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Statistics And Probability With Applications For Engineers And Scientists Using MINITAB R And JMP 2nd Edition Irwin Guttman, Kalanka P. Jayalath, Bhisham C. Gupta - Solutions
12. Consider the data given in Review Problem 11.(a) Use the logistic regression concepts discussed in Section 11.6 (see Section 16.8 for more details) to predict the outcome (i.e., “Status”) using the variable“Age.”(b) Use a cutoff value of 0.5 to obtain the confusion matrix for your
11. Use the following data set to obtain the confusion matrix and report both class 0 and class 1 misclassification rates.Age Status Predicted 38 0 0 43 1 1 43 1 1 39 0 1 43 1 1 36 0 0 31 0 0 39 0 1 39 1 1 39 1 1 43 1 1 42 1 1 35 0 0 44 1 1 49 0 0
10. Consider the data given in Review Problem 4.(a) Obtain a contingency table to investigate the relationship between “SibSp” and“Survival.” Comment on the relationship(s).(b) Obtain a contingency table to investigate the relationship between “Parch” and“Survival.” Comment on the
9. Consider the data given in Review Problem 4.(a) Obtain a contingency table to investigate the relationship between “Sex” and“Survival.” Comment on the relationship(s).(b) Obtain a contingency table to investigate the relationship between “Pclass” and“Survival.” Comment on the
8. Consider the data given in Review Problem 4.(a) Obtain a side-by-side boxplots for “Fare” for “Pclass” and comment on your graphs.(b) Obtain an aesthetic scatter plot for “Age,” “Fare,” “Sex,” and “Survival” and comment on your graph.
7. Consider the data given in Review Problem 4.(a) Obtain a 2D scatter plot of variables “Fare” versus “Age” and identify the nature of survival considering “Survival” as the third variable.(b) Obtain a 3D scatter by plotting variables “Age,” “Fare,” and “Survival” against
6. Conduct a thorough outlier analysis for the appropriate variables given in data in Review Problem 4.
5. Address the precautions one can take to deal with the missing values exist in the data given in Review Problem 4.
4. Consider a part of the famous “Titanic” data set3 listed in “Review Problem 4”(see website: www.wiley.com/college/gupta/statistics2e). It includes the following variables:Passenger ID: Identification number for data analysis purpose.Survival: 0 = No, 1 = Yes.Pclass: Ticket class, 1 =
3. What are the two main ways one can visualize data? Explain.
2. What are the three key points one should consider in the process of data reduction and preparing for data mining? Explain.
1. What is data mining and what is big data?
14. Refer to Problem 12. Suppose that the recording was terminated after the death of the 13th patient at time t = 17. Use MINITAB or JMP to fit Weibull and lognormal models. Construct the probability, reliability (survival), and hazard plots for both models.
13. Refer to Problem 12. Use MINITAB or JMP to find a 95% confidence interval for the mean.
12. The following data give the survival times in months of 20 patients after their brain tumors were removed:10 17 77 27 14 17 17 17 3 2 20 9 18 7 15 7 24 56 30 16 Fit the lognormal model to this data using MINITAB or JMP, and then construct the probability, survival, and hazard plots.
11. Use the data in Problem 10 to fit Weibull and lognormal models. Construct for both models the probability, reliability (survival), and hazard plots.
10. Consider the following times to failure in hundreds of hours of ten systems placed in service under same conditions. The experimenter decided to terminate the experiment after 2900 hours:15 17 20 21 23 26 28 29+ 29+ 29+Fit an exponential model to these data. Prepare the probability, reliability
9. Develop a sequential sampling plan to test the hypothesis H0: μ0 = 3000 and H1: μ1 =2500 with α and β risks 0.05 and 0.10, respectively, when the measured response time to failure of the test items are normally distributed with mean μ and standard deviation 225 (a) when items are replaced
8. Determine the hazard function for the gamma distribution f(t|β, λ) = (λβ/Γ(β))tβ−1e−λt
7. A sample of n = 12 light bulbs are left on test until the fifth bulb has failed. The recorded times to failure in hours are 1284, 1352, 1397, 1473, and 1530.(a) Construct the 95% confidence limits for the mean time between failure (MTBF), and then estimate the MTBF with 95% confidence.(b) What
6. In Problem 5 above, suppose that the test was stopped at the time of the fourth failure. Estimate, with 95% confidence, the reliability at time t = 4000 hours.
