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

A person gets on and off a bathroom scale four times. The four readings (in pounds) are 148, 151, 150, and 152. Each time after the person gets off the scale, the reading is 2 lb. a. Is it possible to estimate the uncertainty in these measurements? If so, estimate it. If not, explain why not. b. Is
Assume that X and Y are independent measurements with uncertainties σx = 0.2 and σy = 0.4. Find the uncertainties in the following quantities: a. 3X b. X − Y c. 2X + 3Y
According to Newton's law of cooling, the temperature T of a body at time t is given by T = Ta + (T0 − Ta)e−kt , where Ta is the ambient temperature, T0 is the initial temperature, and k is the cooling rate constant. For a certain type of beverage container, the value of k is known to be 0.025
Nine independent measurements are made of the length of a rod. The average of the nine measurements is = 5.238 cm, and the standard deviation is s = 0.081 cm. a. Is the uncertainty in the value 5.238 cm closest to 0.009, 0.027, or 0.081 cm? Explain. b. Another rod is measured once by the same
The volume of a rock is measured by placing the rock in a graduated cylinder partially filled with water and measuring the increase in volume. Eight independent measurements are made. The average of the measurements is 87.0 mL, and the standard deviation is 2.0 mL. a. Estimate the volume of the
A certain chemical process is run 10 times at a temperature of 65—¦C and 10 times at a temperature of 80—¦C. The yield at each run was measured as a percent of a theoretical maximum. The data are presented in the following table.a. For each temperature, estimate the mean
The length of a component is to be estimated through repeated measurement. a. Ten independent measurements are made with an instrument whose uncertainty is 0.05 mm. Let X denote the average of these measurements. Find the uncertainty in X. b. A new measuring device, whose uncertainty is only 0.02
The length of a rod is to be measured by a process whose uncertainty is 3 mm. Several independent measurements will be taken, and the average of these measurements will be used to estimate the length of the rod. How many measurements must be made so that the uncertainty in the average will be 1 mm?
In the article "The World's Longest Continued Series of Sea Level Observations" (M. Ekman, Paleogeography, 1988:73-77), the mean annual level of land uplift in Stockholm, Sweden, was estimated to be 4.93 ± 0.23 mm for the years 1774-1884 and to be 3.92 ± 0.19 mm for the years 1885-1984. Estimate
A force of F = 2.2 ± 0.1 N is applied to a block for a period of time, during which the block moves a distance d = 3 m, which is measured with negligible uncertainty. The work W is given by W = Fd. Estimate W, and find the uncertainty in the estimate.
The specific gravity of a substance is given by G = DS/DW, where DS is the density of the substance in kg/m3 and DW is the density of water, which is known to be 1000 kg/m3. The density of a particular substance is measured to be DS = 500 ± 5 kg/m3. Estimate the specific gravity, and find the
Find the uncertainty in Y, given that X = 4.0 ± 0.4 and a. Y = X2 b. Y =√X c. Y = 1/X d. Y = ln X e. Y = eX f. Y = sin X (X is in units of radians)
Convert the following absolute uncertainties to relative uncertainties. a. 37.2 ± 0.1 b. 8.040 ± 0.003 c. 936 ± 37 d. 54.8 ± 0.3
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 2.2 m. The time is measured to be t = 0.67 ± 0.02 s. Estimate g, and find the relative uncertainty in the estimate. (g =
Refer to Exercise 5. a. Assume g = 9.80 m/s2 exactly, and that L = 0.855 ± 0.005 m. Estimate T, and find the relative uncertainty in the estimate. b. Assume L = 0.855 m exactly, and that T = 1.856 ± 0.005 s. Estimate g, and find the relative uncertainty in the estimate
Refer to Exercise 7. Estimate F, and find the relative uncertainty in the estimate, assuming that g = 9.80 m/s2 exactly and that a. d = 0.20 m and l = 35.0 m, both with negligible uncertainty, and h = 4.51 ± 0.03 m. b. h = 4.51 m and l = 35.0 m, both with negligible uncertainty, and d = 0.20 ±
Refer to Exercise 9. Assume that the mass of the rock is 288.2 g with negligible uncertainty, the initial volume of water in the cylinder is 400 ± 0.1 mL, and the volume of water plus rock is 516 ± 0.2 mL. Estimate the density of the rock, and find the relative uncertainty in the estimate.
The volume of a cone is given by V = πr 2h/3, where r is the radius of the base and h is the height. Assume the height is 6 cm, measured with negligible uncertainty, and the radius is r = 5.00 ± 0.02 cm. Estimate the volume of the cone, and find the uncertainty in the estimate.
