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Questions and Answers of
Statistics
Reconsider the situation described in Exercise 7, in which x = accelerated strength of concrete and y = 28day cured strength. Suppose the simple linear regression model is valid for x between 1000
During oil drilling operations, components of the drilling assembly may suffer from sulfide stress cracking. The article "Composition Optimization of High-Strength Steels for Sulfide Cracking
Exercise 16 of Section 12.2 gave data on x = rainfall volume and y = runoff volume (both in m3). Use the accompanying Minitab output to decide whether there is a useful linear relationship between
Exercise 15 of Section 12.2 included Minitab output for a regression of flexural strength of concrete beams on modulus of elasticity. a. Use the output to calculate a confidence interval with a
Electromagnetic technologies offer effective nondestructive sensing techniques for determining characteristics of pavement. The propagation of electromagnetic waves through the material depends on
How does lateral acceleration-side forces experienced in turns that are largely under driver control-affect nausea as perceived by bus passengers? The article "Motion Sickness in Public Road
Mist (airborne droplets or aerosols) is generated when metalremoving fluids are used in machining operations to cool and lubricate the tool and workpiece. Mist generation is a concern to OSHA, which
Magnetic resonance imaging (MRI) is well established as a tool for measuring blood velocities and volume flows. The article "Correlation Analysis of Stenotic Aortic Valve Flow Patterns Using Phase
Refer to the data on x = liberation rate and y = NOx emission rate given in Exercise 19.a. Does the simple linear regression model specify a useful relationship between the two rates? Use the
Carry out the model utility test using the ANOVA approach for the filtration rate-moisture content data of Example 12.6. Verify that it gives a result equivalent to that of the t test?
The accompanying data on y = ammonium concentration (mg/L) and x = transpiration (ml/h) was read from a graph in the article "Response of Ammonium Removal to Growth and Transpiration of Juncus
Use the rules of expected value to show that β0 is an unbiased estimator for β0 (assuming that β1 is unbiased for β1)?
a. Verify that E(1) = β1 by using the rules of expected value from Chapter 5. b. Use the rules of variance from Chapter 5 to verify the expression for V (1) given in this section?
Verify that if each xi is multiplied by a positive constant c and each yi is multiplied by another positive constant d, the t statistic for testing H0: β1 = 0 versus Ha: β1 ≠ 0 is unchanged in
The probability of a type II error for the t test for H0: β1 = β10 can be computed in the same manner as it was computed for the t tests of Chapter 8. If the alternative value
Fitting the simple linear regression model to the n = 27 observations on x = modulus of elasticity and y = flexural strength given in Exercise 15 of Section 12.2 resulted in = 7.592, sY^ = .179 when
Reconsider the filtration rate-moisture content data introduced in Example 12.6 (see also Example 12.7). a. Compute a 90% CI for β1 + 125 β1, true average moisture content when the filtration rate
Astringency is the quality in a wine that makes the wine drinker's mouth feel slightly rough, dry, and puckery. The paper "Analysis of Tannins in Red Wine Using Multiple Methods: Correlation with
The simple linear regression model provides a very good fit to the data on rainfall and runoff volume given in Exercise 16 of Section 12.2. The equation of the least squares line is y = 21.128 +
The catch basin in a storm-sewer system is the interface between surface runoff and the sewer. The catch-basin insert is a device for retrofitting catch basins to improve pollutantremoval properties.
You are told that a 95% CI for expected lead content when traffic flow is 15, based on a sample of n = 10 observations, is (462.1, 597.7). Calculate a CI with confidence level 99% for expected lead
The article "Objective Measurement of the Stretchability of Mozzarella Cheese" (J. of Texture Studies, 1992: 185-194) reported on an experiment to investigate how the behavior of mozzarella cheese
Silicon-germanium alloys have been used in certain types of solar cells. The paper "Silicon-Germanium Films Deposited by Low-Frequency Plasma-Enhanced Chemical Vapor Deposition" (J. of Material Res.,
Refer to Example 12.12 in which x = test track speed and y = rolling test speed. a. Minitab gave S 0 + 0 (45) = .120 and S 0 + 0 (47) = .186. Why is the former estimated standard deviation
Plasma etching is essential to the fine-line pattern transfer in semiconductor processes. The article "Ion Beam- Assisted Etching of Aluminum with Chlorine" (J. of the Electrochem. Soc., 1985:
Consider the following four intervals based on the data of Exercise 12.17 (Section 12.2): a. A 95% CI for mean porosity when unit weight is 110 b. A 95% PI for porosity when unit weight is 110 c. A
The height of a patient is useful for a variety of medical purposes, such as estimating tidal volume of someone in an intensive care who requires artificial ventilation. However, it can be difficult
Verify that V(0 + ix) is indeed given by the expression in the text?
