Questions and Answers of Statistics Engineers Scientists

Let X1, . . . , Xn be a random sample from a N(μ, σ2) population. Find the MLEs of μ and of σ. The likelihood function is a function of two parameters, μ and σ. Compute partial derivatives with
A quality engineer takes a random sample of 100 steel rods from a day’s production, and finds that 92 of them meet specifications.a. Estimate the proportion of that day’s production that meets
The stiffness of a certain type of steel beam used in building construction has mean 30 kN/mm and standard deviation 2 kN/mm. a. Is it possible to compute the probability that the stiffness of a
Among the adults in a large city, 30% have a college degree. A simple random sample of 100 adults is chosen. What is the probability that more than 35 of them have a college degree?
Let X1, . . . , Xn be independent log normal random variables and let a1, . . . , an be constants. Show that the product P = Xa11· · · Xann is lognormal. ln P = a1 ln X1 +· · ·+an ln Xn.
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
A lot of parts contains 500 items, 100 of which are defective. Suppose that 20 items are selected at random. Let X be the number of selected items that are defective.a. Express the quantity P(X = 5)
Someone suggests that the lifetime T (in days) of a certain component can be modeled with the Weibull distribution with parameters α = 3 and β = 0.01.a. If this model is correct, what is P(T ≤
Refer to Exercise 8.a. Assume that P = 224.51 ± 0.04 kPa and V = 11.237 ± 0.002 L. Estimate T, and find the relative uncertainty in the estimate.b. Assume that P = 224.51 ± 0.04 kPa and T =
Refer to Exercise 5. Assume that P1 = 15.3 ± 0.2 MPa and P2 = 25.8 ± 0.1 MPa. Estimate P3, and find the relative uncertainty in the estimate.Refer to Exercise 5.When air enters a compressor at
If X is an unbiased measurement of a true value μX, and U(X) is a nonlinear function of X, then in most cases U is a biased estimate of the true value U(μX ). In most cases this bias is ignored. If
Estimate U, and find the relative uncertainty in the estimate, assuming that X = 5.0 ± 0.2, Y = 10.0 ± 0.5, anda. U = X√Yb. U = 2Y/√Xc. U = X2 + Y2
The lengths of two components will be measured several times. The uncertainty in each measurement of the length of the first component is σ1 = 0.02 cm, and the uncertainty in each measurement of the
Two students want to measure the acceleration a of a cart rolling down an inclined plane. The cart starts at rest and travels a distance s down the plane. The first student estimates the acceleration
The vertical displacement v of a cracked slurry infiltrated fiber concrete member at maximum shear stress is given by v = a + bw, where w is the crack width, and a and b are estimated from data to be
According to Newton€™s law of cooling, the time t needed for an object at an initial temperature T0to cool to a temperature T in an environment with ambient temperature Ta is given by where
Refer to Exercise 6. Assume that c = 448 J/kg◦C and ΔQ = 1210 J are known with negligible uncertainty. Assume the mass is m = 0.75 ± 0.01 kg. Estimate ΔT , and find the relative uncertainty in
A certain scale has an uncertainty of 3 g and a bias of 2 g.a. A single measurement is made on this scale. What are the bias and uncertainty in this measurement?b. Four independent measurements are
A cable is made up of several parallel strands of wire. The strength of the cable can be estimated from the strengths of the individual wires by either of two methods. In the ductile wire method, the
Nine independent measurements are made of the length of a rod. The average of the nine measurements is X̅ = 5.238 cm, and the standard deviation is s = 0.081 cm.a. Is the uncertainty in the value
According to Snell’s law, the angle of refraction θ2 of a light ray traveling in a medium of index of refraction n is related to the angle of incidence θ1 of a ray traveling in a vacuum through
Convert the following relative uncertainties to absolute uncertainties.a. 48.41 ± 0.3%b. 991.7 ± 0.6%c. 0.011 ± 9%d. 7.86 ± 1%
In the article “Influence of Crack Width on Shear Behaviour of SIFCON” (C. Fritz and H. Reinhardt, High Performance Fiber Reinforced Cement Composites: Proceedings of the International RILEM/ACI
A piece of plywood is composed of five layers. The two outer layers are veneers with thickness 0.50 ± 0.02 mm, and the three inner layers each have thickness 6.25 ± 0.05 mm. Assume the thicknesses
