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Miller And Freunds Probability And Statistics For Engineers 9th Edition Richard A. Johnson - Solutions

11.21 The decomposition of the sums of squares into a con- tribution due to error and a contribution due to regres- sion underlies the least squares analysis. Consider the identity y---)=(y-y) Note that y=a+bxy-bx+bx=y+b(x-x) so i-y=b(x-x)Using this last expression, then the definition of b and
11.20 Using the formulas on page 330 for α and β , show that (a) the expression for α is linear in the Yi (b) α is an unbiased estimate of α (c) the expression for β is linear in the Yi (d) β is an unbiased estimate of β
11.19 When the sum of the x values is equal to zero, the calculation of the coefficients of the regression line of Y on x is greatly simplified; in fact, their estimates are given by α = y n and β = x y x2 This simplification can also be attained when the values of x are equally spaced;
11.18 With reference to Exercise 11.16, fit a straight line to the data by the method of least squares, using Mext as the independent variable, and draw its graph on the diagram obtained in part (a) of Exercise 11.16. Note that the two estimated regression lines do not coincide.
11.17 With reference to Exercise 11.16, find a 90% confidence interval for α.
11.16 Recycling concrete aggregate is an important component of green engineering. The strength of any potential material, expressed in terms of its resilient modulus, must meet standards before it is incorporated in the base of new roadways. There are two methods of obtaining the resilient
origin.
11.15 In Exercise 11.4 it would have been entirely reasonable to impose the condition α = 0 before fitting a straight line by the method of least squares. (a) Use the method of least squares to derive a formula for estimating β when the regression line has the form y = βx. (b) With reference
11.14 With reference to Exercise 11.3, express 90% limits of prediction for the tearing strength in terms of the temperature x0. Choosing suitable values of x0, sketch graphs of the loci of upper and lower limits of prediction on the diagram of part (a) of Exercise 11.3. Note that since any two
11.13 With reference to the preceding exercise, find 99% limits of prediction for the level of air pollution when the flow of vehicles is 30%. Also indicate to what extent the width of the interval is affected by the size of the sample and to what extent it is affected by the inherent variability
11.12 The level of pollution because of vehicular emissions in a city is not regulated. Measurements by the local government of the change in flow of vehicles and the change in the level of air pollution (both in percentages) on 12 days yielded the following results: Change in flow Change in level
11.11 With reference to Exercise 11.9, test the null hypothesis β = 0.40 against the alternative hypothesis β > 0.40 at the 0.05 level of significance.
11.10 With reference to Exercise 11.9, construct a 95% confidence interval for α.
11.9 Scientists searching for higher performance flexible structures created a diode with organic and inorganic layers. It has excellent mechanical bending properties. Applying a bending strain to the diode actually leads to higher current density(mA/cm2). Metal curvature molds, each having a
11.8 With reference to Exercise 11.6, find (a) a 90% confidence interval for the average number of classes attended each day by a student present for 15 days; (b) 90% limits of prediction for the number of classes attended each day by a student present for 15 days.
11.7 With reference to the preceding exercise, test the null hypothesis β = 0.75 against the alternative hypothesis β < 0.75 at the 0.10 level of significance.
11.6 The following table shows how many days in December a sample of 6 students were present at their university and the number of lectures each attended on a given day. Number of Number of days present lectures attended x y 12 3 8 2 13 5 10 4 7 1 10 3 (a) Find the equation of the least squares
11.5 With reference to the preceding exercise, (a) construct a 95% confidence interval for β, the elongation per thousand pounds of tensile stress; (b) find 95% limits of prediction for the elongation of a specimen with x = 3.5 thousand pounds.
11.4 In the accompanying table, x is the tensile force applied to a steel specimen in thousands of pounds, and y is the resulting elongation in thousandths of an inch: x 123456 y 14 33 40 63 76 85(a) Graph the data to verify that it is reasonable to assume that the regression of Y on x is linear.
