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inferential statistics
Probability And Statistics For Engineers And Scientists 9th Global Edition Ronald E. Walpole, Raymond Myers, Sharon L. Myers, Keying E. Ye - Solutions
15.12 Consider Exercise 15.11. Suppose there was some experimental difficulty in making the runs. In fact, the total experiment had to be halted after only 4 runs. As a result, the abbreviated experiment is given by Life a 43 b 35 c 44 abc 39 With only these runs, we have the signs for contrasts
15.11 In Myers, Montgomery, and Anderson-Cook(2009), a data set is discussed in which a 23 factorial is used by an engineer to study the effects of cutting speed (A), tool geometry (B), and cutting angle (C)on the life (in hours) of a machine tool. Two levels of each factor are chosen, and
15.10 Consider Exercise 15.3 once again. Threefactor interactions are often not significant, and even if they are, they are difficult to interpret. The interaction ABD appears to be important. To gain some sense of interpretation, show two AD interaction plots, one for B = −1 and the other for B
15.9 Consider Exercise 15.6. Use a +1 and −1 scaling for “high” and “low,” respectively, and do a multiple linear regression with the model Yi = β0 + β1x1i + β2x2i + β12x1ix2i + i , with x1i = reactant concentration (−1, +1) and x2i =feed rate (−1, +1).(a) Compute regression
15.8 Consider Exercise 15.5. Compute all 15 effects and do normal probability plots of the effects.(a) Does it appear as if your assumption of negligible three- and four-factor interactions has merit?(b) Are the results of the effect plots consistent with what you communicated about the importance
15.7 Consider Exercise 15.3. It is of interest to the researcher to learn not only that AD, BC, and possibly AB are important, but also what they mean scientifically.Show two-dimensional interaction plots for all three and give an interpretation.
15.6 It is important to study the effect of the concentration of the reactant and the feed rate on the viscosity of the product from a chemical process. Let the reactant concentration be factor A, at levels 15% and 25%.Let the feed rate be factor B, with levels 20 lb/hr and 30 lb/hr. The experiment
15.5 In the study An X-Ray Fluorescence Method for Analyzing Polybutadiene-Acrylic Acid (PBAA) Propellants(Quarterly Reports, RK-TR-62-1, Army Ordnance Missile Command), an experiment was conducted to determine whether or not there was a significant difference in the amount of aluminum obtained in
15.4 A preliminary experiment is conducted to study the effects of four factors and their interactions on the output of a certain machining operation. Two runs are made at each of the treatment combinations in order to supply a measure of pure experimental error. Two levels of each factor are used,
15.3 In a metallurgy experiment, it is desired to test the effect of four factors and their interactions on the concentration (percent by weight) of a particular phosphorus compound in casting material. The variables are A, percent phosphorus in the refinement; B, percent remelted material; C,
15.2 In an experiment conducted by the Mining and Minerals Engineering Department at Virginia Tech to study a particular filtering system for coal, a coagulant was added to a solution in a tank containing coal and sludge, which was then placed in a recirculation system in order that the coal could
15.1 The following data are obtained from a 23 factorial experiment replicated three times. Evaluate the sums of squares for all factorial effects by the contrast method. Draw conclusions.Treatment Combination Rep 1 Rep 2 Rep 3(1) 12 19 10 a 15 20 16 b 24 16 17 ab 23 17 27 c 17 25 21 ac 16 19 19 bc
14.42 A control valve needs to be very sensitive to the input voltage, thus generating a good output voltage.An engineer turns the control bolts to change the input voltage. The book SN-Ratio for the Quality Evaluation, published by the Japanese Standards Association(1988), described a study on how
14.41 The manufacturer of a certain brand of freezedried coffee hopes to shorten the process time without jeopardizing the integrity of the product. The process engineer wants to use 3 temperatures for the drying chamber and 4 drying times. The current drying time is 3 hours at a temperature of
14.40 In the book SN-Ratio for the Quality Evaluation, published by the Japanese Standards Association(1988), a study on how tire air pressure affects the maneuverability of an automobile was described. Three different tire air pressures were compared on three different driving surfaces. The three
14.39 Exercise 14.25 on page 608 describes an experiment involving the extraction of polyethylene through use of a solvent.Time Solvent Temp. 4 8 16 Ethanol 120 80 94.0 94.0 95.3 95.1 93.8 94.2 94.9 95.3 91.1 90.5 92.5 92.4 Toluene 120 80 94.6 94.5 95.4 95.4 93.6 94.1 95.6 96.0 91.1 91.0 92.1