In the article “Temperature-Dependent Optical Constants of Water Ice in the Near Infrared: New Results and Critical Review of the Available Measurements” (B. Rajaram, D. Glandorf, et al., Applied
The conversion of ammonium cyanide to urea is a second-order reaction. This means that the concentration C of ammonium cyanide at time t is given by 1/C = kt+1/C0, where C0 is the initial
If two gases have molar masses M1 and M2, Graham’s law states that the ratio R of their rates of effusion through a small opening is given by R = √M1/M2. The effusion rate of an unknown gas
The article “Calibration of an FTIR Spectrometer” (P. Pankratz, Statistical Case Studies for Industrial and Process Improvement, SIAM-ASA, 1997: 19–38) describes the use of a spectrometer to
A new and unknown weight is weighed on the same scale that was used in Exercise 8, and the measurement is 127 μg above 1 kg. Using the information in Exercise 8, is it possible to come up with a
The pressure P, temperature T, and volume V of one mole of an ideal gas are related by the equation PV = 8.31T, when P is measured in kilopascals, T is measured in kelvins, and V is measured in
The refractive index n of a piece of glass is related to the critical angle θ by n = 1/ sin θ. Assume that the critical angle is measured to be 0.70 ± 0.02 rad. Estimate the refractive index, and
In a Couette flow, two large flat plates lie one on top of another, separated by a thin layer of fluid. If a shear stress is applied to the top plate, the viscosity of the fluid produces motion in
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. Thus if L and T are measured, we can estimate g with g =
Sixteen independent measurements were made of the resistance of a resistor. The average was 52.37 and the standard deviation was 0.12 Ω.a. Estimate the resistance of this resistor, and find the
The Kilogram is now weighed five times on a different scale. The measurements are as follows, in units of micrograms above 1 kg: 25.6, 26.8, 26.2, 26.8, 25.4.a. Is it possible to estimate the
In a hypothetical scenario, the National Institute of Standards and Technology has received a new replica of The Kilogram. It is weighed five times. The measurements are as follows, in units of
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 change in temperature of an iron bar brought about by a transfer of heat is given by ΔT = ΔQ/mc, where ΔQ is the amount of heat transferred, m is the mass of the bar, and c is the specific
A cylindrical hole is bored through a steel block, and a cylindrical piston is machined to fit into the hole. The diameter of the hole is 20.00 ± 0.01 cm, and the diameter of the piston is 19.90 ±
Let A and B represent two variants (alleles) of the DNA at a certain locus on the genome. Let p represent the proportion of alleles in a population that are of type A, and let q represent the
Refer to Exercise 9.a. Find the mean of the total number of unacceptable bolts (those that are classified as either downgraded or scrap).b. Find the variance of the total number of unacceptable
Curing times in days (x) and compressive strengths in MPa (y) were recorded for several concrete specimens. The means and standard deviations of the x and y values were x̄ = 5, sx = 2, ȳ = 1350, sy
The article €œThe Selection of Yeast Strains for the Production of Premium Quality South African Brandy Base Products€ (C. Steger and M. Lambrechts, Journal of Industrial
In a study of copper bars, the relationship between shear stress in ksi (x) and shear strain in % (y) was summarized by the least-squares line y = −20.00 + 2.56x. There were a total of n = 17
A scatterplot contains four points: (−2,−2), (−1,−1), (0,0), and (1,1). A fifth point, (2,y), is to be added to the plot. Let r represent the correlation between x and y.a. Find the value of
The article ??Approach to Confidence Interval Estimation for Curve Numbers?? (R. McCuen, Journal of Hydrologic Engineering, 2002:43??48) discusses the relationship between rainfall depth and runoff
In the manufacture of synthetic fiber, the fiber is often ??set?? by subjecting it to high temperatures. The object is to improve the shrinkage properties of the fiber. In a test of 25 yarn
An engineer wants to predict the value for y when x = 4.5, using the following data set.a. Construct a scatterplot of the points (x, y).b. Should the least-squares line be used to predict the value
In the following MINITAB output, some of the numbers have been accidentally erased. Recompute them, using the numbers still available. There are n = 20 points in the data set. Predictor Constant