The article "Bone Density and Insertion Torque as Predictors of Anterior Cruciate Ligament Graft Fixation Strength" (The Amer. J. of Sports Med., 2004: 1421-1429) gave the accompanying data on
The article "Behavioural Effects of Mobile Telephone Use During Simulated Driving" (Ergonomics, 1995: 2536-2562) reported that for a sample of 20 experimental subjects, the sample correlation
The Turbine Oil Oxidation Test (TOST) and the Rotating Bomb Oxidation Test (RBOT) are two different procedures for evaluating the oxidation stability of steam turbine oils. The article "Dependence of
Toughness and fibrousness of asparagus are major determinants of quality. This was the focus of a study reported in "Post-Harvest Glyphosphate Application Reduces Toughening, Fiber Content, and
The authors of the paper "Objective Effects of a Six Months' Endurance and Strength Training Program in Outpatients with Congestive Heart Failure" (Medicine and Science in Sports and Exercise, 1999:
The article "Quantitative Estimation of Clay Mineralogy in Fine-Grained Soils" (J. of Geotechnical and Geoenvironmental Engr., 2011: 997-1008) reported on various chemical properties of natural and
Physical properties of six flame-retardant fabric samples were investigated in the article "Sensory and Physical Properties of Inherently Flame-Retardant Fabrics" (Textile Research, 1984: 61-68). Use
The accompanying data on x = UV transparency index and y = maximum prevalence of infection was read from a graph in the article "Solar Radiation Decreases Parasitism in Daphnia" (Ecology Letters,
Torsion during hip external rotation and extension may explain why acetabular labral tears occur in professional athletes. The article "Hip Rotational Velocities During the Full Golf Swing" (J. of
Consider a time series-that is, a sequence of observations X1, X2,... obtained over time-with observed values x1, x2,..., xn. Suppose that the series shows no upward or downward trend over time. An
A sample of n = 500 (x, y) pairs was collected and a test of H0: ( = 0 versus Ha: ( ≠ 0 was carried out. The resulting P-value was computed to be .00032. a. What conclusion would be appropriate at
The appraisal of a warehouse can appear straightforward compared to other appraisal assignments. A warehouse appraisal involves comparing a building that is primarily an open shell to other such
Refer to the previous exercise, which gives data on truss heights for a sample of warehouses and the corresponding sale prices. a. Estimate the true average change in sale price associated with a
The article "Some Field Experience in the Use of an Accelerated Method in Estimating 28-Day Strength of Concrete" (J. of Amer. Concrete Institute, 1969: 895) considered regressing y = 28day
Forensic scientists are often interested in making a measurement of some sort on a body (alive or dead) and then using that as a basis for inferring something about the age of the body. Consider the
Phenolic compounds are found in the effluents of coal conversion processes, petroleum refineries, herbicide manufacturing, and fiberglass manufacturing. These compounds are toxic, carcinogenic, and
The SAS output at the bottom of this page is based on data from the article "Evidence for and the Rate of Denitrification in the Arabian Sea" (Deep Sea Research, 1978: 431-435). The variables under
The presence of hard alloy carbides in high chromium white iron alloys results in excellent abrasion resistance, making them suitable for materials handling in the mining and materials processing
The accompanying data was read from a scatter-plot in the article "Urban Emissions Measured with Aircraft" (J. of the Air and Waste Mgmt. Assoc., 1998: 16-25). The response variable is (NOy, and the
An investigation was carried out to study the relationship between speed (ft/sec) and stride rate (number of steps taken/sec) among female marathon runners. Resulting summary quantities included n =
"Mode-mixity" refers to how much of crack propagation is attributable to the three conventional fracture modes of opening, sliding, and tearing. For plane problems, only the first two modes are
Open water oil spills can wreak terrible consequences on the environment and be expensive to clean up. Many physical and biological methods have been developed to recover oil from water surfaces. The
In Section 12.4, we presented a formula for V(0 + 1x*) and a CI for 0 + 1x*. Taking x* = 0 gives (20 and a CI for 0. Use the data of Example 12.11 to calculate the estimated standard deviation
Show that SSE = Syy - 1 Sxy, which gives an alternative computational formula for SSE?
