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applied statistics and probability for engineers
Applied Statistics For Engineers And Scientists 3rd Edition Jay L. Devore, Nicholas R. Farnum, Jimmy A. Doi - Solutions
=+B that are not disjoint. Shade in the portion of this diagram that corresponds to the event A and B=.
=+4. Draw a Venn diagram depicting two events A and
=+c. Describe, in words, the complement of A.
=+b. Describe, in words, the event A or B.
=+a. Describe, in words, the event A and B.
=+3. Let A and B denote the events A 5 there are more than three defective items in a random sample of ten items and B 5 there are fewer than six defectives in a random sample of ten items.
=+f. Either the plant at site 1 or site 2 or both of the two plants are completed by the contract date.
=+e. Exactly one of the three plants is completed by the contract date.
=+d. Only the plant at site 1 is completed by the contract date.
=+c. None of the plants is completed by the contract date.
=+b. All plants are completed by the contract date.
=+a. At least one plant is completed by the contract date.
=+ Draw a Venn diagram that depicts these three events as intersecting circles. Shade the region on the Venn diagram corresponding to each of the following events (redraw the Venn diagram for each question):
=+2. An engineering firm is constructing power plants at three different sites. Define the events E1, E2, and E3 as follows:Section 5.1 Exercises E15 the plant at site 1 is completed by the contract date E25 the plant at site 2 is completed by the contract date E35 the plant at site 3 is completed
=+1. A random sample, without replacement, of three items is to be selected from a population of five items (labeleda, b,c, d, and e).a. List all possible different samples.b. List the samples that correspond to the event A 5 items a and c are included in the sample.c. List the samples that
=+c. What conclusions can you draw from the results in parts (a) and (b) about the two NOx measuring methods?
=+b. Use the methods of Chapter 3 to fit a regression line to this data, with STC as y and C6M as x.
=+a. Construct a Youden plot of this data.
=+NOx emission rate estimates Lawn mower STC C6M 1 3.03 4.40 2 4.04 4.38 3 5.34 7.64 4 6.42 8.28 5 4.17 7.21 6 1.23 1.43 7 4.10 3.91 8 2.21 1.89 9 6.57 7.14 10 3.80 4.71 11 4.76 6.80 12 .49 .01 13 1.97 2.91 14 1.64 1.23 15 3.26 2.72 16 4.20 6.95 17 .32 .11 18 7.76 8.73 19 4.79 6.75 20 .98 1.12
=+certification), which measures emissions for a 10-sec period, and an experimental method C6M, which is a weighted average of emission rates obtained under six different combinations of running speeds, times, and engine loads.
=+ (nitrogen oxide) emission rates were compared by using both methods on several models of gas-powered lawn mowers. The following table shows NOx emission rates (grams/kWh) for two measuring methods: STC (similar to
=+34. Youden plots are frequently used to compare two different instruments or evaluation methods. In a study of lawn mower exhaust emissions (“Exhaust Emissions from Four-Stroke Lawn Mower Engines,” J. of Air and Waste Management, 1997: 945–950), two methods of measuring NOx
=+c. Air samples were taken at two different locations, an industrial area and an undeveloped commercial site. Samples were collected at each site during six 24-hour sampling periods; wet and dry days were included. What general experimental design principles are illustrated here?
=+b. The authors used ASTM Standard Test Method D5281–92 when measuring the concentrations of Cr(VI). What experimental purpose does using such a standard serve?
=+ What role would such samples play in an experiment to subsequently evaluate emissions at chromite ore plants?
=+a. In the study, background samples of air were selected to be representative of land use in the vicinity of chromite ore processing sites, but not so close that these samples would be affected by emissions from the processing plants.
