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Statistics For Engineers And Scientists 3rd Edition William Navidi - Solutions

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
A cylindrical hole is drilled in a block, and a cylindrical piston is placed in the hole. The clearance is equal to one-half the difference between the diameters of the hole and the piston. The diameter of the hole is normally distributed with mean 15 cm and standard deviation 0.025 cm, and the
The fill volume of cans filled by a certain machine is normally distributed with mean 12.05 oz and standard deviation 0.03 oz. a. What proportion of cans contain less than 12 oz? b. The process mean can be adjusted through calibration. To what value should the mean be set so that 99% of the cans
A fiber-spinning process currently produces a fiber whose strength is normally distributed with a mean of 75 N/m2. The minimum acceptable strength is 65 N/m2. a. Ten percent of the fiber produced by the current method fails to meet the minimum specification. What is the standard deviation of fiber
Let X ∼ N(μ, σ2), and let Z = (X − μ)/σ . Use Equation (4.25) to show that Z ∼ N(0, 1).
A light fixture holds two lightbulbs. Bulb A is of a type whose lifetime is normally distributed with mean 800 hours and standard deviation 100 hours. Bulb B has a lifetime that is normally distributed with mean 900 hours and standard deviation 150 hours. Assume that the lifetimes of the bulbs are
A binary message m, where m is equal either to 0 or to 1, is sent over an information channel. Because of noise in the channel, the message received is X, where X = m + E, and E is a random variable representing the channel noise. Assume that if X ≤ 0.5 then the receiver concludes that m = 0 and
A company receives a large shipment of bolts. The bolts will be used in an application that requires a torque of 100 J. Before the shipment is accepted, a quality engineer will sample 12 bolts and measure the torque needed to break each of them. The shipment will be accepted if the engineer
Let Z ∼ N(0, 1). Find a constant c for which a. P(Z ≥ c) = 0.1587 b. P(c ≤ Z ≤ 0) = 0.4772 c. P(−c ≤ Z ≤ c) = 0.8664 d. P(0 ≤ Z ≤ c) = 0.2967 e. P(|Z| ≥ c) = 0.1470
The lifetime of a battery in a certain application is normally distributed with mean μ = 16 hours and standard deviation σ = 2 hours. a. What is the probability that a battery will last more than 19 hours? b. Find the 10th percentile of the lifetimes. c. A particular battery lasts 14.5 hours.
Scores on a standardized test are approximately normally distributed with a mean of 480 and a standard deviation of 90. a. What proportion of the scores are above 700? b. What is the 25th percentile of the scores? c. If someone's score is 600, what percentile is she on? d. What proportion of the
The lifetime of a lightbulb in a certain application is normally distributed with mean μ = 1400 hours and standard deviation σ = 200 hours. a. What is the probability that a lightbulb will last more than 1800 hours? b. Find the 10th percentile of the lifetimes. c. A particular lightbulb lasts
The lifetime (in days) of a certain electronic component that operates in a high-temperature environment is lognormally distributed with μ = 1.2 and σ = 0.4. a. Find the mean lifetime. b. Find the probability that a component lasts between three and six days. c. Find the median lifetime. d. Find
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, 2001:378-394)
If a resistor with resistance R ohms carries a current of I amperes, the potential difference across the resistor, in volts, is given by V = IR. Suppose that I is lognormal with parameters μI = 1 and σ2I = 0.2, R is lognormal with parameters μR = 4 and σ2R = 0.1, and that I and R are
The article "Withdrawal Strength of Threaded Nails" (D. Rammer, S. Winistorfer, and D. Bender, Journal of Structural Engineering 2001:442 - 449) describes an experiment comparing the ultimate withdrawal strengths (in N/mm) for several types of nails. For an annularly threaded nail with shank
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
Let T ∼ Exp(0.45). Find a. μT b. σ2T c. P(T > 3) d. The median of T
A light fixture contains five lightbulbs. The lifetime of each bulb is exponentially distributed with mean 200 hours. Whenever a bulb burns out, it is replaced. Let T be the time of the first bulb replacement. Let Xi, i = 1, . . . , 5, be the lifetimes of the five bulbs. Assume the lifetimes of the
A catalyst researcher states that the diameters, in microns, of the pores in a new product she has made have the exponential distribution with parameter λ = 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
Refer to Exercise 3. Suppose that the diameters of 10 pores are independent. a. Find the probability that more than seven of them have diameters less than 3 microns. b. Find the probability that exactly one of them has a diameter greater than 11 microns.
A certain type of component can be purchased new or used. Fifty percent of all new components last more than five years, but only 30% of used components last more than five years. Is it possible that the lifetimes of new components are exponentially distributed? Explain.