5. A sample of n = 12 radio receivers is placed on a life test. The times to failure, reported after the receivers are on test for 3000 hours, are 1854, 2110, 2270, and 2695. Given that the hazard rate for these items is a constant, estimate their reliability at t = 4000.Review Practice Problems 475
4. Given the hazard function h(t) = βt, (a) determine the appropriate p.d.f. for f(t) and its reliability function R(t); (b) determine f(t) and R(t) when h(t) = β0 + β1t.
3. Referring to the situation discussed in Section 10.3, determine the maximum likelihood estimate of λ, under the assumption of exponential times, that is f(t) = λ exp(−λt), where μ = 1/λ.
2. The life distribution of certain items is N(μ, σ2) with μ = 65 and σ2 = 100. Determine the force of mortality (or hazard rate) of an item at times μ ± σ, μ, μ ± 2σ.
1. To meet contract specifications, the reliability of a certain type of electrical relay must be at least 0.90 at 1000 days of service. Given that the distribution of time to failure is exponential, what is the MTBF of the relays? What is the hazard rate?
9. Suppose that the time to failure of a part in an airbus gas turbine is lognormal with parameters μ and σ2. A random sample of 10 parts is placed on life test, and the recorded times to failure are 46, 49, 54, 58, 60, 64, 66, 69, 72, and 78 months.Determine the MLE of μ and σ2, and then find
8. Refer to Problems 3, 6, and 7, to construct probability plots (for censored data) using MINITAB for the Weibull, lognormal, and exponential distributions, and then use the Anderson–Darling criterion to decide which distribution is the best fit to these data.
7. Suppose that the time to failure in Problem 3 is modeled by the lognormal distribution.Using MINITAB, find the least-squares and MLE estimates of the mean and the standard deviation for both censored data and uncensored data.
6. Suppose that the time to failure in Problem 3 is modeled by theWeibull distribution.Using MINITAB, find the least-squares and MLE estimates of the mean and the standard deviation for both censored data and uncensored data.
5. In Problem 3, suppose that it was decided to complete the life test at time 4500 hours, so that the seven failures recorded in Problem 3 occurred within that time.(a) Estimate the mean time between failures. (b) Estimate the reliability at t = 8000 hours. (c) Estimate the hazard rate. (d) Find a
4. Refer to Problem 3. (a) Estimate the hazard rate. (b) Find a 99% confidence interval for the mean time between failures. (c) Find a 99% confidence interval for the hazard rate. (d) Estimate the mean time to failure with 95% confidence.
3. Suppose that a random sample of 15 hard drives is placed on life test and that the test was concluded after the seventh failure. The recorded times to failure are 3056, 3245, 3375, 3450, 3524, 3875, and 4380 hours. Estimate the mean time between failures.
2. Referring to Problem 1, develop a sequential test for testing a hypothesis H0: R(100) = 0.95 versus H1: R(100) = 0.80, using (a) α = 0.05, β = 0.10; (b)α = 0.10, β = 0.10.
1. The time to failure (in months) of a component in a supercomputer is appropriately represented by an exponential distribution. Develop a sequential test for testing a hypothesis H0: μ = μ0 = 120 versus H1: μ = μ1 = 72, using (a) α = 0.05, β = 0.05;(b) α = 0.05, β = 0.10.
