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Questions and Answers of
Statistics
Numerous factors contribute to the smooth running of an electric motor ("Increasing Market Share Through Improved Product and Process Design: An Experimental Approach," Quality Engineering, 1991:
An article in the British scientific journal Nature ("Sucrose Induction of Hepatic Hyperplasia in the Rat," August 25, 1972: 461) reports on an experiment in which each of five groups consisting of
Referring to Exercise 38, An article in the British scientific journal Nature ("Sucrose Induction of Hepatic Hyperplasia in the Rat," August 25, 1972: 461) reports on an experiment in which each of
It is common practice in many countries to destroy (shred) refrigerators at the end of their useful lives. In this process material from insulating foam may be released into the atmosphere. The
Refer to Exercise 38. An article in the British scientific journal Nature ("Sucrose Induction of Hepatic Hyperplasia in the Rat," August 25, 1972: 461) reports on an experiment in which each of five
Four laboratories (1-4) are randomly selected from a large population, and each is asked to make three determinations of the percentage of methyl alcohol in specimens of a compound taken from a
The critical flicker frequency (cff) is the highest frequency (in cycles/sec) at which a person can detect the flicker in a flickering light source. At frequencies above the cff, the light source
Let c1, c2,..., cI be numbers satisfying (ci = 0. Then (ciµi = c1µ1 + ...+ cIµI is called a contrast in the µi's. Notice that with c1 = 1, c2 = -1, c3 =... = cI = 0, (ciµi = µ1 - µ2 which
Four types of mortars-ordinary cement mortar (OCM), polymer impregnated mortar (PIM), resin mortar (RM), and polymer cement mortar (PCM)-were subjected to a compression test to measure strength
Suppose the xij's are "coded" by yij = cxij + d. How does the value of the F statistic computed from the yij's compare to the value computed from the xij's? Justify your assertion.
In Example 10.11, subtract x̅i. from each observation in the ith sample (i = 1, ..., 6) to obtain a set of 18 residuals. Then construct a normal probability plot and comment on the plausibility of
Consider the following summary data on the modulus of elasticity (Ã 106 psi) for lumber of three different grades [in close agreement with values in the article "Bending Strength and
The article "Origin of Precambrian Iron Formations" (Econ. Geology, 1964: 1025-1057) reports the following data on total Fe for four types of iron formation (1 = carbonate, 2 = silicate, 3 =
An experiment was carried out to compare electrical resistivity for six different low-permeability concrete bridge deck mixtures. There were 26 measurements on concrete cylinders for each mixture;
A study of the properties of metal plate-connected trusses used for roof support ("Modeling Joints Made with Light-Gauge Metal Connector Plates," Forest Products J., 1979: 39-44) yielded the
Six samples of each of four types of cereal grain grown in a certain region were analyzed to determine thiamin content, resulting in the following data (mg/g):Does this data suggest that at least two
An experiment was carried out to investigate the effect of species (factor A, with I = 4) and grade (factor B, with J = 3) on breaking strength of wood specimens. One observation was made for each
The strength of concrete used in commercial construction tends to vary from one batch to another. Consequently, small test cylinders of concrete sampled from a batch are "cured" for periods up to
For the data of Example 11.5, check the plausibility of assumptions by constructing a normal probability plot of the residuals and a plot of the residuals versus the predicted values, and comment on
Suppose that in the experiment described in Exercise 6 the five houses had actually been selected at random from among those of a certain age and size, so that factor B is random rather than fixed.
a. Show that a constant d can be added to (or subtracted from) each xij without affecting any of the ANOVA sums of squares. b. Suppose that each xij is multiplied by a nonzero constant. c. How does
Use the fact that E(Xij) = µ + ai + βj with (ai = (βj = 0 to show that E(X̅i. - X̅..) = αi, so that = X̅i. - X̅.. is an unbiased estimator for ai.
