Questions and Answers of Statistics Engineers Scientists

An engineer wants to determine the spring constant for a particular spring. She hangs various weights on one end of the spring and measures the length of the spring each time. A scatterplot of length
The article “Mechanistic-Empirical Design of Bituminous Roads: An Indian Perspective” (A. Das and B. Pandey, Journal of Transportation Engineering, 1999:463–471) presents an equation of the
The article "Characteristics and Trends of River Discharge into Hudson, James, and Ungava Bays, 1964−2000" (S. Déry, M. Stieglitz, et al., Journal of Climate, 2005:2540−2557) presents
The article ?Mathematical Modeling of the Argon-Oxygen Decarburization Refining Process of Stainless Steel: Part II. Application of the Model to Industrial Practice? (J.Wei and D. Zhu, Metallurgical
Application to mobile computer networks. Computer scientists often model the movement of a mobile computer as a random path within a rectangle. That is, two points are chosen at random within the
Estimating the value of π. The following figure suggests how to estimate the value of π with a simulation. In the figure, a circle with area equal to π/4 is inscribed in a square whose area is
In the following exercise, compute all partial derivatives.z = √sin(x2 y)
In the following exercise, compute all partial derivatives.v = 2xy3 − 3xy2√xy
In the following exercise, compute all partial derivatives.v = ey2 cos(xz) + ln(x2 y + z)
In the following exercise, compute all partial derivatives.z = ln (x2 + y2)
In the following exercise, compute all partial derivatives.w =√x2 + 4y2 + 3z2
In the following exercise, compute all partial derivatives.v = ex (cos y + sin z)
In the following exercise, compute all partial derivatives.v = exy
In the following exercise, compute all partial derivatives.z = cos x sin y2
In the following exercise, compute all partial derivatives.w = x3 + y3/x2 + y2
In the following exercise, compute all partial derivatives.v = 3x + 2xy4
Refer to Exercise 6. The number of flaws in the 34th sample was 27. Is it possible to determine whether the process was in control at this time? If so, state whether or not the process was in
A component can be manufactured according to either of two designs and with either a more expensive or a less expensive material. Several components are manufactured with each combination of design
During the production of boiler plate, test pieces are subjected to a load, and their elongations are measured. In one particular experiment, five tests will be made, at loads (in MPa) of 11, 37, 54,
Two radon detectors were placed in different locations in the basement of a home. Each provided an hourly measurement of the radon concentration, in units of pCi/L. The data are presented in the
Refer to Exercise 4. Perform a randomization test to determine whether the mileage using regular gasoline has a greater variance than the mileage using premium gasoline. Generate at least 1000 random
A sample of seven concrete blocks had their crushing strength measured in MPa. The results were Ten thousand bootstrap samples were generated from these data, and the bootstrap sample means were
The initial temperature of a certain container is measured to be T0= 20?C. The ambient temperature is measured to be Ta= 4?C. An engineer uses Newton??s law of cooling to compute the time needed to
Refer to Exercise 24. A sample of six repair records for a different type of component was drawn. The repair costs, in dollars, were as follows. Would it be appropriate to compute a 95% confidence
A student measures the acceleration A of a cart moving down an inclined plane by measuring the time T that it takes the cart to travel 1 m and using the formula A = 2/T2. Assume that T = 0.55 ± 0.01
In Example 5.20 the following measurements were given for the cylindrical compressive strength (in MPa) for 11 beams: One thousand bootstrap samples were generated from these data, and the bootstrap
The mass (in kg) of a soil specimen is measured to be X = 1.18 ± 0.02. After the sample is dried in an oven, the mass of the dried soil is measured to be Y = 0.85 ± 0.02. Assume that X
The pressure of air (in MPa) entering a compressor is measured to be X = 8.5 ± 0.2, and the pressure of the air leaving the compressor is measured to be Y = 21.2 ± 0.3. The intermediate pressure is