=+sampling plan was devised to estimate ambient levels of Cr(VI) in the air [“Background Air Concentrations of Cr(VI) in Hudson County, New Jersey:Implications for Setting Health-Based Standards for Cr(VI) in Soil,” J. of Air and Waste Management, 1997: 592–597]. The authors propose using
=+Copyright 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially
=+33. Cr(VI) is a pollutant associated with chromite ore processing. In a study of Cr(VI) concentrations, a Laboratory Replicate 1 Replicate 2 1 11.700 11.502 2 9.790 10.300 3 12.760 12.073 4 10.400 9.800 5 10.648 10.904 6 11.074 11.072 7 10.820 10.746 8 11.473 11.682 9 10.723 11.545 Laboratory
=+b. Generalize your result in part (a). That is, if xtag1 is the number of fish caught and tagged in the first sample and xtag2 is the number of tagged fish found in a second sample of size y, write an equation for the estimated value of T.
=+a. Suppose there are five tagged fish found in the second sample. Because the samples are assumed to be random samples from the entire population of T fish, the proportion of tagged fish in the second sample should be approximately equal to the proportion of tagged fish in the population. Use
=+suppose an initial sample of 100 fish from a lake are caught and tagged. After releasing the fish and allowing sufficient time for them to mix with the rest of the fish in the lake, a second sample of, say, 50 fish are caught. The number of tagged fish in the second sample is counted.
=+32. The method of capture–recapture sampling is often used to estimate the size of wildlife populations(Thompson, S. K., Sampling, John Wiley & Sons, New York, 1992: 212–233). To illustrate the method,
=+31. A common method for selecting a random sample without replacement from the integers 1, 2, 3, . . . , N is to generate a random sample with replacement (using random number tables or a software program) and then discard any duplicate numbers that appear in the sample.Use the sampling rules in
=+30. Consult a published reference, weather bureau, or Internet site to determine the operational definition used by weather forecasters when making statements like “There will be a 30% chance of rain tomorrow.”
=+b. What conclusions can you draw regarding the test procedures being used in these laboratories?Are there any unusual MFI measurements?Supplementary Exercises
=+a. Create a Youden plot of this data.
=+a lack of reference standards for measuring MFI. To address this issue, the authors of “An Interlaboratory Comparison of the Melt Flow Index: Relevant Aspects for the Participant Laboratories” (Polymer Testing, 2007: 576–586) report MFI readings by 24 laboratories for various polypropylene
=+29. The melt flow index (MFI) of a polymer is defined to be the amount of the polymer (in grams) that can flow in 10 minutes through a standard die when subjected to a specified force and temperature. MFI is widely regarded as an important characteristic for commercial polymer processing.
=+b. What type of variation is measured by calculating the sample standard deviation, s, of the six measurements?
=+a. In the language of experimental design, can these six measurements be considered replications?
=+28. After carefully controlling all the chemical reagents and conditions during an experiment, a chemist weighs the amount of reactant produced by an experiment. The chemist weighs the reactant on an electronic balance, then reweighs the reactant five times, being careful to remove and replace
=+its operating range of 250°F to 150°F. What is the maximum absolute error you would expect in a measured reading of 70°F from this thermometer?
=+b. Relative errors are often stated in terms of the maximum relative error to be expected for any measurements within the range of an instrument. Suppose, for example, that a thermometer has a maximum relative error of 64% over
=+a. Calculate the relative errors for each of the five measurements in Exercise 25.
=+27. Many instrument makers report the accuracy of their instruments in terms of relative error as well as absolute error. The relative error in a measurement is defined as (m2x)yx ? 100%, where m is the measured value and x is the true value. Absolute error is given by |m2x|.
=+26. Calibration is the process of comparing an instrument’s measurements to those of a reliable reference standard. If necessary, the instrument is adjusted to bring its measurements into agreement with the reference standard. Explain what effect calibration has on the estimated precision
=+25. To estimate the accuracy and precision of an instrument that measures lengths, a .300-in. gauge block was used as a reference standard and was measured five times. The resulting measurements were: .301,.303, .299, .305, and .304. Calculate estimates of both the accuracy and the precision of
=+(a) is caused by switching brands or is simply due to experimental variation. Describe how you would improve this experiment to obtain an estimate of the experimental error.
