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applied statistics and probability for engineers

Questions and Answers of Applied Statistics And Probability For Engineers

=+18. To see whether the force in drilling is affected by the drilling speed (A), feed rate (B), or material used (C), an experiment using four speeds, three rates, and two materials was performed,
=+b. Use the appropriate F ratios to show that none of the two- or three-factor interactions is significant at 5 .05.c. Which main effects are significant at 5 .05?
=+17. The output of a continuous extruding machine that coats steel pipe with plastic was studied as a function of thermostat temperature profile (A, at three levels), type of plastic (B, at three
=+c. What conclusions can you draw regarding the effects of the two factors on bud rating?
=+b. Create an effects plot for the factors that were found to be significant in part (a).
=+a. Using a significance level of 5%, conduct an ANOVA test for this data. Indicate which factors are significant and whether the interaction term is significant.
=+to three levels of pH to see whether this factor has an effect on virus uptake into the root system. The following table shows data from a 2 3 3 experiment to study both factors:pH 3 5.5 7 1.2,
=+16. 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
=+of the factor combinations. The resulting sums of squares were SSA 5 210.67, SSB 5 132.17, SSC 5 2586.35, SS(AB) 5 57.48, SS(AC) 5 636.84, SS(BC) 5 875.00, SS(ABC) 5 888.52, SSE 5 5416.67 and SST 5
=+finishing outcomes. To see whether average surface roughness (Ra) is affected by the abrasive size (A), abrasive quantity (B), and quill gap (C), an experiment using three sizes, three quantities,
=+a relatively new technology that uses abrasive particles surrounded by magnets that generate a magnetic field around the polishing area, has drawn attention as an alternative finishing method. The
=+15. Highly precise finishing methods are important for the manufacturing of ultraprecision optical parts but conventional polishing methods have proven to be unsatisfactory. Magnetic abrasive
=+c. From your results in parts (a) and (b), what levels of each factor would you select to maximize wetmold strength? What factor levels would you choose to maximize casting hardness?
=+b. Construct an ANOVA table for the effects of these factors on casting hardness. Test for the presence of significant effects using 5 .05.
=+a. Construct an ANOVA table for the effects of these factors on wet-mold strength. Test for the presence of significant effects using 5 .05.
=+14. Experiments often have more than one response value of interest. In the article “Towards Improving the Properties of Plaster Moulds and Castings” (J.Engr. Manuf., 1991: 265–269), a study
=+c. If a significance level of .05 is used for the twoway ANOVA, the interaction effect is significant(just as in general different glues work better with some materials than with others). So now
=+b. Carry out an appropriate analysis of variance and state your conclusions (use a significance level of .01 for any tests). Include any graphs that provide insight.
=+a. Construct a comparative boxplot of the data on the four different treatments and comment.
=+treatment, the next 12 from the OBP-dry treatment, and the last 12 from the OBP-moist treatment.SBP-Dry 56.7 57.4 53.4 54.0 49.9 49.9 56.2 51.9 49.6 45.7 56.8 54.1 SBP-Moist 49.2 47.4 53.7 50.6
=+adhesives—Adper Single Bond Plus (SBP) and OptiBond Solo Plus (OBP)—were used in combination with two different surface conditions. The accompanying data was supplied by the authors of the
=+13. 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
=+b. What possible reasons can you give for an interaction plot that looks like the following one?Small Large Module size No library access Library access Coding time
=+a. If the goal is to reduce coding time, describe the conclusions you can draw from the experiment if the interaction plot looks like this:Small Large Module size No library access Library access
=+the coding time are the size of the module and whether the programmer has access to a library of previously coded submodules. Module size is studied at two levels, large and small, whereas access
=+ 12. Factorial designs have been used to study productivity of software engineers (“Experimental Design and Analysis in Software Engineering,” Software Engineering Notes, 1995: 14–16).
=+different coarse aggregate contents (%), with two observations made for each such combination, resulting in the conductivity data (W/m•K) given here:Coarse Aggregate Content (%)38 41 44 Asphalt
=+11. Lightweight aggregate asphalt mix has been found to have lower thermal conductivity, which is desirable, than a conventional mix would have. The article “Influence of Selected Mix Design
=+10. Draw an interaction plot for the data of Exercise 9.
=+b. Does the yield appear to depend on either the formulation or the speed? (Use 5 .05.)
=+a. Does there appear to be interaction between the two factors? (Use 5 .05.)
=+ A statistical software package gave these results:SS(formulation) 5 2253.44, SS(speed) 5 230.81, SS(interaction) 5 18.58, and SSE 5 71.87.