A windmill is used to generate direct current. Data are collected on 45 different days to determine the relationship between wind speed in mi/h (x) and current in kA (y). The data are presented in
Refer to Exercise 13 in Section 5.6. Can you conclude that the time to freeze-up is more variable in the seventh month than in the first month after installation?Refer to Exercise 13 One month after
Fill in the blank: A 95% confidence interval for μ is (1.2, 2.0). Based on the data from which the confidence interval was constructed, someone wants to test H0 : μ = 1.4 versus H1 : μ = 1.4. The
At a certain genetic locus on a chromosome, each individual has one of three different DNA sequences (alleles). The three alleles are denoted A, B, C. At another genetic locus on the same chromosome,
The article “Magma Interaction Processes Inferred from Fe-Ti Oxide Compositions in the D¨olek and Sari¸ci¸cek Plutons, Eastern Turkey” (O. Karsli, F. Aydin, et al., Turkish Journal of Earth
This exercise requires ideas from Section 2.6. In a two-sample experiment, when each item in one sample is paired with an item in the other, the paired t test (Section 6.8) can be used to test
Refer to Exercise 1 in Section 5.2. Can it be concluded that less than half of the automobiles in the state have pollution levels that exceed the standard?Refer to Exercise 1In a simple random sample
As part of the quality-control program for a catalyst manufacturing line, the raw materials (alumina and a binder) are tested for purity. The process requires that the purity of the alumina be
A copper smelting process is supposed to reduce the arsenic content of the copper to less than 1000 ppm. Let μ denote the mean arsenic content for copper treated by this process, and assume that the
A sample of 25 one-year-old girls had a mean weight of 24.1 pounds with a standard deviation of 4.3 pounds. Assume that the population of weights is normally distributed. A pediatrician claims that
A test has power 0.80 when μ = 3.5. True or false:a. The probability of rejecting H0 when μ = 3.5 is 0.80.b. The probability of making a type I error when μ = 3.5 is 0.80.c. The probability of
A sample of eight repair records for a certain fiber optic component was drawn, and the cost of each repair, in dollars, was recorded. The results were Assume the population of repair records is
Diameters, in mm, were measured for eight specimens of a certain type of ball bearing. The results were Assume the diameters are normally distributed. a. Find a 98% prediction interval for the
The carbon content (in ppm) was measured for each of six silicon wafers. The results were Assume that carbon contents are normally distributed. a. Find a 95% prediction interval for the carbon
The answer to Exercise 19 part (d) is needed for this exercise. A geologist counts 64 emitted particles in one minute from a certain radioactive rock.a. Find a 95% confidence interval for the rate of
Boxes of nails contain 100 nails each. A sample of 10 boxes is drawn, and each of the boxes is weighed. The average weight is 1500 g and the standard deviation is 5 g. Assume the weight of the box
In a study of the lifetimes of electronic components, a random sample of 400 components are tested until they fail to function. The sample mean lifetime was 370 hours and the standard deviation was
In the article “Occurrence and Distribution of Ammonium in Iowa Groundwater” (K. Schilling, Water Environment Research, 2002:177–186), ammonium concentrations (in mg/L) were measured at a large
Refer to Exercise 1. Another molecular biologist repeats the study with a different design. She makes up 12 DNA samples, and then chooses 6 at random to be treated with the enzyme and 6 to remain
Someone claims that the number of hits on his website has a Poisson distribution with mean 20 per hour. Let X be the number of hits in five hours.a. If the claim is true, what is P(X ≤ 95)?b. Based
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
The area covered by 1 L of a certain stain is normally distributed with mean 10m2 and standard deviation 0.2m2.a. What is the probability that 1 L of stain will be enough to cover 10.3m2?b. What is
Use the result of Exercise 17 and Table A.1 to find P(X = 10) where X ∼ Geom(0.3).