A mixture of sucrose and water was heated on a hot plate, and the temperature (in °C) was recorded each minute for 20 minutes by three thermocouples. The results are shown in the following
The article €œPolyhedral Distortions in Tourmaline€ (A. Ertl, J. Hughes, et al., The Canadian Mineralogist, 2002: 153€“162) presents a model for calculating bond length
Use Equation (7.34) (page 545) to show that μβ̂1= β1. Equation (7.34) (X; – X) Bi = E Yi i=1
Use Equation (7.35) (page 545) to show that Î¼Î²Ì‚0= Î²0Equation (7.35). ы п T(x; — х) Bo = E i=1 Ei-1 (x; – X)², Уi
Use Equation (7.34) (page 545) to derive the formula Equation (7.34) .2 E (x; – T)? B1 i=1 ||
Use Equation (7.35) (page 545) to derive the formula Equation (7.35) o? =o? Po .2 E (x; – X)? in n i31 +
The article referred to in Exercise 23 presents values for the dependent and independent variables for 10 additional construction jobs. These values are presented in Tables SE24A and SE24B (page
The following data were collected in an experiment to study the relationship between extrusion pressure (in KPa) and wear (in mg).The least-squares quadratic model is y = ˆ’32.445714 +
The article €œAdvances in Oxygen Equivalence Equations for Predicting the Properties of Titanium Welds€ (D. Harwig, W. Ittiwattana, and H. Castner, The Welding Journal,
In a laboratory test of a new engine design, the emissions rate (in mg/s of oxides of nitrogen, NOx) was measured as a function of engine speed (in rpm), engine torque (in ft · lb), and total
The article €œExperimental Design Approach for the Optimization of the Separation of Enantiomers in Preparative Liquid Chromatography€ (S. Lai and Z. Lin, Separation Science and Technology,
Refer to Exercise 1.a. Find a 95% confidence interval for the coefficient of Manganese.b. Find a 99% confidence interval for the coefficient of Thickness.c. Can you conclude that β1> 3? Perform
Refer to Exercise 1. a. Find a 95% confidence interval for the coefficient of Oxygen. b. Find a 99% confidence interval for the coefficient of Nitrogen. c. Find a 98% confidence interval for the
In the article referred to in Exercise 2, a model was fit to investigate the relationship between the independent variables given in Exercise 2 and the amount of S-isomer collected. The results of a
The article €œInfluence of Freezing Temperature on Hydraulic Conductivity of Silty Clay€ (J. Konrad and M. Samson, Journal of Geotechnical and Geoenvironmental Engineering,
An engineer measures a dependent variable y and independent variables x1, x2, and x3. MINITAB output for the model y = Î²0+ Î²1x1+ Î²2x2+ Î²3x3+
The article “Application of Analysis of Variance to Wet Clutch Engagement” (M. Mansouri, M. Khonsari, et al., Proceedings of the Institution of Mechanical Engineers, 2002:117–125)
An engineer tries three different methods for selecting a linear model. First she uses an informal method based on the F statistic, as described in Section 8.3. Then she runs the best subsets
The article €œSimultaneous Optimization of Mechanical Properties of Steel by Maximizing Exponential Desirability Functions€ (K. J. Kim and D. K. J. Lin, Journal of the Royal
The following table lists values for three variables measured for 60 consecutive eruptions of the geyser Old Faithful in Yellowstone National Park. They are the duration of the eruption (x1), the
The following MINITAB output is for a best subsets regression involving five dependent variables X1, . . . , X5. The two models of each size with the highest values of R2are listed. a. Which
The article ??Earthmoving Productivity Estimation Using Linear Regression Techniques?? (S. Smith, Journal of Construction Engineering and Management, 1999:133??141) presents the following linear
The voltage output (y) of a battery was measured over a range of temperatures (x) from 0°C to 50°C. The following figure is a scatterplot of voltage versus temperature, with three fitted
The data in Table SE6 (page 649) consist of yield measurements from many runs of a chemical reaction. The quantities varied were the temperature in ?C (x1), the concentration of the primary reactant
The following is supposed to be the result of a best subsets regression involving five independent variables X1, . . . , X5. The two models of each size with the highest values of R2are listed.
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
Refer to Exercise 7. a. Find a 95% confidence interval for the coefficient of Last FEV. b. Find a 98% confidence interval for the coefficient of Height. c. Can you conclude that β2 < ??0.08?