Referring to Exercise 7, The article "Some Field Experience in the Use of an Accelerated Method in Estimating 28-Day Strength of Concrete" (J. of Amer. Concrete Institute, 1969: 895) considered
Suppose that x and y are positive variables and that a sample of n pairs results in r ≈ 1. If the sample correlation coefficient is computed for the (x, y2) pairs, will the resulting value also be
Let sx and sy denote the sample standard deviations of the observed x's and y's, respectively [that is, s2x = ((xi - xÌ )2 / (n - 1) and similarly for s2y].a. Show that an alternative
Verify that the t statistic for testing H0: β1 = 0 in Section 12.3 is identical to the t statistic in Section 12.5 for testing H0: ( = 0?
Use the formula for computing SSE to verity that r2 = 1 - SSE / SST?
In biofiltration of wastewater, air discharged from a treatment facility is passed through a damp porous membrane that causes contaminants to dissolve in water and be
Normal hatchery processes in aquaculture inevitably produce stress in fish, which may negatively impact growth, reproduction, flesh quality, and susceptibility to disease. Such stress manifests
The article "Evaluating the BOD POD for Assessing Body Fat in Collegiate Football Players" (Medicine and Science in Sports and Exercise, 1999: 1350-1356) reports on a new air displacement device for
Reconsider the situation of Exercise 73, in which x = retained austenite content using a garnet abrasive and y = abrasive wear loss were related via the simple linear regression model Y =
The flow rate y (m3 / min) in a device used for air-quality measurement depends on the pressure drop x (in. of water) across the device's filter. Suppose that for x values between 5 and 20, the two
Suppose the variables x = commuting distance and y = comuting time are related according to the simple linear regression model with s = 10. a. If n = 5 observations are made at the x values x1 = 5,
a. Show that Σnn=1 ei when the ei's are the residuals from a simple linear regression. b. Are the residuals from a simple linear regression independent of one another, positively correlated, or
a. Express the ith residual Yi - Ŷi (where Ŷi = β̂0 + β̂1xi) in the form ΣjYj, a linear function of the Yj's. Then use rules of variance to verify that V(Yi - Ŷ) id is given by Expression
a. Could a linear regression result in residuals 23, -27, 5, 17, -8, 9, and 15? Why or why not? b. Could a linear regression result in residuals 23, -27, 5, 17, -8, -12, and 2 corresponding to x
Recall that βÌ0 + βÌ1x has a normal distribution with expected value β0 + β1x and varianceso that has a standard normal
If there is at least one x value at which more than one observation has been made, there is a formal test procedure for testing H0: µY.x = β0 + β1x for some values
No tortilla chip aficionado likes soggy chips, so it is important to find characteristics of the production process that produce chips with an appealing texture. The following data on x 5 frying time
Polyester fiber ropes are increasingly being used as components of mooring lines for offshore structures in deep water. The authors of the paper "Quantifying the Residual Creep Life of Polyester
The following data on mass rate of burning x and flame length y is representative of that which appeared in the article "Some Burning Characteristics of Filter Paper" (Combustion Science and
Failures in aircraft gas turbine engines due to high cycle fatigue is a pervasive problem. The article "Effect of Crystal Orientation on Fatigue Failure of Single Crystal Nickel Base Turbine Blade
Thermal endurance tests were performed to study the relationship between temperature and lifetime of polyester enameled wire ("Thermal Endurance of Polyester Enameled Wires Using Twisted Wire
The x values and standardized residuals for the chlorine flow/etch rate data of Exercise 52 (Section 12.4) are displayed in the accompanying table. Construct a standardized residual plot and comment
Exercise 14 presented data on body weight x and metabolic clearance rate/body weight y. Consider the following intrinsically linear functions for specifying the relationship between the two
Mineral mining is one of the most important economic activities in Chile. Mineral products are frequently found in saline systems composed largely of natural nitrates. Freshwater is often used as a
In each of the following cases, decide whether the given function is intrinsically linear. If so, identify x' and y', and then explain how a random error term e can be introduced to yield an