11.3 A textile company, wanting to know the effect of temperature on the tearing strength of a fiber, obtained the data shown in the following table. Temperature Tearing strength (◦C) (g) x y 20 1,600 22 1,700 25 2,100 35 2,500 18 1,550 29 2,600 31 2,550 16 1,100 13 1,050 48 2,650 (a) Draw a
11.2 A motorist found that the efficiency of her engine could be increased by adding lubricating oil to fuel. She experimented with different amounts of lubricating oil and the data are Amount of lubricating oil (ml) Efficiency (%) 0 60 25 70 50 75 75 81 100 84 (a) Obtain the least squares fit of
11.1 A chemical engineer found that by adding different amounts of an additive to gasoline, she could reduce the amount of nitrous oxides (NOx) coming from an automobile engine. A specified amount will be added to a gallon of gas and the total amount of NOx in the exhaust collected. Initially, five
10.62 With reference to Example 13, repeat the analysis after combining the categories below average and average in the training program and the categories poor and average in success. Comment on the form of the dependence.
10.61 The following is the distribution of the daily number of power failures reported in a western city on 300 days: Number of Number power failures of days 0 9 1 43 2 64 3 62 4 42 5 36 6 22 7 14 8 6 9 2 Test at the 0.05 level of significance whether the daily number of power failures in this
10.60 Two hundred tires of each of four brands are individually placed in a testing apparatus and run until failure. The results are obtained the results shown in the following table: Brand A Brand B Brand C Brand D Failed to last 30,000 miles 26 23 15 32 Lasted from 30,000 to 40,000 118 93 116
10.59 Cooling pipes at three nuclear power plants are investigated for deposits that would inhibit the flow of water. From 30 randomly selected spots at each plant, 13 from the first plant, 8 from the second plant, and 19 from the third were clogged. (a) Use the 0.05 level to test the null
10.58 With reference to Exercise 10.57, find a large sample 95% confidence interval for the true difference of the probabilities of failure.
10.57 Two bonding agents, A and B, are available for making a laminated beam. Of 50 beams made with Agent A, 11 failed a stress test, whereas 19 of the 50 beams made with Agent B failed. At the 0.05 level, can we conclude that Agent A is better than Agent B?
10.56 With reference to Exercise 10.55, find a large sample 95% confidence interval for the true difference of probabilities.
10.55 As a check on the quality of eye glasses purchased over the internet, glasses were individually ordered from several different online vendors. Among the 92 lenses with antireflection coating, 61 prescriptions required a thickness at the center greater than 1.9 mm and 31 were thinner. Of the
10.54 Refer to Example 5 but suppose there are two additional design plans B and C for making miniature drones. Under B, 10 of 40 drones failed the initial test and under C 15 of 39 failed. Consider the results for all three design plans. Use the 0.05 level of significance to test the null
10.53 With reference to Exercise 10.52, test the hypothesis p = 0.05 versus the alternative hypothesis p > 0.05 at the 0.05 level.
10.52 In a random sample of 150 trainees at a factory, 12 did not complete the training. Construct a 99% confidence interval for the true proportion of trainees not completing their training using the large sample confidence interval formula.
10.51 With reference to Exercise 10.50, test the null hypothesis p = 0.18 versus the alternative hypothesis p = 0.18 at the 0.01 level.
10.50 In a random sample of 160 workers exposed to a certain amount of radiation, 24 experienced some ill effects. Construct a 99% confidence interval for the corresponding true percentage using the large sample confidence interval formula.
10.49 With reference to Exercise 10.48, test the null hypothesis p = 0.20 versus the alternative hypothesis p < 0.20 at the 0.05 level.
10.48 In a sample of 100 ceramic pistons made for an experimental diesel engine, 18 were cracked. Construct a 95% confidence interval for the true proportion of cracked pistons using the large sample confidence interval formula.
10.47 The procedure in Exercise 10.46 also calculates the chi square test for independence. Do Exercise 10.40 using the computer.