6. Referring to Problem 4, find a 99% confidence interval for the mean time to failure.Also find a 99% confidence interval for the reliability of the part at t = 70 months
5. Referring to Problem 4, estimate the hazard rate and find a 99% confidence interval for the hazard rate.
4. A random sample of 10 automobile voltage regulators is placed on life test, and it is decided to complete the life test after 78 months. The recorded times to failure are 58, 62, 68, 74, and 77 months. Assuming that the time to failure of the voltage 10.3 Hypothesis Testing: Exponential
3. Referring to Problem 1, construct 95% confidence interval for the mean time to failure. Also find a 95% confidence interval for the reliability of the part at t = 1180 hours.
2. Referring to Problem 1, estimate the hazard rate and find a 95% confidence interval for the hazard rate.
1. A random sample of 12 mechanical parts of a system is placed on life test, and the test is concluded after the sixth failure. The recorded times to failure are 964, 1002, 1067, 1099, 1168, and 1260 hours. Assuming that the time to failure of the parts is appropriately represented by an
6. Suppose that in Problem 5, α = 1 and β = 0.5. Find the reliability and the hazard rate at time T = t, where t = 10
5. Suppose that the hazard function of the transmission of a luxury car is given by h(t) = αβtβ−1 1 + αtβ, α>0, β > 0, t ≥ 0 Find the density, reliability, and cumulative hazard function of the life of the transmission.
4. In Problem 2, suppose that the lifetime in months of heart patients after quadruple bypass surgery is modeled by the lognormal distribution with μ = 4 and σ = 1.Find the reliability and the hazard rate function for these patients at six years (72 months).
3. The time T to failure of a computer hard drive is exponentially distributed with mean time to failure μ = 1/λ = 5000 hours. Find the reliability and the hazard rate function for these hard drives at 3000 hours.
2. Suppose that the lifetime, in months, of heart patients after quadruple bypass surgery is modeled by the gamma distribution with shape parameter 3.5 and scale parameter 20. Find the reliability and the hazard rate function for these patients at six years (72 months).
1. Suppose that the lifetime in years of fuel pumps used in an aircraft gas turbine engine is modeled by the Weibull distribution, with threshold parameter 0, shape parameter β = 1.0, and scale parameter α = 30. Find the reliability and the hazard rate function for these pumps at 10 years.
70. Referring to Problem 5 of Section 8.7, determine a 98% confidence interval for the difference between the percentages of persons who favor a nuclear plant in their state and then use it to test at the 2% level of significance the hypothesis H0: p1 − p2 =0 versus H1: p1 − p2 = 0.
69. Referring to Problem 5 of Section 8.7, test at the 5% level of significance the hypothesis that the percentages of persons who favor a nuclear plant in the two states are the same, that is, H0: p1 − p2 = 0 versus H1: p1 − p2 = 0.
68. Use the 99% confidence interval you obtained in Problem 47 of the Review Practice Problems in Chapter 8 to test at the 1% level of significance the hypothesis H0:p1 − p2 = 0 versus H1: p1 − p2 = 0.
67. Referring to Problem 3 of Section 8.7, test at the 5% level of significance the hypothesis that the proportions of the population favoring brand X before and after the advertising campaign are the same, that is, H0: p1 − p2 = 0 versus H1: p1 − p2 = 0.
66. Referring to Problem 2 of Section 8.7, test at the 5% level of significance the hypothesis that the coin is unbiased, that is, H0: p = 0.5 versus H1: p = 0.5.
65. Use the confidence interval you obtained in Problem 1 of Section 8.7 to test, at the 5% level of significance, the hypothesis H0: p = 0.45 versus H1: p = 0.45.442 9 Hypothesis Testing
64. Referring to Problem 1 of Section 8.7, test at the 5% level of significance the hypothesis H0: p = 0.45 versus H1: p < 0.45.