The power curves of Figures 10.5 and 10.6 can be used to obtain β = P(type II error) for the F test in two-factor ANOVA. For fixed values of a1, a2, ... , a1, the quantity (2 = (J/I) (α2i / (2 is
In an experiment to assess the effects of curing time (factor A) and type of mix (factor B) on the compressive strength of hardened cement cubes, three different curing times were used in combination
The article "Towards Improving the Properties of Plaster Moulds and Castings" (J. Engr. Manuf., 1991: 265-269) describes several ANOVAs carried out to study how the amount of carbon fiber and sand
The accompanying data resulted from an experiment to investigate whether yield from a certain chemical process depended either on the formulation of a particular input or on mixer speed.A statistical
A two-way ANOVA was carried out to assess the impact of type of farm (government agricultural settlement, established, individual) and tractor maintenance method (preventive, predictive, running,
Four different coatings are being considered for corrosion protection of metal pipe. The pipe will be buried in three different types of soil. To investigate whether the amount of corrosion depends
The article "Fatigue Limits of Enamel Bonds with Moist and Dry Techniques" (Dental Materials, 2009: 1527-1531) described an experiment to investigate the ability of adhesive systems to bond to
In an experiment to investigate the effect of "cement factor" (number of sacks of cement per cubic yard) on flexural strength of the resulting concrete ("Studies of Flexural Strength of Concrete.
A study was carried out to compare the writing lifetimes of four premium brands of pens. It was thought that the writing surface might affect lifetime, so three different surfaces were randomly
The accompanying data was obtained in an experiment to investigate whether compressive strength of concrete cylinders depends on the type of capping material used or variability in different batches
a. Show that E(X̅i.. - X̅...) = αi, so that X̅i.. - X̅.. is an unbiased estimator for αi (in the fixed effects model). b. With ij = X̅ij.. - X̅i.. - X̅.j. + X̅..., show that ij is an
Show how a 100 (1 - α) % t CI for αi - αi can be obtained. Then compute a 95% interval for α2 - α3 using the data from Exercise 19?
When both factors are random in a two-way ANOVA experiment with K replications per combination of factor levels, the expected mean squares are E(MSE) =(2, EMSA) = (2 + K(2G + JK(2A, E(MSB) = (2 +
The output of a continuous extruding machine that coats steel pipe with plastic was studied as a function of the thermostat temperature profile (A, at three levels), the type of plastic (B, at three
To see whether thrust force in drilling is affected by drilling speed (A), feed rate (B), or material used (C), an experiment using four speeds, three rates, and two materials was performed, with two
The article "Effects of Household Fabric Softeners on Thermal Comfort of Cotton and PolyesterFabrics After Repeated Launderings" (Family and Consumer Science Research J., 2009: 535-549) reported the
An investigation of the machinability of beryllium-copper alloy using two different dielectric mediums and four different working currents resulted in the following data on material removal rate
The following summary quantities were computed from an experiment involving four levels of nitrogen (A), two times of planting (B), and two levels of potassium (C) ("Use and Misuse of Multiple
Nickel titanium (NiTi) shape memory alloy (SMA) has been widely used in medical devices. This is attributable largely to the alloy's shape memory effect (material returns to its original shape after
When factors A and B are fixed but factor C is random and the restricted model is used (see the footnote on page 438; there is a technical complication with the unrestricted model here), and E(MSE) =
Because of potential variability in aging due to different castings and segments on the castings, a Latin square design with N = 7 was used to investigate the effect of heat treatment on aging. With
The article "The Responsiveness of Food Sales to Shelf Space Requirements" (J. Marketing Research, 1964: 63-67) reports the use of a Latin square design to investigate the effect of shelf space on
The article "Variation in Moisture and Ascorbic Acid Content from Leaf to Leaf and Plant to Plant in Turnip Greens" (Southern Cooperative Services Bull., 1951: 13-17) uses a Latin square design in
The article "An Assessment of the Effects of Treatment, Time, and Heat on the Removal of Erasable Pen Marks from Cotton and Cotton/Polyester Blend Fabrics (J. of Testing and Eval., 1991: 394-397)
A four-factor ANOVA experiment was carried out to investigate the effects of fabric (A), type of exposure (B), level of exposure (C), and fabric direction (D) on extent of color change in exposed
The accompanying data resulted from an experiment to study the nature of dependence of welding current on three factors: welding voltage, wire feed speed, and tip-to-work-piece distance. There were
