1.2 mm and standard deviation 0.04 mm.a. Find the mean thickness of an item.b. Find the standard deviation of the thickness of an item.
The force, in N, exerted by gravity on a mass of m kg is given by F = 9.8m. Objects of a certain type have mass whose mean is 2.3 kg with a standard deviation of 0.2 kg. Find the mean and standard deviation of F.
A process that fills plastic bottles with a beverage has a mean fill volume of 2.013 L and a standard deviation of 0.005 L. A case contains 24 bottles. Assuming that the bottles in a case are a simple random sample of bottles filled by this method, find the mean and standard deviation of the
Two batteries, with voltages V1 and V2, are connected in series. The total voltage V is given by V = V1 +V2.Assume that V1 has mean 12 V and standard deviation 1 V, and that V2 has mean 6 V and standard deviation 0.5 V.a. Find ????V .b. Assuming V1 and V2 to be independent, find ????V .
The lifetime of a certain transistor in a certain application has mean 900 hours and standard deviation 30 hours. Find the mean and standard deviation of the length of time that four transistors will last.
The bottom of a cylindrical container has an area of 10 cm2. The container is filled to a height whose mean is 5 cm and whose standard deviation is 0.1 cm. Let V denote the volume of fluid in the container.a. Find ????V .b. Find ????V .
If X and Y are independent random variables with means ????X = 9.5 and ????Y = 6.8, and standard deviations ????X = 0.4 and ????Y = 0.1, find the means and standard deviations of the following:a. 3Xb. Y − Xc. X + 4Y
The reading given by a thermometer calibrated in ice water (actual temperature 0◦C) is a random variable with probability density function f(x) ={k(1 − x2) −1 < x < 1 0 otherwise where k is a constant.a. Find the value of k.b. What is the probability that the thermometer reads above 0◦C?c.
The level of impurity (in %) in the product of a certain chemical process is a random variable with probability density function{(3∕64)x2(4 − x) 0 < x < 4 0 otherwisea. What is the probability that the impurity level is greater than 3%?b. What is the probability that the impurity level is
Particles are a major component of air pollution in many areas. It is of interest to study the sizes of contaminating particles. Let X represent the diameter, in micrometers, of a randomly chosen particle. Assume that in a certain area, the probability density function of X is inversely
The thickness of a washer (in mm) is a random variable with probability density function f(x) ={(3∕52)x(6 − x) 2 < x < 4 0 otherwisea. What is the probability that the thickness is less than 2.5 m?b. What is the probability that the thickness is between 2.5 and 3.5 m?c. Find the mean
The lifetime, in years, of a certain type of pump is a random variable with probability density function f(x) ={ 64(x + 2)5 x > 0 0 x ≤ 0a. What is the probability that a pump lasts more than two years?b. What is the probability that a pump lasts between two and four years?c. Find the mean
The error in the length of a part (absolute value of the difference between the actual length and the target length), in mm, is a random variable with probability density function f(x) ={12(x2 − x3) 0 < x < 1 0 otherwisea. What is the probability that the error is less than 0.2 mm?b. Find the
The diameter of a rivet (in mm) is a random variable with probability density function f(x) ={6(x − 12)(13 − x) 12 < x ≤ 13 0 otherwisea. What is the probability that the diameter is less than 12.5 mm?b. Find the mean diameter.c. Find the standard deviation of the diameters.d. Find the
The lifetime of a transistor in a certain application has a lifetime that is random with probability density function f(t) ={0.1e−0.1t t > 0 0 t ≤ 0a. Find the mean lifetime.b. Find the standard deviation of the lifetimes.c. Find the cumulative distribution function of the lifetime.d. Find the

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