The period T of a simple pendulum is given by T = 2π √L/g where L is the length of the pendulum and g is the acceleration due to gravity. a. Assume g = 9.80 m/s2 exactly, and that L = 0.742 ± 0.005 m. Estimate T, and find the uncertainty in the estimate. b. Assume L = 0.742 m exactly, and that
The friction velocity F of water flowing through a pipe is given by F = √gdh/4l, where g is the acceleration due to gravity, d is the diameter of the pipe, l is the length of the pipe, and h is the head loss. Estimate F, and find the uncertainty in the estimate, assuming that g = 9.80 m/s2
The density of a rock will be measured by placing it into a graduated cylinder partially filled with water, and then measuring the volume of water displaced. The density D is given by D = m/(V1 − V0), where is the mass of the rock, V0 is the initial volume of water, and V1 is the volume of water
Find the uncertainty in U, assuming that X = 10.0 ± 0.5, Y = 5.0 ± 0.1, and a. U = XY2 b. U = X2 + Y2 c. U = (X + Y2)/2
Refer to Exercise 12 in Section 3.2. Assume that τ0 = 50 ± 1 MPa, w = 1.2 ± 0.1 mm, and k = 0.29 ± 0.05 mm−1. a. Estimate τ, and find the uncertainty in the estimate. b. Which would provide the greatest reduction in the uncertainty in τ: reducing the uncertainty in τ0 to 0.1MPa, reducing
Archaeologists studying meat storage methods employed by the Nunamiut in northern Alaska have developed a Meat Drying Index. Following is a slightly simplified version of the index given in the article "A Zooarchaeological Signature for Meat Storage: Rethinking the Drying Utility Index" (T.
A cylindrical wire of radius R elongates when subjected to a tensile force F. Let L0 represent the initial length of the wire and let L1 represent the final length. Young's modulus for the material is given byAssume that F = 800 ± 1 N, R = 0.75 ± 0.1 mm, L0 = 25.0 ± 0.1 mm, and
Refer to Exercise 16. In an experiment to determine the value of k, the temperature T at time t = 10 min is measured to be T = 54.1 ± 0.2◦F. Assume that T0 = 70.1 ± 0.2◦F and Ta = 35.7 ± 0.1◦F. Estimate k, and find the uncertainty in the estimate.
The shape of a bacterium can be approximated by a cylinder of radius r and height h capped on each end by a hemisphere. The volume and surface area of the bacterium are given byIt is known that the rate R at which a chemical is absorbed into the bacterium is R = c(S/V), where c is a constant of
Refer to Exercise 10 in Section 3.2. Assume that τ = 35.2 ± 0.1 Pa, h = 12.0 ± 0.3 mm, and μ = 1.49 Pa ∙ s with negligible uncertainty. Estimate V, and find the relative uncertainty in the estimate.
Refer to Exercise 7. Assume that p = 4.3 ± 0.1 cm and q = 2.1 ± 0.2 cm. Estimate f, and find the relative uncertainty in the estimate.
Refer to Exercise 12. Estimate n, and find the relative uncertainty in the estimate, from the following measurements: θ1 = 0.216 ± 0.003 radians and θ2 = 0.456 ± 0.005 radians.
Refer to Exercise 15. Assume that F = 750 ± 1 N, R = 0.65 ± 0.09 mm, L0 = 23.7 ± 0.2 mm, and L1 = 27.7 ± 0.2 mm. Estimate Y, and find the relative uncertainty in the estimate.
Refer to Exercise 19. Assume that for a certain bacterium, r = 0.8 ± 0.1 μm and h = 1.9 ± 0.1 μm. a. Estimate S, and find the relative uncertainty in the estimate. b. Estimate V, and find the relative uncertainty in the estimate. c. Estimate R, and find the relative uncertainty in the
From a fixed point on the ground, the distance to a certain tree is measured to be s = 55.2 ± 0.1 m and the angle from the point to the top of the tree is measured to be θ = 0.50 ± 0.02 radians. The height of the tree is given by h = s tan θ. a. Estimate h, and find the uncertainty in the
Refer to Exercise 14. Assume that the relative uncertainty in l is 3% and that the relative uncertainty in d is 2%. Find the relative uncertainty in R.