The article €œDrying of Pulps in Sprouted Bed: Effect of Composition on Dryer Performance€ (M. Medeiros, S. Rocha, et al., Drying Technology, 2002:865€“881) presents
In a study to determine the effect of vehicle weight in tons (x1) and engine displacement in in3 (x2) on fuel economy in miles per gallon (y), these quantities were measured for ten automobiles. The
This exercise illustrates a reason for the exceptions to the rule of parsimony.a. A scientist fits the model Y = β1C + ε, where C represents temperature in °C and Y can represent any outcome. Note
A scientist has measured quantities y, x1, and x2. She believes that y is related to x1 and x2 through the equation y = αeβ1 x1+β2 x2 δ, where δ is a random error that is always positive. Find a
The following MINITAB output is for a multiple regression. Something went wrong with the printer, so some of the numbers are missing. Fill in the missing numbers. Predictor Сoef SE Coef P Constant
In rock blasting, the peak particle velocity (PPV) depends both on the distance from the blast and on the amount of charge. The article ??Prediction of Particle Velocity Caused by Blasting for an
The following MINITAB output is for a multiple regression. Some of the numbers got smudged and are illegible. Fill in the missing numbers. Predictor Coef SE Coef T P Constant (a) 1.4553 5.91 0.000
The article ??Permanent Deformation Characterization of Subgrade Soils from RLT Test?? (A. Puppala, L. Mohammad, et al., Journal of Materials in Civil Engineering, 1999:274??282) presents
In the article “Low-Temperature Heat Capacity and Thermodynamic Properties of 1,1,1-trifluoro-2, 2-dichloroethane” (R.Varushchenko and A. Druzhinina, Fluid Phase Equilibria, 2002:109–119), the
The article €œVehicle-Arrival Characteristics at Urban Uncontrolled Intersections€ (V. Rengaraju and V. Rao, Journal of Transportation Engineering, 1995:317€“323)
The following table presents measurements of mean noise levels in dBA (y), roadway width inm(x1), and mean speed in km/h (x2), for 10 locations in Bangkok, Thailand, as reported in the article
The following data were collected in an experiment to study the relationship between the speed of a cutting tool in m/s (x) and the lifetime of the tool in hours (y).The least-squares quadratic model
The article €œElectrical ImpedanceVariation withWater Saturation in Rock€ (Q. Su, Q. Feng, and Z. Shang, Geophysics, 2000:68€“75) reports measurements of permeabilities (in
The article “Models for Assessing Hoisting Times of Tower Cranes” (A. Leung and C. Tam, Journal of Construction Engineering and Management, 1999: 385–391) presents a model constructed by a step
The article €œMultiple Linear Regression for Lake Ice and Lake Temperature Characteristics€ (S. Gao and H. Stefan, Journal of Cold Regions Engineering, 1999:59€“77)
The article ??Low-Temperature Heat Capacity and Thermodynamic Properties of 1,1,1-trifluoro-2, 2-dichloroethane?? (R. Varushchenko and A. Druzhinina, Fluid Phase Equilibria, 2002:109??119) presents
The article ??The Ball-on-Three-Ball Test for Tensile Strength: Refined Methodology and Results for Three Hohokam Ceramic Types?? (M. Beck, American Antiquity, 2002:558??569) discusses the strength
Piecewise linear model: Let xÌƒ be a knownconstant, and suppose that a dependent variable y is related to an independent variable x1as follows:In other words, y and x1 are linearly
The article €œSeismic Hazard in Greece Based on Different Strong Ground Motion Parameters€ (S. Koutrakis, G. Karakaisis, et al., Journal of Earthquake Engineering, 2002:75€“109) presents a
The article ??Estimating Resource Requirements at Conceptual Design Stage Using Neural Networks?? (A. Elazouni, I. Nosair, et al., Journal of Computing in Civil Engineering, 1997:217??223) suggests
Copper wires are coated with a thin plastic coating. Samples of four wires are taken every hour, and the thickness of the coating (in mils) is measured. The data from the last 30 samples are
Repeat Exercise 3, using the S chart in place of the R chart. Repeat Exercise 3 The means are X̅ = 2.505, R̅ = 0.1395, and s̅ = 0.057. a. Calculate the 3σ control limits for the R̅ chart. Is
Someone constructs an X̅ chart where the control limits are at ±2.0σX̅ rather than at ±3σX̅ .a. If the process is in control, what is the ARL for this chart?b. If the process mean shifts by
The target fill weight for a box of cereal is 350 g. Each day a sample of 300 boxes is taken, and the number that are underweight is counted. The number of underweight boxes for each of the last 25

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