The number of traffic accidents at a certain intersection is thought to be well modeled by a Poisson process with a mean of 3 accidents per year. a. Find the mean waiting time between accidents. b. Find the standard deviation of the waiting times between accidents. c. Find the probability that more
A person arrives at a certain bus stop each morning. The waiting time, in minutes, for a bus to arrive is uniformly distributed on the interval (0, 15). a. Find the mean waiting time. b. Find the standard deviation of the waiting times. c. Find the probability that the waiting time is between 5 and
A system consists of two components connected in series. The system will fail when either of the two components fails. Let T be the time at which the system fails. Let X1 and X2 be the lifetimes of the two components. Assume that X1 and X2 are independent and that each has theWeibull distribution
Let X ∼ U(a, b). Use the definition of the variance of a continuous random variable (Equation 2.36 or 2.37) to show that σ2X = (b − a)2/12.
Let X ˆ¼ U(a, b).a. Show that if x ‰¤ a then P(X ‰¤ x) = 0.b. Show that if x > b then P(X ‰¤ x) = 1.c. Show that if a (x ˆ’ a)/(b ˆ’ a). Conclude that the cumulative distribution function of X is
Let T ∼ (4, 0.5). a. Find μT. b. Find σT. c. Find P(T ≤ 1). d. Find P(T ≥ 4).
Let T ∼ (r, λ). If μT = 8 and r = 16, find λ and σ2.
Let T ∼ Weibull (0.5, 3). a. Find μT. b. Find σT. c. Find P(T < 1). d. Find P(T > 5).
The lifetime of a cooling fan, in hours, that is used in a computer system has the Weibull distribution with α = 1.5 and β = 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
Let X ∼ Geom(p). Find the MLE of p.
Maximum likelihood estimates possess the property of functional invariance, which means that if is the MLE of θ, and h(θ) is any function of θ, then h() is the MLE of h(θ). a. Let X ∼ Bin(n, p) where n is known and p is unknown. Find the MLE of the odds ratio p/(1 − p). b. Use the result
Let X1, . . . , Xn be a random sample from a N(0, σ2) population. Find the MLE of σ.
An airplane has 100 seats for passengers. Assume that the probability that a person holding a ticket appears for the flight is 0.90. If the airline sells 105 tickets, what is the probability that everyone who appears for the flight will get a seat?
In a certain process, the probability of producing an defective component is 0.07. a. In a sample of 250 randomly chosen components, what is the probability that fewer than 20 of them are defective? b. In a sample of 10 randomly chosen components, what is the probability that one or more of them is
Yeast cells are suspended in a liquid medium. A 2 mL sample of the suspension is withdrawn. A total of 56 yeast cells are counted. a. Estimate the concentration of yeast cells per mL of suspension, and find the uncertainty. b. What volume of suspension must be withdrawn to reduce the uncertainty to
Thicknesses of shims are normally distributed with mean 1.5 mm and standard deviation 0.2 mm. Three shims are stacked, one atop another. a. Find the probability that the stack is more than 5 mm thick. b. Find the 80th percentile of the stack thickness. c. What is the minimum number of shims to be
The lifetime of a bearing (in years) follows the Weibull distribution with parameters α = 1.5 and β = 0.8. a. What is the probability that a bearing lasts more than 1 year? b. What is the probability that a bearing lasts less than 2 years?
A cereal manufacturer claims that the gross weight (including packaging) of a box of cereal labeled as weighing 12 oz has a mean of 12.2 oz and a standard deviation of 0.1 oz. You gather 75 boxes and weigh them all together. Let S denote the total weight of the 75 boxes of cereal. a. If the claim
A distribution sometimes used to model the largest item in a sample is the extreme value distribution. This distribution has cumulative distribution function F(x) = e−e−x Let X be a random variable with this distribution. a. Find P(X ≤ 0). b. Find P(X > ln 2). c. Find the median of X.
A distribution that has been used to model tolerance levels in bioassays is the logistic distribution with parameters α and β. The cumulative distribution function of the logistic distribution is F(x) = [1 + e−(x−α)/β ]−1 The parameter α may be any real number; the parameter β may be
Let X ∼ Geom(p). Let s ≥ 0 be an integer. a. Show that P(X > s) = (1− p)s. b. Let t ≥ 0 be an integer. Show that P(X > s + t | X > s) = P(X > t). This is called the lack of memory property. c. A penny and a nickel are both fair coins. The penny is tossed three times and comes up tails each
Let X represent the lifetime of a component, in weeks. Let Y represent the lifetime of the component in days, so Y = 7X. Suppose X ∼ Exp(λ). a. Let FY be the cumulative distribution function of Y and let FX be the cumulative distribution function of X. Show that FY (y) = 1 − e−λy/7. b. Show
Let X ˆ¼ Bin(n, p).a. Show that if x is an integer between 1 and n inclusive, thenb. Show that if X ˆ¼ Bin(n, p), the most probable value for X is the greatest integer less than or equal to np + p.