14.38 To ascertain the number of tellers needed during peak hours of operation, data were collected by an urban bank. Four tellers were studied during three“busy” times: (1) weekdays between 10:00 and 11:00 A.M., (2) weekday afternoons between 2:00 and 3:00 P.M., and (3) Saturday mornings
14.37 In the experiment of Review Exercise 14.35, cake volume was also used as a response. The units are cubic inches. Test for interaction between factors and discuss main effects. Assume that both factors are fixed effects.Sweetener Flour Concentration All-Purpose Cake 050 75 100 4.48 3.98 4.42
14.36 An experiment was conducted in the Department of Food Science and Technology at Virginia Tech.It was of interest to characterize the texture of certain types of fish in the herring family. The effect of sauce types used in preparing the fish was also studied. The response in the experiment
14.35 The Laboratory for Interdisciplinary Statistical Analysis at Virginia Tech at Virginia Tech was involved in analyzing a set of data taken by personnel in the Human Nutrition, Foods, and Exercise Department in which it was of interest to study the effects of flour type and percent sweetener on
14.34 Personnel in the Materials Science and Engineering Department at Virginia Tech conducted an experiment to study the effects of environmental factors on the stability of a certain type of copper-nickel alloy.The basic response was the fatigue life of the material.The factors are level of
14.33 A study was made to determine if humidity conditions have an effect on the force required to pull apart pieces of glued plastic. Three types of plastic were tested using 4 different levels of humidity. The results, in kilograms, are as follows:Humidity Plastic Type 30% 50% 70% 90%A 39.0 33.1
14.32 A process engineer wants to determine if the power setting on the machines used to fill certain types of cereal boxes results in a significant effect on the actual weight of the product. The study consists of 3 randomly chosen types of cereal manufactured by the company and 3 fixed power
14.31 A manufacturer of latex house paint (brand A)would like to show that its paint is more robust to the material being painted than that of its two closest competitors. The response is the time, in years, until chipping occurs. The study involves the three brands of paint and three randomly
14.30 A defense contractor is interested in studying an inspection process to detect failure or fatigue of transformer parts. Three levels of inspections are used by three randomly chosen inspectors. Five lots are used for each combination in the study. The factor levels are given in the data. The
14.29 Consider the following analysis of variance for a random effects experiment:Source of Degrees of Mean Variation Freedom Square AB CAB AC BC ABC Error 3123626 24 140 480 325 15 24 18 25 Total 47 Test for significant variance components among all main effects and interaction effects at the 0.01
14.28 A plant manager would like to show that the yield of a woven fabric in the plant does not depend on machine operator or time of day and is consistently high. Four randomly selected operators and 3 randomly selected hours of the day are chosen for the study. The yield is measured in yards
14.27 To estimate the various components of variability in a filtration process, the percent of material lost in the mother liquor is measured for 12 experimental conditions, with 3 runs on each condition. Three filters and 4 operators are selected at random for use in the experiment.(a) Test the
14.26 Assuming a random effects experiment for Exercise 14.6 on page 596, estimate the variance components for brand of orange juice concentrate, for number of days from when orange juice was blended until it was tested, and for experimental error.
14.25 In the book Design of Experiments for Quality Improvement, published by the Japanese Standards Association (1989), a study is reported on the extraction of polyethylene by using a solvent and how the amount of gel (proportion) is influenced by three factors:the type of solvent, extraction
14.24 A scientist collects experimental data on the radius of a propellant grain, y, as a function of powder temperature, extrusion rate, and die temperature.Results of the three-factor experiment are as follows:Powder Temp 150 190 Die Temp Die Temp Rate 220 250 220 250 12 82 124 88 129 24 114 157
14.23 Consider combinations of three factors in the removal of dirt from standard loads of laundry. The first factor is the brand of the detergent, X, Y , or Z.The second factor is the type of detergent, liquid or powder. The third factor is the temperature of the water, hot or warm. The experiment
14.22 Consider the data set in Exercise 14.21.(a) Construct an interaction plot for any two-factor interaction that is significant.(b) Do a normal probability plot of residuals and comment.
