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. Part 3: Effects of Variation in Testing Procedure,” Proceedings, ASTM,1957: 1127–1139), I = 3 different factor values were used, J = 5 different batches of cement were selected, and K = 2 beams were cast from each cement factor/batch combination. Sums of squares include SSA = 22,941.80, SSB = 22,765.53, SSE = 15,253.50,and SST = 64,954.70. Construct the ANOVA table. Then, assuming a mixed model with cement factor (A) fixed and batches (B) random, test the three pairs of hypotheses of interest at level .05.
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Chapter 11 Solutions
Probability and Statistics for Engineering and the Sciences
- The specification for the pull strength of a wire that connects an integrated circuit to its frame is 10 g or more. Units made with aluminum wire have a defect rate of 10%. A redesigned manufacturing process, involving the use of gold wire, is being investigated. The goal is to reduce the rate of defects to 5% or less. Out of the first 100 units manufactured with gold wire, only 4 are defective. True or false: a) Since only 4% of the 100 units were defective, we can conclude that the goal has been reached. b) Although the sample percentage is under 5%, this may represent sampling variation, so the goal may not yet be reached. c) There is no use in testing the new process, because no matter what the result is, it could just be due to sampling variation. d) If we sample a large enough number of units, and if the percentage of defective units is far enough below 5%, then it is reasonable to conclude that the goal has been reached.arrow_forwardA study is made of the effect of curing temperature on the compressive strength of a certain type of concrete. Five concrete specimens are cured at each of four temperatures, and the compressive strength of each specimen is measured (in MPa). The results are as follows: Temperature (°C) Strengths 31.2 29.6 30.8 30.0 31.4 10 30.0 27.7 31.1 31.3 30.6 20 35.9 36.8 35.0 34.6 36.5 30 38.3 37.0 37.5 36.1 38.4 Construct an ANOVA table. You may give a range for the P-value. Can you conclude that the mean strength differs with temperature? a. b.arrow_forwardAn engineer wants to compare the tensile strengths of steel bars that are produced using a conventional method and an experimental method. (The tensile strength of a metal is a measure of its ability to resist tearing when pulled lengthwise.) To do so, the engineer randomly selectoarrow_forward
- The quality control engineer at Palmer Industries is interested in estimating the tensile strength of steel wire based on its outside diameter and the amount of molybdenum in the steel. As an experiment, she selected 25 pieces of wire, measured the outside diameters, and determined the molybdenum content. Then she measured the tensile strength of each piece. The results of the first four are recorded in the table. Tensile Strength Outside Diameter Amount of Molybdenum Place (PSI ) Y (mm) X1 (Units) X2 A 11 0.3 6 B 9 0.2 5 C 16 0.4 8 D 12 0.3 7 Using a statistical software package, the QC engineer determined the multiple regression equation to be Y’=-0.5+20X1+1X2. a) Based on the equation, what is the estimated tensile strength of a steel wire having an outside diameter of .35 mm and 6.4 units of molybdenum? b) Interpret the value of b1 in the equation.arrow_forwardThe article “An Ivestigation into the Ball Burnishing of Aluminium Alloy 6061-T6" (M. El-Axir, J Engineering Manufacture, 2007:1733-1742) presents the results of study that investigated the effects of three bumishing factors on the reduction in diameter of the workpiece (in um). The factors are A: Bumishing speed, B: Burnishing force, and C: Bumishing feed. The results presented in the following table form a 23 factorial design (some additional results are omitted). Reduction 570 A -1 -1 -1 -1 -1 353 -1 778 -1 769 -1 544 -1 319 -1 651 625 Compute estimates of the main effects and the interactions. b. Is it possible to compute an error sum of squares? Explain. Are any of the interactions among the larger effects? If so, which ones? d. Someone claims that the additive model holds. Do the results tend to support this statement? Explain.arrow_forwardAn experiment to compare the tension bond strength of polymer latex modified mortar (Portland cement mortar to which polymer latex emulsions have been added during mixing) to that of unmodified mortar resulted in x = 18.19 kgf/cm? for the modified mortar (m = 42) and y = 16.85 kgf/cm? for the unmodified mortar (n = 30). Let u, and u, be the true average tension bond strengths for the modified and unmodified mortars, respectively. Assume that the bond strength distributions are both normal. Assuming that o, = 1.6 and o, = 1.3, test Hn: 4, - H, = 0 versus H: u, - u, > 0 at level 0.01. Calculate the test statistic and determine the P-value. (Round your test statistic to two decimal places and your P-value to four decimal places.) P-value = Compute the probability of a type II error for the test of part (a) when 4 - Hz = 1. (Round your answer to four decimal places.) Suppose the investigator decided to use a level 0.05 test and vwished B = 0.10 when u, - uz = 1. If m = 42, what value of n…arrow_forward