Pea plants contain two genes for seed color, each of which may be Y (for yellow seeds) or G (for green seeds). Plants that contain one of each type of gene are called heterozygous. According to the Mendelian theory of genetics, if two heterozygous plants are crossed, each of their offspring will
The Environmental Protection Agency (EPA) has contracted with your company for equipment to monitor water quality for several lakes in your water district. A total of 10 devices will be used. Assume that each device has a probability of 0.01 of failure during the course of the monitoring period. a.
Medication used to treat a certain condition is administered by syringe. The target dose in a particular application is μ. Because of the variations in the syringe, in reading the scale, and in mixing the fluid suspension, the actual dose administered is normally distributed with mean μ and
The intake valve clearances on new engines of a certain type are normally distributed with mean 200 μm and standard deviation 10 μm. a. What is the probability that the clearance is greater than 215 μm? b. What is the probability that the clearance is between 180 and 205 μm? c. An engine has
Find the value of zα/2 to use in expression (5.1) to construct a confidence interval with level a. 95% b. 98% c. 99% d. 80%
A supplier sells synthetic fibers to a manufacturing company. A simple random sample of 81 fibers is selected from a shipment. The average breaking strength of these is 29 lb, and the standard deviation is 9 lb. a. Find a 95% confidence interval for the mean breaking strength of all the fibers in
Refer to Exercise 6. a. Find a 98% lower confidence bound for the mean time. b. Someone says that the mean time is greater than 134.3 minutes. With what level of confidence can this statement be made?
Refer to Exercise 8. a. Find a 99% upper confidence bound for the mean porosity. b. The claim is made that the mean porosity is less than 22.7%.With what level of confidence can this statement be made?
Refer to Exercise 10. a. Find a 98% lower confidence bound for the mean time to complete the step. b. An efficiency expert says that the mean time is greater than 70 seconds. With what level of confidence can this statement be made?
An investigator computes a 95% confidence interval for a population mean on the basis of a sample of size 70. If she wishes to compute a 95% confidence interval that is half as wide, how large a sample does she need?
Based on a large sample of capacitors of a certain type, a 95% confidence interval for the mean capacitance, in μF, was computed to be (0.213, 0.241). Find a 90% confidence interval for the mean capacitance of this type of capacitor?
A large box contains 10,000 ball bearings. A random sample of 120 is chosen. The sample mean diameter is 10 mm, and the standard deviation is 0.24 mm. True or false: a. A 95% confidence interval for the mean diameter of the 120 bearings in the sample is 10 ± (1.96) (0.24) / √120. b. A 95%
As the confidence level goes up, the reliability goes and the precision goes Options: up, down?
In a sample of 100 steel wires the average breaking strength is 50 kN, with a standard deviation of 2 kN. a. Find a 95% confidence interval for the mean breaking strength of this type of wire. b. Find a 99% confidence interval for the mean breaking strength of this type of wire. c. An engineer
The capacities (in ampere-hours) were measured for a sample of 120 batteries. The average was 178 and the standard deviation was 14. a. Find a 95% confidence interval for the mean capacity of batteries produced by this method. b. Find a 99% confidence interval for the mean capacity of batteries
In a sample of 80 ten-penny nails, the average weight was 1.56 g and the standard deviation was 0.1 g. a. Find a 95% confidence interval for the mean weight of this type of nail. b. Find a 98% confidence interval for the mean weight of this type of nail. c. What is the confidence level of the
In a simple random sample of 70 automobiles registered in a certain state, 28 of them were found to have emission levels that exceed a state standard. a. What proportion of the automobiles in the sample had emission levels that exceed the standard? b. Find a 95% confidence interval for the
The article "Leachate from Land Disposed Residential Construction Waste" (W. Weber, Y. Jang, et al., Journal of Environmental Engineering, 2002: 237-245) presents a study of contamination at landfills containing construction and demolition waste. Specimens of leachate were taken from a test site.