63. The monthly returns in percentage of dollars of two investment portfolios were recorded for one year. The results obtained are Portfolio I 2.1 1.2 −1.5 1.9 0.7 2.5 3.0 −2.2 1.8 0.5 2.0 1.5 Portfolio II 2.9 3.5 −2.8 1.0 −3.0 2.6 −3.5 4.5 1.5 2.3 −1.0 0.8 Assume that the monthly
62. Pull-strength tests were applied on soldered lead in an electronic apparatus for each of two independent random samples of 12. The lead soldering in the two samples were done using two different techniques. The test results indicate the force required in pounds to break the bond. The data
61. In Problem 44, test at the 5% level of significance the hypothesis H0: σ2 1 = σ2 2 against H1: σ2 1 < σ2 2
60. An endocrinologist measured the serum levels of lipid peroxides (LP) among subjects with type I diabetes and also among normal subjects. These data produced the following summary statistics.Diabetic subjects : n1 = 25, ¯X1 = 2.55, S2 1 = 1.475 Normal subjects : n2 = 36, ¯X2 = 2.25, S2 2 =
59. A machine is calibrated to fill bottles with 16 oz of orange juice. A random sample of 12 bottles was selected, and the actual amount of orange juice in each bottle was measured. The data are as follows:15.0 15.9 15.4 16.1 15.2 15.8 16.4 15.7 15.8 16.3 16.5 16.2 Assuming normality, test at the
58. A random sample of 18 observations from a normal population produced a sample mean of 37.4 and a sample variance of 15.6. Do the data provide sufficient evidence to indicate that σ2 < 20? Use the 5% level of significance.Review Practice Problems 441
57. A six-sigma black belt quality control engineer found that in a random sample of 140 printed circuit boards, 18 are defective due to the result of certain nonconformity tests. At the 5% level of significance, test that the percentage of defective printed circuit boards is 10% against the
56. A manufacturer of brass bolts has two plants. A random sample of 300 bolts from plant I showed that 21 of them were defective. Another random sample of 425 bolts from plant II showed that 24 of them were defective. Testing at the 5% level of significance, can you conclude that the proportions
55. A patron of a casino doubts that the dice used in the casino are balanced. During a visit, she rolled a die 100 times and got an even number only 30 times.(a) Formulate the hypothesis you would use to test whether the die is fair.(b) At what observed level of significance would this null
54. An owner of two stores wants to evaluate the customer service at his/her stores. In order to do this, the owner took a random sample of 400 customers from store I and of 500 customers from store II. He asked the customers to rate the service at the store as excellent or not excellent and found
53. The following data shows the weight gain (in pounds) in one week for a sample of 10 pigs before and after they were given a type of hormone:Pig number 1 2 3 4 5 6 7 8 9 10 Before 10 8 9 11 7 8 6 12 8 9 After 15 13 12 10 11 9 11 15 12 16 Assuming normality:(a) Formulate a hypothesis to test the
52. The following are the numbers of defective parts produced in a shift by 10 workers before and after going through a very rigorous training program (assuming normality):Worker 1 2 3 4 5 6 7 8 9 10 Before 15 12 16 14 18 12 13 17 10 16 After 8 5 10 5 14 4 6 6 3 12 440 9 Hypothesis Testing(a) Do
51. A medical school professor claims that medical students study (including the time in class) on average at least 16 hours a day. To verify the professor’s claim, a random sample of 19 students was taken, and each student was asked about the number of hours he or she spends studying each day.