The accompanying data resulted from a 23 experiment with three replications per combination of treatments designed to study the effects of concentration of detergent (A), concentration of sodium
In an experiment to see whether the amount of coverage of light-blue interior latex paint depends either on the brand of paint or on the brand of roller used, one gallon of each of four brands of
In a study of processes used to remove impurities from cellulose goods ("Optimization of Rope-Range Bleaching of Cellulosic Fabrics," Textile Research J., 1976: 493-496), the following data resulted
As with many dried products, sun-dried tomatoes can exhibit an undesirable discoloration during the drying and storage process. A replicated 23 experiment was conducted in an effort to relate color
The following data on power consumption in electric furnace heats (kW consumed per ton of melted product) resulted from a 24 factorial experiment with three replicates ("Studies on a 10-cwt Arc
The article "Statistical Design and Analysis of Qualification Test Program for a Small Rocket Engine" (Industrial Quality Control, 1964: 14-18) presents data from an experiment to assess the effects
a. In a 24 experiment, suppose two blocks are to be used, and it is decided to confound the ABCD interaction with the block effect. Which treatments should be carried out in the first block [the one
a. An experiment was carried out to investigate the effects on audio sensitivity of varying resistance (A), two capacitances (B, C), and inductance of a coil (D) in part of a television circuit. If
In an experiment involving four factors (A, B, C, and D) and four blocks, show that at least one main effect or two-factor interaction effect must be confounded with the block effect?
a. In a seven-factor experiment (A,..., G), suppose a quarter-replicate is actually carried out. If the defining effects are ABCDE and CDEFG, what is the third non-estimable effect, and what
The article "Applying Design of Experiments to Improve a Laser Welding Process" (J. of Engr. Manufacture, 2008: 1035-1042) included the results of a half replicate of a 24 experiment. The four
A half-replicate of a 25 experiment to investigate the effects of heating time (A), quenching time (B), drawing time (C), position of heating coils (D), and measurement position (E) on the hardness
In an experiment to assess the effect of the angle of pull on the force required to cause separation in electrical connectors, four different angles (factor A) were used, and each of a sample of five
The results of a study on the effectiveness of line drying on the smoothness of fabric were summarized in the article "Line-Dried vs. Machine-Dried Fabrics: Comparison of Appearance, Hand, and
The water absorption of two types of mortar used to repair damaged cement was discussed in the article "Polymer Mortar Composite Matrices for Maintenance-Free, Highly Durable Ferrocement" (J. of
Four plots were available for an experiment to compare clover accumulation for four different sowing rates ("Performance of Overdrilled Red Clover with Different Sowing Rates and Initial Grazing
In an automated chemical coating process, the speed with which objects on a conveyor belt are passed through a chemical spray (belt speed), the amount of chemical sprayed (spray volume), and the
Coal-fired power plants used in the electrical industry have gained increased public attention because of the environmental problems associated with solid wastes generated by large-scale combustion
Impurities in the form of iron oxides lower the economic value and usefulness of industrial minerals, such as kaolins, to ceramic and paper-processing industries. A 24 experiment was conducted to
Factorial designs have been used in forestry to assess the effects of various factors on the growth behavior of trees. In one such experiment, researchers thought that healthy spruce seedlings should
One property of automobile air bags that contributes to their ability to absorb energy is the permeability (ft3/ft2/ min) of the woven material used to construct the air bags. Understanding how
A chemical engineer has carried out an experiment to study the effects of the fixed factors of vat pressure (A), cooking time of pulp (B), and hardwood concentration (C) on he strength of paper. The
The bond strength when mounting an integrated circuit on a metalized glass substrate was studied as a function of factor A = adhesive type, factor B = curve time, and factor C = conductor material
A particular county employs three assessors who are responsible for determining the value of residential property in the county. To see whether these assessors differ systematically in their
The article "Effect of Cutting Conditions on Tool Performance in CBN Hard Turning" (J. of Manuf. Processes, 2005: 10-17) reported the accompanying data on cutting speed (m/s), feed (mm/rev), depth of
Analogous to a Latin square, a Greco-Latin square design can be used when it is suspected that three extraneous factors may affect the response variable and all four factors (the three extraneous