=+c. Because this experiment does not provide any estimate of the experimental error expected in successive experimental runs, it is impossible to know whether the estimated change in part
=+b. Calculate an estimate of how much plastic hardness is increased or decreased by switching from machine 1 to machine 2.
=+a. Calculate an estimate of how much plastic hardness is increased or decreased by switching from the brand A resin to the brand B resin.
=+24. Refer to Example 4.15 and Figure 4.2(b). Suppose the hardness measurements (in Mohs) of plastics in four test runs are as follows:2.6 2.8 3.2 3.6 Brand A Machine 1 Machine 2 Brand B
=+What two basic experimental design principles are violated by this experimental procedure?
=+be repeated up to six times in any given day. Consequently, one lab assistant is assigned to set up the lab equipment and then conduct six runs one day. The next day a second lab assistant conducts another six runs using the same lab setup from the previous day.
=+A lengthy lab equipment setup, followed by a tedious experimental procedure, allows the experiment to
=+23. A complex chemical experiment is conducted and, because the amount of precipitate produced is expected to vary, the experiment is repeated several times.
=+b. If it is suspected that there could be significant differences in the growing conditions among the four main subplots, is one of the two designs preferable over the other? Why?
=+a. If care were taken to ensure that there are no significant differences in the growing conditions(soil type, irrigation, drainage, sunlight, etc.)among the four large subplots, is one of these designs preferable over the other? Why?
=+are randomly assigned to the large subplots, whereas in experiment B, all four fertilizers are randomly assigned to the subplots of the four large plots. An illustration of both experimental designs follows.M1 Experiment A M1 M1 M1 M4 M4 M4 M4 M3 M3 M3 M3 M2 M2 M2 M2 M1 Experiment B M4 M3 M2 M3
=+divided into four more squares. Two experimental methods are proposed for applying the fertilizers to the subplots. In experiment A, the four fertilizers
=+seedlings. A square plot of land is subdivided into four equal-size square plots, each planted with the same amount, by weight, of seedlings. Before the fertilizers are applied, each square subplot is itself
=+22. In a study of the ratio of nitrogen, phosphoric acid, and potash in fertilizers, four different mixtures(M1, M2, M3, M4) of the three chemicals are to be tested for their effects on the rate of growth of grass
=+c. What changes would you make to the experiment to increase the generalizability of the experimental results?
=+b. What operational definitions would you suggest that the researcher incorporate into this experiment?
=+a. What is the purpose of measuring efficiency every 100 miles? Why not just measure efficiency at the end of the 500-mile course?
=+proposes that a car be driven for a total of 500 miles and that at the end of each 100-mile segment the fuel efficiency be measured and recorded.
=+21. A researcher wants to test the effectiveness of a new fuel additive for increasing the fuel efficiency (miles per gallon, mpg) of automobiles. The researcher
=+b. Suppose it is known that, even for a fixed photoresist thickness, pit depths can vary substantially.Answer the question posed in part (a) for this situation.
=+with little or no replication and several photoresist thickness levels or (2) an experiment with more replication, but fewer photoresist thickness levels?
=+a. Suppose that it is known that, for any fixed photoresist thickness, there tends to be little, if any, variation in the pit depths on a DVD.Which would be better: (1) an experiment
=+depth of the holes etched on the surface of the DVD.The experiment must be conducted under a fixed budget and time constraint that allows the researcher to analyze a sample of at most 20 DVDs.
=+a researcher decides to examine several different photoresist thicknesses used in making the plates from which plastic DVDs are stamped. As a response variable, the researcher decides to measure the average pit
=+20. In a study of factors that affect the ability of the laser in a DVD player to read the information on a DVD,
=+19. What primary purpose do replicated measurements serve in an experimental design?
=+how you would design an experiment that fairly compares the word processing programs while simultaneously accounting for possible differences in users’ computer proficiency.
=+18. Four new word processing software programs are to be compared by measuring the speed with which various standard tasks can be completed. Before conducting the tests, researchers note that the level of a person’s computer experience is likely to have a large influence on the test results.