10.46 A chi square test is easily implemented on a computer. With the counts 31 42 22 25 19 8 28 25 from Example 8 in columns 1–4, the MINITAB commands Dialog box: Stat > Tables > Chi-square Test for Association Pull down Summarized data in a two-way table. Type C1 − C4 in columns containing
10.45 Among 100 purification filters used in an experiment, 46 had a service life of less than 20 hours, 19 had a service life of 20 or more but less than 40 hours, 17 had a service life of 40 or more but less than 60 hours, 12 had a service life of 60 or more but less than 80 hours, and 6 had a
10.44 The following is the distribution of the hourly number of trucks arriving at a company’s warehouse: Trucks arriving per hour Frequency 0 52 1 151 2 130 3 102 4 45 5 12 6 5 7 1 8 2 Find the mean of this distribution, and using it (rounded to one decimal place) as the parameter λ, fit a
10.43 With reference to Exercise 10.42, verify that the mean of the observed distribution is 1.6, corresponding to 40% of the cars requiring repairs. Then look up the probabilities for n = 5 and p = 0.25 in Table 1, calculate the expected frequencies, and test at the 0.05 level of significance
10.42 An engineer takes samples on a daily basis of n = 5 cars coming to a workshop to be checked for repairs and on 250 consecutive days the data summarized in the following table are obtained: Number requiring Number repairs of days 0 25 1 112 2 63 3 68 4 12 To test the claim that 20% of all cars
10.41 Tests of the fidelity and the selectivity of 190 digital radio receivers produced the results shown in the following table: Fidelity Low Average High Low 6 12 32 Selectivity Average 33 61 18 High 13 15 0 Use the 0.01 level of significance to test whether there is a relationship (dependence)
10.40 A large electronics firm that hires many workers with disabilities wants to determine whether their disabilities affect such workers’ performance. Use the level of significance α = 0.05 to decide on the basis of the sample data shown in the following table whether it is reasonable to
10.39 Referring to Example 12 and the data on repair, use the 0.05 level of significance to test whether there is homogeneity among the shops’ repair distributions.
10.38 Verify that the square of the Z statistic on page 314 equals the χ2 statistic on page 312 for k = 2.
10.37 Verify that if the expected frequencies are determined in accordance with the rule on page 312, the sum of the expected frequencies for each row and column equals the sum of the corresponding observed frequencies.
10.36 Verify that the formulas for the χ2 statistic on page 311 (with p substituted for the pi) and on page 312 are equivalent.
10.35 With reference to part (b) of Exercise 10.34, find a large sample 99% confidence interval for the true difference of the proportions.
10.34 To test the null hypothesis that the difference between two population proportions equals some constant δ0, not necessarily 0, we can use the statistic Z = X1 n1 − X2 n2 − δ0 X1 n1 1 − X1 n1 n1 + X2 n2 1 − X2 n2 n2 which, for large samples, is a random variable having the
10.33 With reference to Exercise 10.32, find a large sample 99% confidence interval for the true difference of the proportions.
10.32 Photolithography plays a central role in manufacturing integrated circuits made on thin disks of silicon. Prior to a quality-improvement program, too many rework operations were required. In a sample of 200 units, 26 required reworking of the photolithographic step. Following training in the
10.31 With reference to the preceding exercise, verify that the square of the value obtained for Z in part (b) equals the value obtained for χ2 in part (a).
10.30 A factory owner must decide which of two alternative electric generator systems should be installed in their factory. If each generator is tested 175 times and the first generator fails to work (does not start or does not transmit electricity) 15 times and the second generator fails to work
10.29 The following data come from a study in which random samples of the employees of three government agencies were asked questions about their pension plan: Agency 1 Agency 2 Agency 3 For the pension plan 67 84 109 Against the pension plan 33 66 41 Use the 0.01 level of significance to test
10.28 A study showed that 64 of 180 persons who saw a photocopying machine advertised during the telecast of a baseball game and 75 of 180 other persons who saw it advertised on a variety show remembered the brand name 2 hours later. Use the χ2 statistic to test at the 0.05 level of significance
10.27 Tests are made on the proportion of defective castings produced by 5 different molds. If there were 14 defectives among 100 castings made with Mold I, 33 defectives among 200 castings made with Mold II, 21 defectives among 180 castings made with Mold III, 17 defectives among 120 castings
10.26 Refer to Exercise10.25. Suppose a sample of 650 moderate machine bearings yielded 550 bearings that had a work life of more than 5 years. Obtain a 90% confidence interval for the difference in proportions.
10.24 Suppose that 4 of 13 undergraduate engineering students are going on to graduate school. Test the dean’s claim that 60% of the undergraduate students will go on to graduate school, using the alternative hypothesis p < 0.60 and the level of significance α = 0.05. [Hint: Use Table 1 to
10.23 An airline claims that only 6% of all lost luggage is never found. If, in a random sample, 17 of 200 pieces of lost luggage are not found, test the null hypothesis p = 0.06 against the alternative hypothesis p > 0.06 at the 0.05 level of significance.