50. Two random samples from two normal populations with standard deviations σ1 = 4.5 and σ2 = 6.2, respectively, produced the following data:Sample from population I 20 30 31 28 34 35 32 26 24 38 25 40 Sample from population II 34 36 49 52 41 44 30 33 47 49 39(a) Test at the 2% level of
49. Repeat Problem 48, assuming that the population variances are not equal.
48. In a study of pregnancy-induced hypertension, two randomly selected groups of women with this diagnosis were selected. One group was treated for a certain period with an aspirin-based medicine and the other group was given a placebo. After the period of treatment, their arterial blood pressures
47. A health insurance company wants to find the average amounts of benefits it pays for a typically insured family of four (a couple with two children). The company selected a random sample of 16 such families and found that it paid on average $4858 with a standard deviation of $575. Assuming
46. Observations of a random sample from a normal population with unknown mean μand unknown standard deviation σ are 25 20 23 28 26 21 30 29 23 29 Review Practice Problems 439(a) Test at the 1% level of significance H0: μ = 25 versus H1: μ = 25.(b) Find the p-value for the test in (a).
45. A random sample of 64 cigarettes of a particular brand yielded mean tar content per cigarette of 15.5 milligrams (mg) and a standard deviation of 1.4 mg.Assuming normality:(a) Test a hypothesis H0: μ = 15 versus H1: μ > 15 at the 0.01 level of significance.(b) Find β the probability of the
44. A manager of a large bank wants to compare the loan amounts of two of her loan officers. The loans issued by the two sales managers during the past three months furnished the following summary statistics:n1 = 55, ¯X1 = $68,750, S1 = $4,930; n2 = 60, ¯X2 = $74,350, S2 = $5,400 Assuming
43. A chemical manufacturing company is interested in increasing the yield of a certain chemical. To achieve this goal, a chemical engineer decides to use the catalyst at two different temperatures, 300 and 350 ◦C. Two samples, each of size n = 49, are produced at each of these temperatures, and
42. The piston rings of certain diameter in mm for automobile engines are being manufactured at two plants. Two random samples of piston rings, one sample from each 438 9 Hypothesis Testing plant, are taken, and the diameters of the rings are measured. These data produce the following sample
41. A rod used in auto engines is required to have a diameter of 18 millimeters (mm).A random sample of 64 rods produced a sample mean of 18.2mm and a standard deviation of 1.2 mm. Assuming normality, test the hypothesis H0: μ = 18 versus H1:μ = 18 at the 5% level of significance. Find the
40. The lifetime (in hours) of AAA batteries used in TI-83 series calculators is assumed to be normally distributed. A random sample of 100 such batteries produced a sample mean of 58.7 hours and a sample standard deviation of 2.5 hours. Test at the 1% level of significance the hypotheses:(a) H0:
39. In Problem 38, find the p-value for the hypotheses in (a) and (b).
38. The mean and the standard deviation of daily intake of vitamin D for a random sample of 36 girls between the ages 16 and 20 years is 450 and 50 mg, respectively. Assuming normality, test the following at the 5% level of significance:(a) H0: μ = 500 against H1: μ < 500(b) H0: μ = 500 against
37. An advising committee at a university is interested in finding the mean time spent by all the university students in watching sports on television. The time spent in watching sports on television by a random sample of 49 students produced a sample mean ¯X of 7.5 hours and a standard deviation
36. If a random variable X has probability density function f(x) given by f(x) =1 √2πσ2 e−(1/2σ2)(x−μ0)2 where μ0 is the known value of the population mean, describe with α = 0.10, β = 0.05, how sequentially to test the hypothesis H0: σ2 = σ2 0 versus H1: σ2 = σ2 1 > σ2 0.
35. If X is a random variable with probability function p(x) =n xpx(1 − p)n−x, x= 0, 1, . . . , n, describe, with α = 0.10, β = 0.05, how to test sequentially the hypothesis H0: p = 0.10 versus H1: p = 0.20.
34. If X is a random variable with probability function f(x) = e−λλx x! , x= 0, 1, 2, . . .describe with α = 0.05, β = 0.01, how to test sequentially the hypothesis H0: λ =λ0 = 1.5 versus H1: λ = λ1 = 2.0.