=+17. In stratified sampling, what value would you use in place of 1.96 if you wanted the confidence level to be 99% rather than 95%? What is the consequence of using the higher confidence level on the necessary sample size?
=+16. In stratified sampling, explain why the number of strata, k, should not exceed ny2, where n 5 n1 1 n2 1 n3 1 1 nk is the total sample size and ni denotes the number of sampled items selected from stratum Si(i 5 1, 2, 3, . . . , k).
=+b. In the case where all unit sampling costs are equal and all strata variances are equal, show algebraically that the resulting weights give the formulas for n and ni specified by the “proportional” allocation.
=+a. In the case where all unit sampling costs are equal, show that the resulting weights give the formulas for n and ni specified by the Neyman allocation.
=+15. When the per unit cost of sampling from stratum i is ci, it can be shown that the optimal weights for allocating the total sample size are given by wi 5 Ni i 1ci N1 1 1c1 1N2 2 1c2 1N3 3 1c3 1 1 Nk k 1ck
=+14. Of the elements of a certain population 20% are grouped into stratum S1 and the rest of the population elements comprise stratum S2. Suppose that the variances of the characteristic being measured are the same for each stratum, but it costs twice as much to obtain a sampled item from stratum
=+d. Calculate the standard error associated with the estimate in part (c).
=+c. Using the sample sizes in part (b), the results of the study showed the following numbers of pinholes per sample:Sample #: 1 2 3 4 5 6 7 8 9 10 Pinholes: 5 4 7 6 3 9 5 6 2 8 Calculate the stratified estimate of the proportion of pinholes on the entire IC.
=+b. Using a confidence level of 90% and a bound on the error of estimation of B 5 .03 (i.e., 63%), calculate the required sample size n and its allocation n1, n2, n3, . . . , n10 to the ten strata.Round all sample sizes to the nearest integer.
=+a. Calculate the population size N.
=+of pinholes on an IC, its entire surface was first divided into 10 equal areas (strata), each of which was further subdivided into 1000 smaller rectangles that served as the elements to be sampled. It was also assumed that the unit costs and variances of the numbers of pinholes were equal from
=+A stratified estimate of the overall proportion of defects can be used to help estimate the eventual yield of the IC manufacturing process.In one such study, to estimate the proportion
=+IC. The area of the IC is first divided into smaller areas (i.e., strata) and then small sample areas are selected from the strata and examined for defects.
=+13. Integrated circuits (ICs) consist of thousands of small circuits, electronic subcomponents (e.g., resistors), and connections. An important factor in the manufacture of ICs is the yield, the percentage of manufactured ICs that function correctly. Stratified sampling has recently been used
=+b. Show that the weighted average in part (a)simplifies to (x1 1 x2 1 x3 1 1 xk)y(n1 1 n2 1 n3 1 1 nk).
=+a. Write an expression for the weighted average of the sample proportions, using the stratum sizes as weights.
=+yni for each i).
=+12. A population of items is partitioned into k strata of sizes N1, N2, . . . , Nk. Using proportional allocation, random samples of size n1, n2, n3, . . . , nk are selected from the strata and the numbers x1, x2, x3, . . . , xk of items having a specified characteristic are determined. Sample
=+11. Explain how to use the5RANDBETWEEN function in Excel™ to generate a random sample from the integers 1 through 1000. Does the 5RANDBETWEEN function generate samples with or without replacement?
=+10. In stratified sampling, explain why it is best to choose strata such that the objects within any stratum are relatively homogeneous.
=+on the four sides of the hill, the hill should be divided into four quadrants and trees should be randomly sampled from each quadrant. What is the name for this type of sampling procedure?
=+b. Suppose that the group of trees is grown on the top and sides of a small hill. A researcher suggests that, because growing conditions(e.g., daily amounts of sunlight) are different
=+a. Without performing any calculations, do you think that both methods are capable of generating random samples from the block of trees?Justify your answer using the rules for random samples listed in this section.
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