10.22 In a random sample of 600 cars making a right turn at a certain intersection, 157 pulled into the wrong lane. Test the null hypothesis that actually 30% of all drivers make this mistake at the given intersection, using the alternative hypothesis p = 0.30 and the level of significance (a) α
10.21 To check on an ambulance service’s claim that at least 40% of its calls are life-threatening emergencies, a random sample was taken from its files, and it was found that only 49 of 150 calls were life-threatening emergencies. Can the null hypothesis p = 0.40 be rejected against the
10.20 A supplier of imported vernier calipers claims that 90% of their instruments have a precision of 0.999. Testing the null hypothesis p= 0.90 against the alternative hypothesis p = 0.90, what can we conclude at the level of significance α = 0.10, if there were 665 calipers out of 700 with a
10.19 A manufacturer of submersible pumps claims that at most 30% of the pumps require repairs within the first 5 years of operation. If a random sample of 120 of these pumps includes 47 which required repairs within the first 5 years, test the null hypothesis p = 0.30 against the alternative
10.18 An international corporation needed several millions of words, from thousands of documents and manuals, translated. The work was contracted to a company that used computer-assisted translation, along with some human checks. The corporation conducted its own quality check by sampling the
10.17 A chemical laboratory was facing issues with the concentration of the sulfuric acid they prepared. The first step was to collect data on the magnitude of the problem. Of 5,186 recently supplied acid vials, 846 had concentration issues that could easily be detected by a basic chemical test.
10.16 Use the formula of Exercise 10.15 to rework Exercise 10.3.
10.15 Show that the inequality on page 304 leads to the following (1 − α)100% confidence limits: x + 1 2 z 2 α/2 ± zα/2 x ( n − x) n + 1 4 z 2 α/2 n + z2 α/2
10.14 Use Exercise 10.13 or other software to obtain the interval requested in Exercise 10.3.
10.13 MINITAB determination of confidence interval for p When the sample size is not large, the confidence interval for a proportion p can be obtained using the following commands. We illustrate the case n = 20 and x = 4. Stat > Basic Statistics > 1-Proportion. Choose Summarized outcomes. Type 4 in
10.12 Refer to Example 1. How large a sample of wind turbines is needed to ensure that, with at least 95% confidence, the error in our estimate of the sample proportion is at most 0.06 if (a) nothing is known about the population proportion? (b) the population proportion is known not to exceed
10.11 Suppose that we want to estimate what percentage of all bearings wears out due to friction within a year of installation. How large a sample will we need to be at least 90% confident that the error of our estimate, the sample percentage, is at most 2.25%?
10.10 With reference to Exercise 10.9, how would the required sample size be affected if it is known that the proportion to be estimated is at least 0.75?
10.9 What is the size of the smallest sample required to estimate an unknown proportion of customers who would pay for an additional service, to within a maximum error of 0.06 with at least 95% confidence?
10.8 New findings suggest many persons possess symptoms of motion sickness after watching a 3D movie. One scientist administered a questionnaire to n = 451 adults after they watched a 3D movie of their choice. Based on these self-reported results, 247 are determined to have some motion sickness.
10.7 Among 100 fish caught in a large lake, 18 were inedible due to the pollution of the environment. If we use 18 100 = 0.18 as an estimate of the corresponding true proportion, with what confidence can we assert that the error of this estimate is at most 0.065?
10.6 In an experiment, 85 of 125 processors were observed to process data at a speed of 4,700 MIPS. If we estimate the corresponding true proportion as 85 125 = 0.68, what can we say with 99% confidence about the maximum error?
10.5 In a random sample of 140 observations of workers on a site, 25 were found to be idle. Construct a 99% confidence interval for the true proportion of workers found idle, using the large sample confidence interval formula.
10.4 With reference to Exercise 10.3, what can we say with 95% confidence about the maximum error if we use the sample proportion to estimate the corresponding true proportion?
10.3 In a random sample of 400 industrial accidents, it was found that 231 were due at least partially to unsafe working conditions. Construct a 99% confidence interval for the corresponding true proportion using the large sample confidence interval formula.