A sociologist is interested in surveying workers in computer-related jobs to estimate the proportion of such workers who have changed jobs within the past year. a. In the absence of preliminary data, how large a sample must be taken to ensure that a 95% confidence interval will specify the
For major environmental remediation projects to be successful, they must have public support. The article "Modelling the Non-Market Environmental Costs and Benefits of Biodiversity Using Contingent Value Data" (D. Macmillan, E. Duff, and D. Elston, Environmental and Resource Economics,
A soft-drink manufacturer purchases aluminum cans from an outside vendor. A random sample of 70 cans is selected from a large shipment, and each is tested for strength by applying an increasing load to the side of the can until it punctures. Of the 70 cans, 52 meet the specification for puncture
The article "The Functional Outcomes of Total Knee Arthroplasty" (R. Kane, K. Saleh, et al., Journal of Bone and Joint Surgery, 2005:1719-1724) reports that out of 10,501 surgeries, 859 resulted in complications within six months of surgery. a. Find a 95% confidence interval for the proportion of
Refer to Exercise 2. Find a 98% upper confidence bound for the proportion of residences that reduced their water consumption?
A random sample of 400 electronic components manufactured by a certain process are tested, and 30 are found to be defective. a. Let p represent the proportion of components manufactured by this process that are defective. Find a 95% confidence interval for p. b. How many components must be sampled
Find the value of tn−1,α/2 needed to construct a two sided confidence interval of the given level with the given sample size: a. Level 90%, sample size 12. b. Level 95%, sample size 7. c. Level 99%, sample size 2. d. Level 95%, sample size 29.
The article "Effect of Granular Subbase Thickness on Airfield Pavement Structural Response" (K. Gopalakrishnan and M. Thompson, Journal of Materials in Civil Engineering, 2008:331-342) presents a study of the effect of the subbase thickness on the amount of surface deflection caused by aircraft
Twelve samples of water from a particular source have an average lead concentration of 12.5μg/L with a standard deviation of 2.0μg/L. Find a 95% confidence interval for the lead concentration in water from this source.
The following MINITAB output presents a confidence interval for a population mean, but some of the numbers got smudged and are now illegible. Fill in the missing numbers for (a), (b), and (c).
The article "Filtration Rates of the Zebra Mussel (Dreissena polymorpha) on Natural Seston from Saginaw Bay, Lake Huron" (D. Fanslow, T. Nalepa, and G. Lang, Journal of Great Lakes Research 1995:489:500) reports measurements of the rates (in mL/mg/h) at which mussels filter seston (particulate
Find the level of a two-sided confidence interval that is based on the given value of tn−1,α/2 and the given sample size? a. t = 2.776, sample size 5. b. t = 2.718, sample size 12. c. t = 5.841, sample size 4. d. t = 1.325, sample size 21. e. t = 1.746, sample size 17.
The article "Ozone for Removal of Acute Toxicity from Logyard Run-off" (M. Zenaitis and S. Duff, Ozone Science and Engineering, 2002: 83-90) presents chemical analyses of runoff water from sawmills in British Columbia. Included were measurements of pH for six water specimens: 5.9, 5.0, 6.5, 5.6,
The article "An Automatic Visual System for Marble Tile Classification" (L. Carrino, W. Polini, and S. Turchetta, Journal of Engineering Manufacture, 2002:1095-1108) describes a measure for the shade of marble tile in which the amount of light reflected by the tile is measured on a scale of 0-255.
Eight independent measurements are taken of the diameter of a piston. The measurements (in inches) are: 3.236, 3.223, 3.242, 3.244, 3.228, 3.253, 3.253, and 3.230. a. Make a dotplot of the eight values. b. Should the Student's t distribution be used to find a 99% confidence interval for the
To study the effect of curing temperature on shear strength of a certain rubber compound, 80 specimens were cured at 150◦C and 95 were cured at 130◦C. The specimens cured at 150◦C had an average shear strength of 620 psi, with a standard deviation of 20 psi. Those cured at 130◦C had an
In a study of the effect of cooling rate on the hardness of welded joints, 50 welds cooled at a rate of 10◦C/s had an average Rockwell (B) hardness of 91.1 and a standard deviation of 6.23, and 40 welds cooled at a rate of 30◦C/s had an average hardness of 90.7 and a standard deviation of
The article "Inconsistent Health Perceptions for US Women and Men with Diabetes" (M. McCollum, L. Hansen, et al., Journal of Women's Health, 2007:1421-1428) presents results of a survey of adults with diabetes. The average body mass index (BMI) in a sample of 1559 men was 30.4, with a standard
The article "Vehicle-Arrival Characteristics at Urban Uncontrolled Intersections" (V. Rengaraju and V. Rao, Journal of Transportation Engineering, 1995: 317-323) presents data on traffic characteristics at 10 intersections in Madras, India. At one particular intersection, the average speed for a
In experiments to determine the effectiveness of drugs such as anti-fungal ointments that are applied to the skin, the concentration of the drug in a patient's skin must be measured. The article "Determining Bioequivalence of Topical Dermatological Drug Products by Tape-Stripping" (W. Navidi, A.