=+9. The following data was obtained in an experiment to investigate whether the yield from a certain chemical process depends on either the chemical formulation of the input materials or the mixer
=+ 8. A chemical engineer conducts an experiment to test the effects of gas flow rate (factor A) and liquid flow rate (factor B) on the gas film heat transfer coefficient (in Btu/hr ft2). Four
=+ 7. A fixed effects model is used to analyze two factors, each of which has five levels. Three replicated measurements are available for each combination of factor levels. Complete the following
=+6. In the example discussed on page 454, perform the necessary hypothesis tests to show that neither factor B nor the two-factor AB interaction is significant(using 5 .05).
=+5. Why do parallel line segments in effects plots indicate that there is no interaction between two factors?
=+c. Sketch some of the contours using the equation(s)in part (b). Using these results, determine from these contours the approximate coordinates of the point(s) at which the response surface is at
=+b. Find an equation that describes the contours of the response surface.
=+4. Suppose that the response surface for a two-factor experiment can be described by the function f (x, y) 5 e2(x2y)2.
=+d. Sketch some of the contours using your answer to part (b). From this sketch, determine the approximate coordinates of the point at which the response surface is at its maximum from these
=+c. Find an equation that describes the typical contour of the response surface.
=+b. From the graph in part (a), determine the approximate coordinates of the point (x, y) at which the response surface is at its maximum.
=+3. Suppose that the response surface for a two-factor experiment can be described by the function f (x, y) 5 e(21y2)[(x22)2 1(y25)2].a. Use a computer package to create a graph of the response
=+b. Calculate an estimate of the effect of changing factor B from level 1 to level 2.
=+a. Calculate an estimate of the effect of changing factor A from level 1 to level 2.
=+2. Factors A and B are thought to have an effect on a certain response value, y. The following table contains data on the response variable measured at each combination of the two levels of
=+1. What statistical purpose does replication serve in an experimental design?
=+b. Using a significance level of .05, can you conclude that there is a difference between the monthly mean kilowatt-hours of electricity used by the four types of air conditioners?
=+randomly selected to receive one of the four airconditioning systems. The resulting data is given in the table.Type of home 1 2 3 4 5 1 116 118 97 101 115 Airconditioning system 2 171 131 105 107
=+Because of the many differences that can exist between residences (e.g., floor space, type of insulation, type of roof, etc.), five different groups of homes were identified for study. From each
=+52. A consumer protection organization carried out a study to compare the electricity usage for four different types of residential air-conditioning systems. Each system was installed in five
=+b. Using a significance level of .05, can you conclude that there is a difference between the mean smoothness scores for the five drying methods?
=+for nine types of fabric and five different drying methods. Because the different types of fabric were expected to have large differences in smoothness, regardless of drying method, each of the
=+51. The results on the effectiveness of line drying on the smoothness of fabric were studied in the paper “Line-Dried vs. Machine-Dried Fabrics: Comparison on Appearance, Hand, and Consumer
=+c. Conduct the appropriate test you identified in part (b), using 5 .01, and compare your answer to the answer in part (a).
=+b. Which test procedure in Chapter 8 could have been used on this data in place of the ANOVA test in part (a)?
=+a. Using 5 .01, conduct an ANOVA test to determine whether there is a difference in the average focus settings between the two groups of pilots.
=+focus settings were then measured with a dioptometer (“Oculomotor Responses with Aviator HelmetMounted Displays and Their Relation to In-Flight Symptoms,” Human Factors, 1995: 699–710). The
=+In a study of HMDs, researchers tested Apache helicopter pilots to determine whether the presence of in-flight vision problems has an effect on a pilot’s ability to focus the HMD panel. Thirteen
=+50. Helmet-mounted displays (HMDs) are computer displays that are presented on see-through screens attached to the helmets of helicopter pilots.HMDs are normally employed to aid night flights.
=+49. In Exercise 47, suppose that the three sample means are 10, 15, and 20. Can you now find a value of SSE that satisfies the two conditions in Exercise 47?
=+Minimum Significant Difference = 56.989 Means with the same letter are not significantly different.Tukey Grouping Mean N trt A 336.00 21 M2 AA 301.00 21 M4 B 171.43 21 M3 BB 155.71 21 M1
=+a post hoc analysis by applying Tukey’s procedure(as the authors did) using the following output from the SAS software:Alpha = 0.05 df = 60 MSE = 4883.488 Critical Value of Studentized Range =
=+48. For the data referenced in Exercise 39, the article reported that there was a difference in RPN means for the four design methods (M1, M2, M3, M4). Perform
=+(1) The calculated F statistic is larger than the tabled value of F for 5 .05, df1 5 2, and df2 5 12, so the hypothesis H0: 1 5 2 5 3 is rejected at 5 .05.(2) When Tukey’s procedure is
=+47. Consider a single-factor ANOVA in which samples of size 5 each are measured at each of three levels of a certain factor. The means of the three samples are 10, 12, and 20. Find a value of SSE
=+b. Apply Tukey’s procedure to this data with 5 .05. Compare your results to the conclusion obtained in part (a).