14.21 Electronic copiers make copies by gluing black ink on paper, using static electricity. Heating and gluing the ink on the paper comprise the final stage of the copying process. The gluing power during this final process determines the quality of the copy. It is postulated that temperature,
14.20 For a study of the hardness of gold dental fillings, five randomly chosen dentists were assigned combinations of three methods of condensation and two types of gold. The hardness was measured. (See Hoaglin, Mosteller, and Tukey, 1991.) Let the dentists play the role of blocks. The data are
14.19 Corrosion fatigue in metals has been defined as the simultaneous action of cyclic stress and chemical attack on a metal structure. In the study Effect of Humidity and Several Surface Coatings on the Fatigue Life of 2024-T351 Aluminum Alloy, conducted by the Department of Mechanical
14.18 Consider an experimental situation involving factors A, B, and C, where we assume a three-way fixed effects model of the form yij k l = μ + αi + βj +γk +(βγ)j k +ij k l . All other interactions are considered to be nonexistent or negligible. The data are presented here.B1 B2 C1 C2 C3
14.17 The following data are measurements from an experiment conducted using three factors A, B, and C, all fixed effects:C1 C2 C3 B1 B2 B3 B1 B2 B3 B1 B2 B3 A1 15.0 14.8 15.9 16.8 14.2 13.2 15.8 15.5 19.2 18.5 13.6 14.8 15.4 12.9 11.6 14.3 13.7 13.5 22.1 12.2 13.6 14.3 13.0 10.1 13.0 12.6 11.1 A2
14.16 The method of X-ray fluorescence is an important analytical tool for determining the concentration of material in solid missile propellants. In the paper An X-ray Fluorescence Method for Analyzing Polybutadiene Acrylic Acid (PBAA) Propellants (Quarterly Report, RK-TR-62-1, Army Ordinance
14.15 The purpose of the study The Incorporation of a Chelating Agent into a Flame Retardant Finish of a Cotton Flannelette and the Evaluation of Selected Fabric Properties, conducted at Virginia Tech, was to evaluate the use of a chelating agent as part of the flame retardant finish of cotton
14.14 Consider the data set in Exercise 14.10 and answer the following questions.(a) Both factors, copper and time, are quantitative in nature. As a result, a regression model may be of interest. Describe what might be an appropriate model using x1 = copper content and x2 = time.Fit the model to
14.13 In Myers, Classical and Modern Regression with Applications (Duxbury Classic Series, 2nd edition, 1990), an experiment is described in which the Environmental Protection Agency seeks to determine the effect of two water treatment methods on magnesium uptake. Magnesium levels in grams per
14.12 Two factors in a manufacturing process for an integrated circuit are studied in a two-factor experiment.The purpose of the experiment is to learn their effect on the resistivity of the wafer. The factors are implant dose (2 levels) and furnace position (3 levels).Experimentation is costly so
14.11 A study was done to determine the impact of two factors, method of analysis and the laboratory doing the analysis, on the level of sulfur content in coal.Twenty-eight coal specimens were randomly assigned to 14 factor combinations, the structure of the experimental units represented by
14.10 In an experiment conducted in the Civil Engineering Department at Virginia Tech, growth of a certain type of algae in water was observed as a function of time and the dosage of copper added to the water.The data are as follows. Response is in units of algae.Time in Days Copper 5 12 18 1 0.30
14.9 An engineer is interested in the effects of cutting speed and tool geometry on the life in hours of a machine tool. Two cutting speeds and two different geometries are used. Three experimental tests are accomplished at each of the four combinations. The data are as follows.Tool Cutting Speed
14.8 In Myers, Montgomery, and Anderson-Cook(2009), a scenario is discussed involving an auto bumper plating process. The response is the thickness of the material. Factors that may impact the thickness include amount of nickel (A) and pH (B). A two-factor experiment is designed. The plan is a
14.7 The extraction rate of a certain polymer is known to depend on the reaction temperature and the amount of catalyst used. An experiment was conducted at four levels of temperature and five levels of the catalyst, and the extraction rate was recorded in the following table.Amount of Catalyst
14.6 To ascertain the stability of vitamin C in reconstituted frozen orange juice concentrate stored in a refrigerator for a period of up to one week, the study Vitamin C Retention in Reconstituted Frozen Orange Juice was conducted by the Department of Human Nutrition, Food and Exercise at Virginia