10.2 With reference to Exercise 10.1, what can we say with 95% confidence about the maximum error if we use the sample proportion as an estimate of the true proportion of complaints filed against this construction company where the proportion of sand exceeds 75 percent?
10.1 In a random sample of 150 complaints filed against a construction company for mixing excess sand in their concrete mixture, 95 complaints showed that the proportion of sand in the mix exceeded 75 percent. Construct a 90% confidence interval for the true proportion of complaints filed against
9.24 A bioengineering company manufactures a device for externally measuring blood flow. Measurements of the electrical output (milliwatts) on a sample of 16 units yields the data 11 1 5 3 2 23 37 5 18 7 1 11 2 2 30 3 plotted in Figure 9.5. (a) Should you report the 95% confidence interval for σ
9.23 MINITAB calculation of tα, χ2 ν, and Fα The software finds percentiles, so to obtain Fα, we first convert from α to 1 − α. We illustrate with the calculation of F0.025(4, 7), where 1 − 0.025 = 0.975. Dialog box: Calc> Probability distributions > F. Choose Inverse cumulative
9.22 With reference to the Example 8, Chapter 8, find a 98% confidence interval for the ratio of variances of the two aluminum alloys.
9.21 With reference to the Example 8, Chapter 8, test the equality of the variances for the two aluminum alloys. Use the 0.02 level of significance.
9.20 Thermal resistance tests on 13 samples of Enterococcus hirae, present in milk, yield the following results in degrees Celsius: 65.5 65.8 68.1 67.9 66.6 66.2 65.7 67.8 65.4 67.5 66.8 65.2 67.8 Another set of seven samples of milk was tested after pasteurization to determine whether thermal
9.19 Past data indicate that the variance of measurements made on sheet metal stampings by experienced quality-control inspectors is 0.18 (inch)2. Such measurements made by an inexperienced inspector could have too large a variance (perhaps because of inability to read instruments properly) or
9.18 If 44 measurements of the refractive index of a diamond have a standard deviation of 2.419, construct a 95% confidence interval for the true standard deviation of such measurements. What assumptions did you make about the population?
9.17 With reference to Example 20, Chapter 7, construct a 95% confidence interval for the true standard deviation of the lead content.
9.16 With reference to Exercise 8.6, where we had n1 = 40, n2 = 30,s1 = 15.2, and s2 = 18.7, use the 0.05 level of significance to test the claim that there is a greater variability in the number of cars which make left turns approaching from the south between 4 p.m. and 6 p.m. at the second
9.15 Two different computer processors are compared by measuring the processing speed for different operations performed by computers using the two processors. If 12 measurements with the first processor had a standard deviation of 0.1 GHz and 16 measurements with the second processor had a
9.14 With reference to Exercise 8.10, use the 0.10 level of significance to test the assumption that the two populations have equal variances.
9.13 Explore the use of the two sample t test in Exercise 8.9 by testing the null hypothesis that the two populations have equal variances. Use the 0.02 level of significance.
9.12 The fire department of a city wants to test the null hypothesis that σ = 10 minutes for the time it takes a fire truck to reach a fire site against the alternative hypothesis σ = 10 minutes. What can it conclude at the 0.05 level of significance if a random sample of size n = 48 yields s
9.11 Playing 10 rounds of golf on his home course, a golf professional averaged 71.3 with a standard deviation of 2.64. (a) Test the null hypothesis that the consistency of his game on his home course is actually measured by σ = 2.40, against the alternative hypothesis that he is less consistent.
9.10 Use the 0.01 level of significance to test the null hypothesis that σ = 0.015 inch for the diameters of certain bolts against the alternative hypothesis that σ = 0.015 inch, given that a random sample of size 15 yielded s2 = 0.00011.
9.9 With reference to Exercise 8.5, test the null hypothesis that σ = 0.75 hours for the time that is required for repairs of the second type of bulldozer against the alternative hypothesis that σ > 0.75 hours. Use the 0.10 level of significance and assume normality.
9.8 If 15 determinations of the purity of gold have a standard deviation of 0.0015, test the null hypothesis that σ = 0.002 for such determinations. Use the alternative hypothesis σ = 0.002 and the level of significance α = 0.05.

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