In a study to compare two different corrosion inhibitors, specimens of stainless steel were immersed for four hours in a solution containing sulfuric acid and a corrosion inhibitor. Forty-seven specimens in the presence of inhibitor A had a mean weight loss of 242 mg and a standard deviation of 20
In a test of the effect of dampness on electric connections, 100 electric connections were tested under damp conditions and 150 were tested under dry conditions. Twenty of the damp connections failed and only 10 of the dry ones failed. Find a 90% confidence interval for the difference between the
Angioplasty is a medical procedure in which an obstructed blood vessel is widened. In some cases, a wire mesh tube, called a stent, is placed in the vessel to help it remain open. The article "Long-term Outcomes of Patients Receiving Drug-eluting Stents" (A. Philpott, D. Southern, et al., Canadian
Crash testing is a highly expensive procedure to evaluate the ability of an automobile to withstand a serious accident. A simple random sample of 12 small cars were subjected to a head-on collision at 40 miles per hour. Of them 8 were "totaled," meaning that the cost of repairs is greater than the
A mobile computer network consists of a number of computers (called nodes) that communicate with each other while moving throughout a region. A node that is out of transmission range of the other nodes, so that it is unable to communicate, is said to be partitioned. In studies carried out at the
In a study comparing various methods of gold plating, 7 printed circuit edge connectors were gold-plated with control-immersion tip plating. The average gold thickness was 1.5 μm, with a standard deviation of 0.25 μm. Five connectors were masked, then plated with total immersion plating. The
The breaking strength of hockey stick shafts m de of two different grphite-Kevl r composites yield the following results (in newtons):Find a 98% confidence interval for the difference between the mean breaking strengths of hockey stick hafts made of the two materials?
A computer system administrator notices that computers running a particular operating system seem to freeze up more often as the installation of the operating system ages. She measures the time (in minutes) before freeze-up for seven computers one month after installation, and for nine computers
The article "Effects of Waste Glass Additions on the Properties and Durability of Fired Clay Brick" (S. Chidiac and L. Federico, Can J Civ Eng, 2007: 1458-1466) reports that the average initial rate of absorption (in g/min·cm2) for a sample of five clay bricks with fine glass particles added was
In an experiment involving the breaking strength of a certain type of thread used in personal flotation devices, one batch of thread was subjected to a heat treatment for 60 seconds and another batch was treated for 120 seconds. The breaking strengths (in N) of ten threads in each batch were
The article "Differences in Susceptibilities of Different Cell Lines to Bilirubin Damage" (K. Ngai, C. Yeung, and C. Leung, Journal of Paediatric Child Health, 2000:36-45) reports an investigation into the toxicity of bilirubin on several cell lines. Ten sets of human liver cells and 10 sets of
During the spring of 1999, many fuel storage facilities in Serbia were destroyed by bombing. As a result, significant quantities of oil products were spilled and burned, resulting in soil pollution. The article "Mobility of Heavy Metals Originating from Bombing of Industrial Sites" (B. krbi, J.
The article "The Frequency Distribution of Daily Global Irradiation at Kumasi" (F. Akuffo and A. Brew-Hammond, Solar Energy, 1993:145-154) defines the daily clearness index for a location to be the ratio of daily global irradiation to extraterrestrial irradiation. Measurements were taken in the
The article "Simulation of the Hot Carbonate Process for Removal of CO2 and H2S from Medium Btu Gas" (K. Park and T. Edgar, Energy Progress, 1984:174-180) presents an equation used to estimate the equilibrium vapor pressure of CO2 in a potassium carbonate solution. The actual equilibrium pressure
Transepidermal water loss (TEWL) is a measure of the rate that water crosses the skin through diffusion and evaporation. In general, damaged skin has a higher TEWL than non-damaged skin. A report submitted to the U.S. Food and Drug Administration (Therapeutic Equivalence of Topical Products A.
A group of five individuals with high blood pressure were given a new drug that was designed to lower blood pressure. Systolic blood pressure was measured before and after treatment for each individual, with the following results:Find a 90% confidence for the mean reduction in systolic blood
For a sample of nine automobiles, the mileage (in 1000s of miles) at which the original front brake pads were worn to 10% of their original thickness was measured, as was the mileage at which the original rear brake pads were worn to 10% of their original thickness. The results are given in the

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