14.5 To determine which muscles need to be subjected to a conditioning program in order to improve one’s performance on the flat serve used in tennis, a study was conducted by the Human Nutrition, Foods, and Exercise Department at Virginia Tech. Five different muscles 1: anterior deltoid 4:
14.4 An experiment was conducted to determine whether additives increase the adhesiveness of rubber products. Sixteen products were made with the new additive and another 16 without the new additive. The observed adhesiveness was as recorded below.Temperature (◦C)50 60 70 80 2.3 3.4 3.8 3.9
14.3 Three strains of rats were studied under 2 environmental conditions for their performance in a maze test. The error scores for the 48 rats were recorded.Strain Environment Bright Mixed Dull Free 28 12 33 83 101 94 22 23 36 14 33 56 25 10 41 76 122 83 36 86 22 58 35 23 Restricted 72 32 60 89
14.2 Corrosion fatigue in metals has been defined as the simultaneous action of cyclic stress and chemical attack on a metal structure. A widely used technique for minimizing corrosion fatigue damage in aluminum involves the application of a protective coating. A study conducted by the Department
14.1 An experiment was conducted to study the effects of temperature and type of oven on the life of a particular component. Four types of ovens and 3 temperature levels were used in the experiment.Twenty-four pieces were assigned randomly, two to each combination of treatments, and the following
13.56 Group Project: It is of interest to determine which type of sports ball can be thrown the longest distance.The competition involves a tennis ball, a baseball, and a softball. Divide the class into teams of five individuals. Each team should design and conduct a separate experiment. Each team
13.55 For the randomized block design with k treatments and b blocks, show that E(SSB) = (b − 1)σ2 + kb j=1β2 j . Dependent Variable: gasket The GLM Procedure Source Model DF Sum of Squares 5 1.68122778 Error 12 0.05273333 Mean Square F Value 0.33624556 0.00439444 76.52 Pr F F material 2
13.54 Show that the computing formula for SSB, in the analysis of variance of the randomized complete block design, is equivalent to the corresponding term in the identity of Theorem 13.3.
13.53 Prove Theorem 13.2.
13.52 Four different locations in the northeast were used for collecting ozone measurements in parts per million. Amounts of ozone were collected in 5 samples at each location.Location 1 2 3 4 0.09 0.15 0.10 0.10 0.10 0.12 0.13 0.07 0.08 0.17 0.08 0.05 0.08 0.18 0.08 0.08 0.11 0.14 0.09 0.09(a) Is
13.51 Show that the mean square error s2 = SSE k(n − 1)for the analysis of variance in a one-way classification is an unbiased estimate of σ2 .
13.50 A study is conducted to compare gas mileage for 3 competing brands of gasoline. Four different automobile models of varying size are randomly selected.The data, in miles per gallon, follow. The order of testing is random for each model.Gasoline Brand Model A B C A 32.4 35.6 38.7 B 28.8 28.6
13.49 An experiment was conducted to compare three types of paint for evidence of differences in their wearing qualities. They were exposed to abrasive action and the time in hours until abrasion was noticed was observed. Six specimens were used for each type of paint. The data are as follows.Paint
13.48 A company that stamps gaskets out of sheets of rubber, plastic, and cork wants to compare the mean number of gaskets produced per hour for the three types of material. Two randomly selected stamping machines are chosen as blocks. The data represent the number of gaskets (in thousands)
13.47 Four laboratories are being used to perform chemical analysis. Samples of the same material are sent to the laboratories for analysis as part of a study to determine whether or not they give, on the average, the same results. The analytical results for the four laboratories are as
13.46 An experiment was designed for personnel in the Department of Animal and Poultry Science at Virginia Tech to study urea and aqueous ammonia treatment of wheat straw. The purpose was to improve nutritional value for male sheep. The diet treatments were control, urea at feeding, ammonia-treated
13.44 An analysis was conducted by the Laboratory for Interdisciplinary Statistical Analysis at Virginia Tech in conjunction with the Department of Forestry.A certain treatment was applied to a set of tree stumps in which the chemical Garlon was used with the purpose of regenerating the roots of
13.43 Consider the aggregate data of Example 13.1.Perform Bartlett’s test, at level α = 0.1, to determine if there is heterogeneity of variance among the aggregates.