When air enters a compressor at pressure P1 and leaves at pressure P2, the intermediate pressure is given by P3 = √P1P2. Assume that P1 = 10.1 ± 0.3 MPa and P2 = 20.1 ± 0.4 MPa. a. Estimate P3, and find the uncertainty in the estimate. b. Which would provide a greater reduction in the
The lens equation says that if an object is placed at a distance p from a lens, and an image is formed at a distance q from the lens, then the focal length f satisfies the equation 1/ f = 1/p +1/q. Assume that p = 2.3 ± 0.2 cm and q = 3.1 ± 0.2 cm. a. Estimate f, and find the uncertainty in the
The Beer-Lambert law relates the absorbance A of a solution to the concentration C of a species in solution by A = MLC, where L is the path length and M is the molar absorption coefficient. Assume that C = 1.25 ± 0.03 mol/cm3, L = 1.2 ± 0.1 cm, and A = 1.30 ± 0.05. a. Estimate M and find the
Assume that X, Y, and Z are independent measurements with X = 25 ± 1, Y = 5.0 ± 0.3, and Z = 3.5 ± 0.2. Find the uncertainties in each of the following quantities: a. X + Y Z b. X/(Y − Z) c. X√Y + eZ d. X ln(Y2 + Z)
A laminated item is made up of six layers. The two outer layers each have a thickness of 1.25 ± 0.10mm and the four inner layers each have a thickness of 0.80 ± 0.05 mm. Assume the thicknesses of the layers are independent. Estimate the thickness of the item, and find the uncertainty in the
In the article "Measurements of the Thermal Conductivity and Thermal Diffusivity of Polymer Melts with the Short-Hot-Wire Method" (X. Zhang, W. Hendro, et al., International Journal of Thermo-physics, 2002:1077-1090), the thermal diffusivity of a liquid measured by the transient short-hot-wire
Refer to Exercise 14.Acable is composed of 16 wires. The strength of each wire is 5000 ± 20 lb. a. Will the estimated strength of the cable be the same under the ductile wire method as under the brittle wire method? b. Will the uncertainty in the estimated strength of the cable be the same under
The flow rate of water through a cylindrical pipe is given by Q = πr2v, where r is the radius of the pipe and v is the flow velocity. a. Assume that r = 3.00 ± 0.03 m and v = 4.0 ± 0.2 m/s. Estimate Q, and find the uncertainty in the estimate. b. Assume that r = 4.00 ± 0.04 m and v = 2.0 ± 0.1
The decomposition of nitrogen dioxide (NO2) into nitrogen monoxide (NO) and oxygen is a second-order reaction. This means that the concentration C of NO2 at time t is given by 1/C = kt + 1/C0, where C0 is the initial concentration and k is the rate constant. Assume the initial concentration is
A track has the shape of a square capped on two opposite sides by semicircles. The length of a side of the square is measured to be 181.2 ± 0.1 m. a. Compute the area of the square and its uncertainty. b. Compute the area of one of the semicircles and its uncertainty. c. Let S denote the area of
If X1, X2, . . . , Xn are independent and unbiased measurements of true values Î¼1, Î¼2, . . . , Î¼n , and U(X1, X2, . . . , Xn) is a nonlinear function of 1, X2, . . . , Xn , then in general U(X1, X2, . . . , Xn) is a biased estimate of the true value
An item is to be constructed by laying two components end to end. The length of each component will be measured. a. If the uncertainty in measuring the length of each component is 0.1 mm, what is the uncertainty in the combined length of the two components? b. If it is desired to estimate the
5. The Darcy-Weisbach equation states that the power-generating capacity in a hydroelectric system that is lost due to head loss is given by P = ηγ QH, where η is the efficiency of the turbine, γ is the specific gravity of water, Q is the flow rate, and H is the head loss. Assume that η = 0.85
The heating capacity of a calorimeter is known to be 4 kJ/oC, with negligible uncertainty. The number of dietary calories (kiloCalories) per gram of a substance is given by C = cH(∆T )/m, where C is the number of dietary calories, H is the heating capacity of the calorimeter, ∆T is the increase
The article "Insights into Present-Day Crustal Motion in the Central Mediterranean Area from GPS Surveys" (M. Anzidei, P. Baldi, et al., Geophysical Journal International, 2001:98-100) reports that the components of velocity of the earth's crust in Zimmerwald, Switzerland, are 22.10 ± 0.34 mm/year
After scoring a touchdown, a football team may elect to attempt a two-point conversion, by running or passing the ball into the end zone. If successful, the team scores two points. For a certain football team, the probability that this play is successful is 0.40. a. Let X = 1 if successful, X = 0
When a certain glaze is applied to a ceramic surface, the probability is 5% that there will be discoloration, 20% that there will be a crack, and 23% that there will be either discoloration or a crack, or both. Let X = 1 if there is discoloration, and let X = 0 otherwise. Let Y = 1 if there is a
A penny and a nickel are tossed. Both are fair coins. Let X = 1 if the penny comes up heads, and let X = 0 otherwise. Let Y = 1 if the nickel comes up heads, and let Y = 0 otherwise. Let Z = 1 if both the penny and nickel come up heads, and let Z = 0 otherwise. a. Let pX denote the success
Let X and Y be Bernoulli random variables. Let Z = XY. a. Show that Z is a Bernoulli random variable. b. Show that if X and Y are independent, then pZ = pX pY.