=+a. Perform the F test for this single-factor ANOVA at 5 .05.
=+46. Consider the following data on plant growth after the application of five different types of growth hormone:Data A 13 17 7 14 B 21 13 20 17 C 18 15 20 17 D 7 11 18 10 E 6 11 15 8
=+a. Verify this relationship by looking up F.05(df1 5 1, df2 5 10) and t.025(df 5 10) in the F and t tables, Appendix Tables VIII and IV, respectively.b. For 5 .05, the values of t@2 approach
=+an F distribution with df1 5 1 and any value of df2 and for a t distribution with df 5 df2. The subscripts and @2 on F and t@2 denote right-tail areas of and @2 under the density curves for
=+45. In the special case where df1 5 1, the right-tail areas associated with an F distribution are related to similar areas under a t distribution’s density curve.In particular, it can be shown
=+indicate harder metals).Pulse Current: 100 100 100 120 120 120 140 140 140 Hardness: 326 296 312 245 273 276 299 296 282 Use 5 .05 to conduct the test for whether there are any differences in the
=+44. In the study described in Exercise 12, the authors also investigated how pulse current affects the hardness of the SDSS welds. Hardness is measured in HV (known as the Vickers number; higher
=+c. Using 5 .05, can you conclude that there are any differences between the average lumen outputs for the three brands?
=+b. Compute each of the entries in the ANOVA table for this experiment.
=+a. State the hypotheses of interest. Describe, in words, the parameters that appear in the hypotheses.
=+43. The lumen output was determined for three different brands of 60-watt soft-white light bulbs, with eight bulbs of each brand tested. From the resulting lumen measurements, the following sums
=+ Use 5 .05 to conduct the test for whether there are any differences in the true average MRR that may be attributable to the different pulse times.
=+accompanying data resulted from an EDM process using an oil dielectric medium where researchers applied four different EDM pulse times (s) and recorded the corresponding material removal rate(MRR,
=+42. The authors of “Statistical Analysis and Optimization Study on the Machinability of BerylliumCopper Alloy in Electro Discharge Machining” (J. of Engr. Manuf., 2012: 1847–1861)
=+c. Suppose that you ignore the fact that the batches are blocks in this experiment and that you simply run a one-factor ANOVA test, treating the three columns of data as three random samples.Using
=+b. Can you conclude that there are differences between the batch means? (Use 5 .05.)
=+a. Using a significance level of 5%, can you conclude that there is a difference in mean concrete strength between the three curing methods?
=+41. Example 4.15 (Chapter 4) describes a randomized block experiment for comparing three different methods (A, B, and C) of curing concrete. Different batches of concrete are used as the blocks in
=+d. Explain why your conclusions about wood types in this experiment differ from the conclusions reached in Exercise 20.
=+c. Suppose that wood with a large bending strength is needed for a particular structure and that any wood grade is acceptable. Which type and grade of wood is best for such a structure?
=+b. Can you conclude that there are differences between the mean bending strengths for the three grades of wood? (use 5 .05.)
=+a. Using the three wood grades as blocks, can you conclude that there is a difference between the mean bending strengths of the three species of wood? (use 5 .05.)
=+The following table shows bending strengths from testing wood samples of each type and grade:Wood grade SS Grade 2 Grade3 Species Douglas Fir 65 43 41 Hem-Fir 45 38 32 Spruce-Pine-Fir 42 35 30
=+40. In the study described in Exercise 20, the wood grade is known to affect wood strength. To incorporate this information, three wood grades were studied: SS (select structural), grade 2, and
=+c. Do the person-to-person differences in RPN seem to be confirmed by the data? Explain.
=+b. Using 5 .05, can it be concluded that there is a difference in the true average RPN among the four design methods?
=+a. Fill in the missing values in the table above.
=+(in random order) by all 21 individuals in the study.The data was analyzed by the R software, giving the following output. Note that the format of the ANOVA table in R is very similar to the one
=+that compared four design methods (M1, M2, M3, M4)in preproduction trials of the upper range for a particular casting valve. The design methods are applied by human operators, which introduces
=+given the highest priority in carrying out further analyses.The article “Continuous Quality Improvement in Investment Castings: An Experimental Study using a Modified FMEA Approach Called
=+of the consequences of failure, (2) likelihood of failure occurrence, and (3) likelihood that failure would not be detected. The product of these scores is the risk priority number (RPN) for the
=+39. To assess the potential risks associated with failure of a particular process, investigators often perform a failure modes and effects analysis (FMEA). An FMEA identifies opportunities for

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