13.42 Three catalysts are used in a chemical process;a control (no catalyst) is also included. The following are yield data from the process:Catalyst Control 1 2 3 74.5 77.5 81.5 78.1 76.1 82.0 82.3 80.2 75.9 80.6 81.4 81.5 78.1 84.9 79.5 83.0 76.2 81.0 83.0 82.1 Use Dunnett’s test at the α =
13.41 A textile company weaves a certain fabric on a large number of looms. The managers would like the looms to be homogeneous so that their fabric is of uniform strength. It is suspected that there may be significant variation in strength among looms. Consider the following data for 4 randomly
13.40 An experiment is conducted in which 4 treatments are to be compared in 5 blocks. The data are given below.Block Treatment 1 2 3 4 5 1 12.8 10.6 11.7 10.7 11.0 2 11.7 14.2 11.8 9.9 13.8 3 11.5 14.7 13.6 10.7 15.9 4 12.6 16.5 15.4 9.6 17.1(a) Assuming a random effects model, test the
13.39 The following data show the effect of 4 operators, chosen randomly, on the output of a particular machine.Operator 1 2 3 4 175.4 168.5 170.1 175.2 171.7 162.7 173.4 175.7 173.0 165.0 175.7 180.1 170.5 164.1 170.7 183.7 (a) Perform a random effects analysis of variance at the 0.05 level of
13.38 Five “pours” of metals have had 5 core samples each analyzed for the amount of a trace element. The data for the 5 randomly selected pours are as follows:Pour Core 1 2 3 4 5 1 0.98 0.85 1.12 1.21 1.00 2 1.02 0.92 1.68 1.19 1.21 3 1.57 1.16 0.99 1.32 0.93 4 1.25 1.43 1.26 1.08 0.86 5 1.16
13.37 Testing patient blood samples for HIV antibodies, a spectrophotometer determines the optical density of each sample. Optical density is measured as the absorbance of light at a particular wavelength. The blood sample is positive if it exceeds a certain cutoff value that is determined by the
13.36 In the paper “Self-Control and Therapist Control in the Behavioral Treatment of Overweight Women,” published in Behavioral Research and Therapy(Vol. 10, 1972), two reduction treatments and a control treatment were studied for their effects on the weight change of obese women. The two
13.35 In the book Design of Experiments for the Quality Improvement, published by the Japanese Standards Association (1989), a study on the amount of dye needed to get the best color for a certain type of fabric was reported. The three amounts of dye, 13% wof (13%of the weight of a fabric), 1% wof,
13.34 An experiment was conducted to compare three types of coating materials for copper wire. The purpose of the coating is to eliminate “flaws” in the wire. Ten different specimens of length 5 millimeters were randomly assigned to receive each coating, and the thirty specimens were subjected
13.33 Scientists in the Department of Plant Pathology, Physiology, and Weed Science at Virginia Tech devised an experiment in which 5 different treatments were applied to 6 different locations in an apple orchard to determine if there were significant differences in growth among the treatments.
13.32 Organic arsenicals are used by forestry personnel as silvicides. The amount of arsenic that the body takes in when exposed to these silvicides is a major health problem. It is important that the amount of exposure be determined quickly so that a field worker with a high level of arsenic can
13.31 In a study conducted by the Department of Human Nutrition, Foods and Exercise at Virginia Tech, 3 diets were assigned for a period of 3 days to each of 6 subjects in a randomized complete block design. The subjects, playing the role of blocks, were assigned the following 3 diets in a random
13.30 A nuclear power facility produces a vast amount of heat, which is usually discharged into aquatic systems. This heat raises the temperature of the aquatic system, resulting in a greater concentration of chlorophylla, which in turn extends the growing season.To study this effect, water samples
13.29 In a study on The Periphyton of the South River, Virginia: Mercury Concentration, Productivity, and Autotropic Index Studies, conducted by the Department of Environmental and Water Resources Engineering at Virginia Tech, the total mercury concentration in periphyton total solids was measured
13.28 Three varieties of potatoes are being compared for yield. The experiment is conducted by assigning each variety at random to one of 3 equal-size plots at each of 4 different locations. The following yields for varieties A, B, and C, in 100 kilograms per plot, were recorded:Location 1 Location