- An article in the ACI Materials Journal (Vol. 84, 1987, pp. 213-216) describes several experiments investigating the rodding of concrete to remove trapped air. A 3-inch x 6-inch cylinder was used, and the number of times this rod was used is the design variable. The resulting compressive strength of the concrete specimen is the response. The data are shown in the following table. ... Compressive Strength (psi) Rodding Level Observations 10 1530 1530 1440 15 1610 1650 1500 20 1560 1730 1530 25 1500 1490 1510 Calculate the test statistic fo. Input answer up to 2 decimal places. Test Statisticf =1.68 Blank 1 1.68arrow_forwardThe City Council proposed to utilize government-owned land with an area of 12,150 square meters. One-third of the area will be used to plant mango trees while the rest of the land is where an emergency field hospital shall be built. As the city engineer, you were tasked to answer the problem: Please give me a detailed illustration and explanation about the representation of variables whereas this is the given: a. the rate at which the total average number of COVID cases is increasing at x=10 tents and dxdt=1 tent per day, given that if 30 tents are built, the average number of COVID cases per tent will be 7 cases while the average number of cases will increase by 2 per tent for each additional tent on the same area due to overcrowding.Thank you so much! Please give me a detailed illustration and explanation about the representation of the variables for the given problemarrow_forwardCoke is a solid fuel made by heating coal in the absence of air so that the volatile components are driven off. For screened coke, the porosity factor is measured by the difference in weight between dry and soaked coke. A certain supply of screened coke from a supplier is claimed to have a porosity factor of 1.8 kilograms. Ten samples of the screened coke obtained from this supplier are tested for porosity factors and the results are as follows : 1.7 , 1.9 , 1.8 , 1.9 , 2.1 , 2.1 , 2.0 , 1.8 , 1.7 , 2.0 . Is there sufficient evidence to indicate that the actual coke from the supplier is more porous than what is claimed? Use Assume that the porosity factor is a normally distributed variable. Use the following format in your presentation. Show the values of the mean and the standard deviation which you calculate using the formulas or directly through your calculator stat or through Excel. Ho : HA : Test Statistic Critical Value : Test Statistic Calculated Value : (show the…arrow_forward
- An experiment to compare the tension bond strength of polymer latex modified mortar (Portland cement mortar to which polymer latex emulsions have been added during mixing) to that of unmodified mortar resulted in x = 18.13 kgf/cm? for the modified mortar (m = 42) and y = 16.85 kgf/cm2 for the unmodified mortar (n = 32). Let u, and u, be the true average tension bond strengths for the modified and unmodified mortars, respectively. Assume that the bond strength distributions are both normal. (a) Assuming that o, = 1.6 and o, = 1.3, test Ho: 4, - H, = 0 versus H: u, - µ, > 0 at level 0.01. Calculate the test statistic and determine the P-value. (Round your test statistic to two decimal places and your p-value to four decimal places.) z = 3.80 P-value = 0.0001 State the conclusion in the problem context. O Fail to reject H,. The data suggests that the difference in average tension bond strengths exceeds 0. O Fail to reject Ho: The data does not suggest that the difference in average…arrow_forwardWild irises are beautiful flowers found throughout the United States, Canada, and northern Europe. This problem concerns the length of the sepal (leaf-like part covering the flower) of different species of wild iris. Data are based on information taken from an article by R. A. Fisher in Annals of Eugenics (Vol. 7, part 2, pp. 179 -188). Measurements of sepal length in centimeters from random samples of Iris setosa (I), Iris versicolor (II), and Iris virginica (III) are as follows below. I II III 5.5 5.2 6.8 4.6 6.5 5.3 5.1 6.1 4.4 5.5 4.1 7.9 4.1 5.1 5.9 5.4 6.1 6.9 5.4 5.1 6.6 Shall we reject or not reject the claim that there are no differences among the population means of sepal length for the different species of iris? Use a 5% level of significance. (a) What is the level of significance?State the null and alternate hypotheses. Ho: ?1 = ?2 = ?3; H1: Exactly two means are equal.Ho: ?1 = ?2 = ?3; H1: Not all the means are equal. Ho: ?1 = ?2 = ?3; H1:…arrow_forwardWild irises are beautiful flowers found throughout the United States, Canada, and northern Europe. This problem concerns the length of the sepal (leaf-like part covering the flower) of different species of wild iris. Data are based on information taken from an article by R. A. Fisher in Annals of Eugenics (Vol. 7, part 2, pp. 179 -188). Measurements of sepal length in centimeters from random samples of Iris setosa (I), Iris versicolor (II), and Iris virginica (III) are as follows below. I II III 5.7 5.1 6.5 4.7 6.2 5.1 4.7 6.6 4.7 5.8 4.9 7.5 4.6 5.2 5.3 5.3 6.2 6.2 5.4 5.8 6.4 (b) Find SSTOT, SSBET, and SSW and check that SSTOT = SSBET + SSW. (Use 3 decimal places.) SSTOT = SSBET = SSW = Find d.f.BET, d.f.W, MSBET, and MSW. (Use 4 decimal places for MSBET, and MSW.) dfBET = dfW = MSBET = MSW = Find the value of the sample F statistic. (Use 2 decimal places.)What are the degrees of freedom? (numerator) (denominator)arrow_forward
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