Below are the durations (in minutes) of 40 eruptions of the geyser Old Faithful in Yellowstone National Park.Construct a normal probability plot for these data. Do the data appear to come from an approximately normal distribution?
Construct a normal probability plot for the PM data in Table 1.2 (page 21). Do the PM data appear to come from a normal population?In Table 1.2
Bottles filled by a certain machine are supposed to contain 12 oz of liquid. In fact the fill volume is random with mean 12.01 oz and standard deviation 0.2 oz. a. What is the probability that the mean volume of a random sample of 144 bottles is less than 12 oz? b. If the population mean fill
In a process that manufactures bearings, 90% of the bearings meet a thickness specification. A shipment contains 500 bearings. A shipment is acceptable if at least 440 of the 500 bearings meet the specification. Assume that each shipment contains a random sample of bearings. a. What is the
Radioactive mass A emits particles at a mean rate of 20 per minute, and radioactive mass B emits particles at a mean rate of 25 per minute. a. What is the probability that fewer than 200 particles are emitted by both masses together in a five minute time period? b. What is the probability that mass
The concentration of particles in a suspension is 50 per mL. A 5 mL volume of the suspension is withdrawn. a. What is the probability that the number of particles withdrawn will be between 235 and 265? b. What is the probability that the average number of particles per mL in the withdrawn sample is
A new process has been designed to make ceramic tiles. The goal is to have no more than 5% of the tiles be nonconforming due to surface defects. A random sample of 1000 tiles is inspected. Let X be the number of nonconforming tiles in the sample. a. If 5% of the tiles produced are nonconforming,
Seventy percent of rivets from vendor A meet a certain strength specification, and 80% of rivets from vendor B meet the same specification. If 500 rivets are purchased from each vendor, what is the probability that more than 775 of the rivets meet the specifications?
A simple random sample of 100 men is chosen from a population with mean height 70 in. and standard deviation 2.5 in. What is the probability that the average height of the sample men is greater than 69.5 in?
The breaking strength (in kg/mm) for a certain type of fabric has mean 1.86 and standard deviation 0.27. A random sample of 80 pieces of fabric is drawn. a. What is the probability that the sample mean breaking strength is less than 1.8 kg/mm? b. Find the 80th percentile of the sample mean breaking
A sample of 225 wires is drawn from the population of wires described in Example 4.70. Find the probability that fewer than 110 of these wires have no flaws.
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 = 0.5oC. How many readings must be taken so that the probability is 0.90 that the average is within ±0.1oC of the actual temperature?
Vendor A supplies parts, each of which has probability 0.03 of being defective. Vendor B also supplies parts, each of which has probability 0.05 of being defective. You receive a shipment of 100 parts from each vendor. a. Let X be the number of defective parts in the shipment from vendor A and let
The age of an ancient piece of organic matter can be estimated from the rate at which it emits beta particles as a result of carbon-14 decay. For example, if X is the number of particles emitted in 10 minutes by a 10,000-year-old bone fragment that contains 1 g of carbon, then X has a Poisson
Rectangular plates are manufactured whose lengths are distributed N(2.0, 0.12) and whose widths are distributed N(3.0, 0.22). Assume the lengths and widths are independent. The area of a plate is given by A = XY. a. Use a simulated sample of size 1000 to estimate the mean and variance of A. b.
A system consists of components A and B connected in series, as shown in the following schematic illustration. The lifetime in months of component A is lognormally distributed with Î¼ = 1 and Ïƒ = 0.5, and the lifetime in months of component B is lognormally distributed with
Let X ∼ Bin(8, 0.4). Find a. P(X = 2) b. P(X = 4) c. P(X < 2) d. P(X > 6) e. μX f. σ2X
In a random sample of 100 parts ordered from vendor A, 12 were defective. In a random sample of 200 parts ordered from vendor B, 10 were defective. a. Estimate the proportion of parts from vendor A that are defective, and find the uncertainty in the estimate. b. Estimate the proportion of parts
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 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
Ak 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 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.