13.27 The following data are the percents of foreign additives measured by 5 analysts for 3 similar brands of strawberry jam, A, B, and C:Analyst 1 Analyst 2 Analyst 3 Analyst 4 Analyst 5 B: 2.7 C: 3.6 A: 3.8 C: 7.5 A: 1.6 B: 5.2 B: 2.8 A: 2.7 C: 6.4 A: 1.7 B: 1.9 C: 2.6 C: 8.1 A: 2.0 B: 4.8
13.26 The following data represent the final grades obtained by 5 students in mathematics, English, French, and biology:Subject Student Math English French Biology 1 68 57 73 61 2 83 94 91 86 3 72 81 63 59 4 55 73 77 66 5 92 68 75 87 Test the hypothesis that the courses are of equal difficulty.Use
13.25 Four kinds of fertilizer f1, f2, f3, and f4 are used to study the yield of beans. The soil is divided into 3 blocks, each containing 4 homogeneous plots.The yields in kilograms per plot and the corresponding treatments are as follows:Block 1 Block 2 Block 3 f1 = 42.7 f3 = 48.5 f4 = 32.8 f2 =
13.24 The following table (from A. Hald, Statistical Theory with Engineering Applications, John Wiley &Sons, New York, 1952) gives tensile strengths (in deviations from 340) for wires taken from nine cables to be used for a high-voltage network. Each cable is made from 12 wires. We want to know
13.23 In a biological experiment, four concentrations of a certain chemical are used to enhance the growth of a certain type of plant over time. Five plants are used at each concentration, and the growth in each plant is measured in centimeters. The following growth data are taken. A control (no
13.22 Do Duncan’s test for paired comparisons for the data of Exercise 13.2 on page 538. Discuss the results.
13.21 The printout in Figure 13.5 on page 552 gives information on Duncan’s test, using PROC GLM in SAS, for the aggregate data in Example 13.1. Give conclusions regarding paired comparisons using Duncan’s test results.
13.20 The financial structure of a firm refers to the way the firm’s assets are divided into equity and debt, and the financial leverage refers to the percentage of assets financed by debt. In the paper The Effect of Financial Leverage on Return, Tai Ma of Virginia Tech claims that financial
13.19 It is suspected that the environmental temperature at which batteries are activated affects their life.Thirty homogeneous batteries were tested, six at each of five temperatures, and the data are shown below(activated life in seconds). Analyze and interpret the data. (From C. R. Hicks,
13.18 The following data are values of pressure (psi)in a torsion spring for several settings of the angle between the legs of the spring in a free position:Angle (◦)67 71 75 79 83 83 84 86 87 89 90 85 85 87 87 90 92 85 88 88 90 86 88 88 91 86 88 89 87 90 Compute a one-way analysis of variance
13.17 In the study An Evaluation of the Removal Method for Estimating Benthic Populations and Diversity conducted by Virginia Tech on the Jackson River, 5 different sampling procedures were used to determine the species counts. Twenty samples were selected at random, and each of the 5 sampling
13.16 The study Loss of Nitrogen Through Sweat by Preadolescent Boys Consuming Three Levels of Dietary Protein was conducted by the Department of Human Nutrition, Food and Exercise at Virginia Tech to determine perspiration nitrogen loss at various dietary protein levels. Twelve preadolescent boys
13.15 Use Tukey’s test, with a 0.05 level of significance, to analyze the means of the five different brands of headache tablets in Exercise 13.6 on page 539.
13.14 An investigation was conducted to determine the source of reduction in yield of a certain chemical product. It was known that the loss in yield occurred in the mother liquor, that is, the material removed at the filtration stage. It was thought that different blends of the original material
13.13 The purpose of the study The Incorporation of a Chelating Agent into a Flame Retardant Finish of a Cotton Flannelette and the Evaluation of Selected Fabric Properties conducted at Virginia Tech was to evaluate the use of a chelating agent as part of the flame-retardant finish of cotton
13.12 Consider the data of Review Exercise 13.47 on page 575. Make significance tests on the following contrasts:(a) B versus A, C, and D;(b) C versus A and D;(c) A versus D.
13.11 Use Bartlett’s test at the 0.05 level of significance to test for homogeneity of variances in Exercise 13.2 on page 538.
13.10 For the data set in Exercise 13.7, use Bartlett’s test to check whether the variances are equal. Useα = 0.05.
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