The manufacture of a certain part requires two different machine operations. The time on machine 1 has mean 0.5 hours and standard deviation 0.4 hours. The time on machine 2 has mean 0.6 hours and
Let U ∼ U(0, 1). Let a and b be constants with a < b, and let X = (b − a)U + a.a. Find the cumulative distribution function of Ub. Show that P(X ≤ x) = P(U ≤ (x −a)/(b−a)).c. Use the
A battery manufacturer claims that the lifetime of a certain type of battery has a population mean of 40 hours and a standard deviation of 5 hours. Let X represent the mean lifetime of the batteries
The lifetime of a microprocessor is exponentially distributed with mean 3000 hours.a. What proportion of microprocessors will fail within 300 hours?b. What proportion of microprocessors will function
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
Grandma is trying out a new recipe for raisin bread. Each batch of bread dough makes three loaves, and each loaf contains 20 slices of bread.a. If she puts 100 raisins into a batch of dough, what is
The amount of paint required to paint a surface with an area of 50 m2 is normally distributed with mean 6 L and standard deviation 0.3 L.a. If 6.2 L of paint are available, what is the probability
A thermocouple placed in a certain medium produces readings within 0.1°C of the true temperature 70% of the time, readings more than 0.1°C above the true temperature 10% of the time, and readings
At a certain fast-food restaurant, 25% of drink orders are for a small drink, 35% for a medium, and 40% for a large. A random sample of 20 orders is selected for audit.a. What is the probability that
The concentration of particles in a suspension is 30 per mL.a. What is the probability that a 2 mL sample will contain more than 50 particles?b. Ten 2 mL samples are drawn. What is the probability
A plate is attached to its base by 10 bolts. Each bolt is inspected before installation, and the probability of passing the inspection is 0.9. Only bolts that pass the inspection are installed. Let X
Gears produced by a grinding process are categorized either as conforming (suitable for their intended purpose), downgraded (unsuitable for the intended purpose but usable for another purpose), or
The probability that a certain radioactive mass emits no particles in a one-minute time period is 0.1353. What is the mean number of particles emitted per minute?
Shafts manufactured for use in optical storage devices have diameters that are normally distributed with mean μ = 0.652 cm and standard deviation σ = 0.003 cm. The specification for the shaft
Let X ∼ U(a, b). Use the definition of the mean of a continuous random variable (Equation 2.35) to show that μX = (a + b)/2.
Of customers ordering a certain type of personal computer, 20% order an upgraded graphics card, 30% order extra memory, 15% order both the upgraded graphics card and extra memory, and 35% order
A random sample of size 8 is taken from a Exp(λ) distribution, where λ is unknown. The sample values are 2.74, 6.41, 4.96, 1.65, 6.38, 0.19, 0.52, and 8.38. This exercise shows how to use the
A machine produces 1000 steel O-rings per day. Each ring has probability 0.9 of meeting a thickness specification.a. What is the probability that on a given day, fewer than 890 O-rings meet the
A process that polishes a mirrored surface leaves an average of 2 small flaws per 5 m2 of surface. The number of flaws on an area of surface follows a Poisson distribution.a. What is the probability
Two-dimensional Poisson process. The number of plants of a certain species in a certain forest has a Poisson distribution with mean 10 plants per acre. The number of plants in T acres therefore has a
Specifications for an aircraft bolt require that the ultimate tensile strength be at least 18 kN. It is known that 10% of the bolts have strengths less than 18.3 kN and that 5% of the bolts have
A random sample will be drawn from a normal distribution, for the purpose of estimating the population mean μ. Since μ is the median as well as the mean, it seems that both the sample median m and
The temperature of a solution will be estimated by taking n independent readings and averaging them. Each reading is unbiased, with a standard deviation of σ = 0.5°C. How many readings must be