Suppose x and y are related according to a probabilistic exponential model Y = aeβx.e, with V(e) a constant independent of x (as was the case in the simple linear model Y = β0 + β1x + e). Is V(Y)
Kyphosis refers to severe forward flexion of the spine following corrective spinal surgery. A study carried out to determine risk factors for kyphosis reported the accompanying ages (months) for 40
The article "Acceptable Noise Levels for Construction Site Offices" (Building Serv. Engr. Res. Tech., 2009: 87-94) analyzed responses from a sample of 77 individuals, each of whom was asked to say
The article "Physical Properties of Cumin Seed" (J. of Agric. Engr. Res., 1996: 93-98) considered a quadratic regression of y 5 bulk density on x 5 moisture content. Data from a graph in the article
The following data on y = glucose concentration (g/L) and x = fermentation time (days) for a particular blend of malt liquor was read from a scatterplot in the article "Improving Fermentation
The viscosity (y) of an oil was measured by a cone and plate viscometer at six different cone speeds (x). It was assumed that a quadratic regression model was appropriate, and the estimated
High-alumina refractory castables have been extensively investigated in recent years because of their significant advantages over other refractory brick of the same class-lower production and
Example 12.6 presented the residuals from a simple linear regression of moisture content y on filtration rate x. a. Plot the residuals against x. Does the resulting plot suggest that a straight-line
The accompanying data was extracted from the article "Effects of Cold and Warm Temperatures on Springback of Aluminum-Magnesium Alloy 5083- H111" (J. of Engr. Manuf., 2009: 427-431). The response
The accompanying data on y = energy output (W) and x 5 temperature difference (°K) was provided by the authors of the article "Comparison of Energy and Exergy Efficiency for Solar Box and
The following data is a subset of data obtained in an experiment to study the relationship between x = soil pH and y = A1 Concentration/EC ("Root Responses of Three Gramineae Species to Soil Acidity
In many polynomial regression problems, rather than fitting a "centered" regression function using x' = x - xÌ , computational accuracy can be improved by using a function of the
The following data resulted from an experiment to assess the potential of unburnt colliery spoil as a medium for plant growth. The variables are x 5 acid extractable cations and y = exchangeable
The article "The Respiration in Air and in Water of the Limpets Patella caerulea and Patella lusitanica" (Comp. Biochemistry and Physiology, 1975: 407-411) proposed a simple power model for the
Cardiorespiratory fitness is widely recognized as a major component of overall physical well-being. Direct measurement of maximal oxygen uptake (VO2max) is the single best measure of such fitness,
A trucking company considered a multiple regression model for relating the dependent variable y = total daily travel time for one of its drivers (hours) to the predictors x1 = distance traveled
Let y = wear life of a bearing, x1 = oil viscosity, and x2 = load. Suppose that the multiple regression model relating life to viscosity and load is Y = 125.0 + 7.75x1 + .0950x2 - .0090x1x2 + e a.
Let y = sales at a fast-food outlet (1000s of $), x1 = number of competing outlets within a 1-mile radius, x2 = population within a 1-mile radius (1000s of people), and x3 be an indicator variable
The accompanying data on y = normalized energy (J/m2) and x = intraocular pressure (mmHg) appeared in a scatterplot in the article "Evaluating the Risk of Eye Injuries: Intraocular Pressure During
The article cited in Exercise 49 of Chapter 7 gave summary information on a regression in which the dependent variable was power output (W) in a simulated 200-m race and the predictors were x1 = arm
Researchers carried out a study to see how y = ultimate deflection (mm) of reinforced ultrahigh toughness cementitious composite beams were influenced by x1 = shear span ratio and x2 = splitting
As the air temperature drops, river water becomes supercooled and ice crystals form. Such ice can significantly affect the hydraulics of a river. The article "Laboratory Study of Anchor Ice Growth"
When the model Y = β0 = β1x1 + β2x2 + β3x21 + β4x22 + β5x1x2 + ε is fit to the data of Exercise 49, the resulting value of SSE is 28.947. Given that the predictors x1, x2, and x1x2 remain in
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