The accompanying data resulted from an experiment involving three different brands of lathe in combination with three different operators (the blocking factor). The response variable is the
The paper "Exercise Thermoregulation and Hyperprolac-tinaemia" (Ergonomics, 2005: 1547-1557) discussed how various aspects of exercise capacity might depend on the temperature of the environment. The
The article "The Effects of a Pneumatic Stool and a One-Legged Stool on Lower Limb Joint Load and Muscular Activity During Sitting and Rising" (Ergonomics, 1993: 519-535) gives the accompanying data
The efficiency ratio for a steel specimen immersed in a phosphating tank is the weight of the phosphate coating divided by the metal loss (both in mg/ft2). The article "Statistical Process Control of
Suppose the expected cost of a production run is related to the size of the run by the equation y = 4000 + 10x. Let Y denote an observation on the cost of a run. If the 1ariables' size and cost are
Suppose that in a certain chemical process the reaction time y (hr) is related to the temperature (°F) in the chamber in which the reaction takes place according to the simple linear regression
Refer back to the data in Exercise 4, in which y = ammonium concentration (mg/L) and x = transpiration (ml/h). Summary quantities include n = 13, (xi = 303.7, (yi = 52.8, Sxx = 1585.230769, Sxy =
The accompanying data on x = current density (mAycm2) and y = rate of deposition smmymin d appeared in the article "Plating of 60/40 Tin/Lead Solder for Head Termination Metallurgy" (Plating and
Refer to the tank temperature-efficiency ratio data given in Exercise 1. a. Determine the equation of the estimated regression line. b. Calculate a point estimate for true average efficiency ratio
Values of modulus of elasticity (MOE, the ratio of stress, i.e., force per unit area, to strain, i.e., deformation per unit length, in GPa) and flexural strength (a measure of the ability to resist
The article "Characterization of Highway Runoff in Austin, Texas, Area" (J. of Envir. Engr., 1998: 131-137) gave a scatter-plot, along with the least squares line, of x = rain-full volume (m3) and y
No-fines concrete, made from a uniformly graded coarse aggregate and a cement-water paste, is beneficial in areas prone to excessive rainfall because of its excellent drainage properties. The article
For the past decade, rubber powder has been used in asphalt cement to improve performance. The article "Experimental Study of Recycled Rubber-Filled High-Strength Concrete" (Magazine of Concrete
The following data is representative of that reported in the article "An Experimental Correlation of Oxides of Nitrogen Emissions from Power Boilers Based on Field Data" (J. of Engr. for Power, July
The article "Exhaust Emissions from Four-Stroke Lawn Mower Engines" (J. of the Air and Water Mgmnt. Assoc., 1997: 945-952) reported data from a study in which both a baseline gasoline mixture and a
The bond behavior of reinforcing bars is an important determinant of strength and stability. The article "Experimental Study on the Bond Behavior of Reinforcing Bars Embedded in Concrete Subjected to
Wrinkle recovery angle and tensile strength are the two most important characteristics for evaluating the performance of cross-linked cotton fabric. An increase in the degree of cross-linking, as
Calcium phosphate cement is gaining increasing attention for use in bone repair applications. The article "Short-Fibre Reinforcement of Calcium Phosphate Bone Cement" (J. of Engr. in Med., 2007:
a. Obtain SSE for the data in Exercise 19 from the defining formula [SSE = ((yi - i)2], and compare to the value calculated from the computational formula. b. Calculate the value of total sum of
The invasive diatom species Didymosphenia geminate has the potential to inflict substantial ecological and economic damage in rivers. The article "Substrate Characteristics Affect Colonization by the
Show that b1 and b0 of expressions (12.2) and (12.3) satisfy the normal equations?
Show that the "point of averages" (x̅, y̅) lies on the estimated regression line?
Suppose an investigator has data on the amount of shelf space x devoted to display of a particular product and sales revenue y for that product. The investigator may wish to fit a model for which the
a. Consider the data in Exercise 20. Suppose that instead of the least squares line passing through the points (x1, y1),..., (xn, yn), we wish the least squares line passing through (x1 x2, y1) ,...,
Consider the following three data sets, in which the variables of interest are x = commuting distance and y = commuting time. Based on a scatter-plot and the values of s and r2, in which situation
Bivariate data often arises from the use of two different techniques to measure the same quantity. As an example, the accompanying observations on x = hydrogen concentration1(ppm) using a gas
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