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,
Refer to Exercise 4. What is the probability that in a sequence of 10 days, four green lights, one yellow light, and five red lights are encountered?Refer to Exercise 4.A traffic light at a certain
The lifetime of a laser (in hours) is lognormally distributed with μ = 8 and σ2 = 2.4. Two such lasers are operating independently.a. Use a simulated sample of size 1000 to estimate the probability
Among all the income-tax forms filed in a certain year, the mean tax paid was $2000 and the standard deviation was $500. In addition, for 10% of the forms, the tax paid was greater than $3000. A
Below are the durations (in minutes) of 40 time intervals between eruptions of the geyser Old Faithful in Yellowstone National Park.Construct a normal probability plot for these data. Do they appear
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
Of the items manufactured by a certain process, 20% are defective. Of the defective items, 60% can be repaired.a. Find the probability that a randomly chosen item is defective and cannot be
Refer to Exercise 5. Assume that the relative uncertainty in P1 is 5% and the relative uncertainty in P2 is 2%. Find the relative uncertainty in P3.Refer to Exercise 5.When air enters a compressor at
Given that X and Y are related by the given equation, and that X = 3.0 ± 0.1, estimate Y and its uncertainty.a. XY = 1b. Y/X = 2c. √XY = 3d. Y√X = 4
Refer to Exercise 16. Assume that T0= 73.1 ? 0.1?F, Ta = 37.5 ? 0.2?F, k = 0.032 min??1 with negligible uncertainty, and T = 50?F exactly. Estimate t, and find the relative uncertainty in the
In a chemical reaction run at a certain temperature, the concentration C of a certain reactant at time t is given by 1/C = kt+1/C0, where C0 is the initial concentration and k is the rate constant.
Convert the following absolute uncertainties to relative uncertainties.a. 20.9 ± 0.4b. 15.1 ± 0.8c. 388 ± 23d. 2.465 ± 0.009
Refer to Exercise 10 in Section 3.2. Assume that τ = 30.0 ± 0.1 Pa, h = 10.0 ± 0.2 mm, and μ = 1.49 Pa · s with negligible uncertainty.a. Estimate V and find the uncertainty in the estimate.b.
The velocity V of sound in air at temperature T is given by V = 20.04√T , where T is measured in kelvins (K) and V is in m/s. Assume that T = 300 ± 0.4 K. Estimate V, and find the uncertainty in
The volume of a cone is given by V = πr2h/3, where r is the radius of the base and h is the height. Assume the radius is 5 cm, measured with negligible uncertainty, and the height is h = 6.00 ±
The volume of a cone is given by V = πr2h/3, where r is the radius of the base and h is the height. Assume the height is measured to be h = 6.00 ± 0.01 cm and the radius is r = 5.00 ± 0.02 cm.a.
Find the uncertainty in Y, given that X = 2.0 ± 0.3 anda. Y = X3b. Y = √2Xc. Y = 3/Xd. Y = ln Xe. Y = eXf. Y = cos X (X is in units of radians)
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
Refer to Exercise 26.a. Find Î¼X.b. Find Ïƒ2X.c. Find Cov(X,Y).d. Find ÏX,Y.Refer to Exercise 26. 0
The number of customers in line at a supermarket express checkout counter is a random variable whose probability mass function is given in the following table.For each customer, the number of items
If X is a random variable, prove that Cov(X, X) = σ2X.
A lot of 1000 components contains 300 that are defective. Two components are drawn at random and tested. Let A be the event that the first component drawn is defective, and let B be the event that
The lifetimes, in months, of two components in a system, denoted X and Y , have joint probability density function.a. What is the probability that both components last longer than one month?b. Find
Refer to Example 2.54.a. Find Cov(X,Y).b. Find ρX,Y.
For continuous random variables X and Y with joint probability density function.a. Find P(X > 1 and Y > 1).b. Find the marginal probability density functions fX (x) and fY (y).c. Are X and Y
A computer sends a packet of information along a channel and waits for a return signal acknowledging that the packet has been received. If no acknowledgment is received within a certain time, the

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