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Questions and Answers of Statistics

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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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 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.
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
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
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
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
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,
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
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
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,
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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.
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
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%
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
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
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
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
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
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
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
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
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
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
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
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,
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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 "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
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
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
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
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

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