33. Using the data of Problem 25, test at the 1% level of significance the hypothesisσ2 I = σ2 II .Review Practice Problems 437
32. Using the data of Problem 24, test at the 1% level of significance the hypothesisσ2A= σ2B.
31. In Problem 20, is the assumption of equality of variances valid? Use the significance level 0.01.
30. In Problem 14, is the assumption that σ2A= σ2B warranted on the basis of the data?Use the significance level of 0.05.
29. In Problem 2 of Section 9.7, it was assumed that σ2A= σ2B. Test this assumption at the 1% level of significance.
28. The dependability of analysts is occasionally measured by the variability of their work. Two analysts A and B each make 10 determinations of percent of iron content in a batch of prepared ore from a certain deposit. The sample variances obtained are S2A= 0.4322 and S2B= 0.5006. Are the analysts
27. Two different methods of storing chicken are contrasted by applying technique 1 (a freezing technique) to one-half of a chicken and technique 2 (a wrapping technique)to the other half of the same chicken. Both halves are stored for three weeks, and a certain “tenderness of the meat” test is
26. Over a long period of time, 10 patients selected at random are given two treatments for a specific form of arthritis. The results (in coded units) are given below:Patients Treatment 1 Treatment 2 1 47 52 2 38 35 3 50 52 4 33 35 5 47 46 6 23 27 7 40 45 8 42 41 9 15 17 10 36 41 Is there a
25. Analysts I and II each make a determination of the melting point in degrees centigrade of hydroquinine on each of eight specimens of hydroquinine with the results shown below:Specimen number Analyst I (◦C) Analyst II (◦C)1 174.0 173.0 2 173.5 173.0 3 173.0 172.0 4 173.5 173.0 5 171.5 171.0
23. Two randomly selected groups of 70 trainees each are taught a new assembly line operation by two different methods, with the following results when the groups are tested:Group 1 : n1 = 70, ¯X1 = 268.8, S1 = 20.2 Group 2 : n2 = 70, ¯X2 = 255.4, S2 = 26.8(a) Assuming normality, test the
22. The systolic blood pressure of a group of 70 patients yielded ¯X1 = 145 and S1 = 14.A second group of 70 patients, after being given a certain drug, yielded ¯X2 = 140 and S2 = 9.Review Practice Problems 435(a) Assuming normality, test at the 5% level of significance H0: σ2 1 = σ2 2 versus
21. Orange juice cans are filled using two methods. Two random samples one from each method produced the following results:Method 1 : n1 = 40, ¯X1 = 21.78, S2 1 = 3.11 Method 2 : n2 = 40, ¯X2 = 20.71, S2 2 = 2.40(a) Assuming normality, test at the 5% level of significance H0: σ2 1 = σ2 2 versus
20. A sample of 220 items turned out (during a given week by a certain process) to have average weight of 2.46 lb and standard deviation of 0.57 lb. During the next week, a different lot of raw material was used, and the average weight of a sample of 205 items turned out that week to be 2.55 lb and
19. It has been suspected for some time that the morning shift is more efficient than the afternoon shift. Random observations yield the following data:Morning shift : n1 = 5, ¯X1 = 22.9, S2 1 = 6.75 Afternoon shift : n2 = 7, ¯X2 = 21.5, S2 2 = 7.25 Assuming normality:(a) Test the hypothesis σ2
18. Viewing times of members of households in two different types of communities are sampled, with the following results:Community 1 : n1 = 40, ¯X1 = 19.2 hours/week, S2 1 = 6.4 Community 2 : n2 = 50, ¯X2 = 15.9 hours/week, S2 2 = 3.2(a) Assuming normality, test the hypothesis σ2 1 = σ2 2 at
17. A comparison of yields of marigolds from control plots and treated plots is carried out. Samples from eight control plots and eight treated plots yield the following data:Treated (A) : nA = 8, ¯xA = 128.4, s2 A = 117.1 Nottreated (B) : nB = 8, ¯xB = 96.5, s2 B = 227.7 Assuming normality:(a)
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