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
Porcelain figurines are sold for $10 if flawless, and for $3 if there are minor cosmetic flaws. Of the figurines made by a certain company, 90% are flawless and 10% have minor cosmetic flaws. In a sample of 100 figurines that are sold, let Y be the revenue earned by selling them and let X be the
Find the following probabilities: a. P(X = 3) when X ∼ Bin(5, 0.2) b. P(X ≤ 2) when X ∼ Bin(10, 0.6) c. P(X ≥ 5) when X ∼ Bin(9, 0.5) d. P(3 ≤ X ≤ 4) when X ∼ Bin(8, 0.8)
Of all the registered automobiles in a certain state, 10% violate the state emissions standard. Twelve automobiles are selected at random to undergo an emissions test. a. Find the probability that exactly three of them violate the standard. b. Find the probability that fewer than three of them
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
Several million lottery tickets are sold, and 60% of the tickets are held by women. Five winning tickets will be drawn at random. a. What is the probability that three or fewer of the winners will be women? b. What is the probability that three of the winners will be of one gender and two of the
Let X ∼ Poisson(4). Find a. P(X = 1) b. P(X = 0) c. P(X < 2) d. P(X > 1) e. μX f. σX
A microbiologist wants to estimate the concentration of a certain type of bacterium in a wastewater sample. She puts a 0.5 mL sample of the wastewater on a microscope slide and counts 39 bacteria. Estimate the concentration of bacteria, per mL, in this wastewater, and find the uncertainty in the
The number of defective components produced by a certain process in one day has a Poisson distribution with mean 20. Each defective component has probability 0.60 of being repairable. a. Find the probability that exactly 15 defective components are produced. b. Given that exactly 15 defective
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?
Mom and Grandma are each baking chocolate chip cookies. Each gives you two cookies. One of Mom's cookies has 14 chips in it and the other has 11. Grandma's cookies have 6 and 8 chips. a. Estimate the mean number of chips in one of Mom's cookies. b. Estimate the mean number of chips in one of
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
In a certain city, the number of potholes on a major street follows a Poisson distribution with a rate of 3 per mile. Let X represent the number of potholes in a two-mile stretch of road. Find a. P(X = 4) b. P(X ≤ 1) c. P(5 ≤ X < 8) d. μX e. σX
A sensor network consists of a large number of microprocessors spread out over an area, in communication with each other and with a base station. In a certain network, the probability that a message will fail to reach the base station is 0.005. Assume that during a particular day, 1000 messages are
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 5 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 3
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 3. 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
Twenty air-conditioning units have been brought in for service. Twelve of them have broken compressors, and eight have broken fans. Seven units are chosen at random to be worked on. What is the probability that three of them have broken fans?
In a lot of 10 microcircuits, 3 are defective. Four microcircuits are chosen at random to be tested. Let X denote the number of tested circuits that are defective. a. Find P(X = 2). b. Find μX. c. Find σX.
Ten items are to be sampled from a lot of 60. If more than one is defective, the lot will be rejected. Find the probability that the lot will be rejected in each of the following cases. a. The number of defective items in the lot is 5. b. The number of defective items in the lot is 10. c. The
A certain make of car comes equipped with an engine in one of four sizes (in liters): 2.8, 3.0, 3.3, or 3.8. Ten percent of customers order the 2.8 liter engine, 40% order the 3.0, 30% order the 3.3, and 20% order the 3.8. A random sample of 20 orders is selected for audit. a. What is the
Let X ∼ Geom(p), let n be a non-negative integer, and let Y ∼ Bin(n, p). Show that P(X = n) = (1 / n) P(Y = 1).
The probability that a computer running a certain operating system crashes on any given day is 0.1. Find the probability that the computer crashes for the first time on the twelfth day after the operating system is installed.
Refer to Exercise 4. Let Y denote the number of days up to and including the third day on which a red light is encountered. a. Find P(Y = 7). b. Find μY. c. Find σ2Y.
A system is tested for faults once per hour. If there is no fault, none will be detected. If there is a fault, the probability is 0.8 that it will be detected. The tests are independent of one another. a. If there is a fault, what is the probability that it will be detected in 3 hours or less? b.
Find the area under the normal curve a. To the right of z = −0.85. b. Between z = 0.40 and z = 1.30. c. Between z = −0.30 and z = 0.90. d. Outside z = −1.50 to z = −0.45.

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