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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
9. A sequence of 100 measurements on a process, supposedly under statistical control, was analyzed for runs above and below the median. The total number of runs above 620 14 Nonparametric Tests and below the median was observed to be 39. Is this a significantly low value at the 5% level of
8. The total thicknesses X of the four pads of 36 half-ring (H-R) mounts for aircraft engines, taken periodically from the production line, were found to be as shown below.Determine whether the total number of runs above and below the median of this sequence of values of X is significantly
7. Twenty-four samples of four insecticide dispensers were taken periodically during a production period. The average charge weights (in grams) of the 24 samples are shown below. Use the median test on the sample means to test for randomness. Use α = 0.05.Sample no. ¯X Sample no. ¯X 1 471.5 13
6. Use the Wald–Wolfowitz run test for the data of Problem 9 of Review Practice Problems in Chapter 8 to test the null hypothesis that the samples come from populations having identical continuous c.d.f.s F1(x) and F2(x). Use α = 0.05.Review Practice Problems 619
5. Two samples of 30 observations each from populations A and B are such that when the Mann-Whitney W test for two samples is used, the sum of the ranks in the pooled sample of the X measurements (from population B) is found to be 1085. Hence, the Mann–Whitney test statistic has value
4. Use the sign test for Problem 24 in Review Practice Problems in Chapter 9.
3. The following data show measurements of the corrosion effects of various soils for coated and uncoated steel pipe (from Hoel, 1954). Use the sign test to test the hypothesis that the particular coating used has no effect on corrosion. Use α = 0.05.Uncoated, x: 42 37 61 74 55 57 44 55 37 70 52
2. A consumer group rated 12 manufactured products imported from a single source on a scale of 1–10. The data are shown below. Do the data provide sufficient evidence to indicate that the median score is greater than 5? Use the sign test at the 1% level of significance.2 9 6 7 8 5 10 2 8 5 3 4
1. Fifteen diabetic patients are given 1000 mg of Metformin (500 mg twice a day) and two weeks later their serum sugar levels were as shown below. Do the data give sufficient evidence to indicate that the patients on Metformin have median serum sugar level of 140 mg/dL? Use the Sign test at the 5%
8. The following data give the white blood cell counts and the duration of follow-up(in months) after the operations of some cancer patients. Determine the Spearman rank correlation and test at the 5% level of significance that there is a negative correlation between the white blood cell counts and
7. The following data give systolic blood pressure readings taken by a doctor and her nurse on the last 10 patients who visited the doctor’s office. Determine the Spearman rank correlation and test at the 5% level of significance that there is a positive correlation between the systolic blood
6. A manager of a manufacturing company wished to study the years of service and productivity of a group of technicians. The following data give the years of service and the productivity for 10 technicians. Determine the Spearman rank correlation and test at the 5% level of significance that there
5. The following data give the IQ level and the scores obtained in a standardized test for 12 candidates. Determine the Spearman rank correlation and test at the 5% level of significance that there is a positive correlation between the IQ level and the test scores.IQ: 92 117 100 90 130 108 121 130
4. In a study of the relationship between age and LDL (mg/dL), data were obtained on 10 subjects between 40 and 70 years. The following data give the ages and LDL for each of the 10 subjects. The experimenter wants to know if we can conclude at the 5% level of significance that age and LDL are
3. The following data show heights in centimeters (cm) and average scores per game of 12 randomly selected basketball players. Do these data provide sufficient evidence to indicate an association between the heights and the average scores per game? Useα = 0.10.Heights: 201 199 194 188 198 201 202
2. Ten female candidates are ranked in a beauty competition by two judges. These ranks are shown below. Do these data provide sufficient evidence to indicate an association between the two ranks? Use α = 0.05.JudgeI: 4 7 9 5 10 2 8 3 1 6 JudgeII: 2 10 3 6 5 1 7 8 9 4
1. The following data give the scxores in the final exams on differential equations and quantum mechanics that were received by 10 students randomly selected from a freshmen engineering class. Do these data provide sufficient evidence to indicate an association between the two scores? Use α =
6. The following data give the total yields of a chemical produced by using the same catalyst at two different temperatures. Use the Mann-Whitney test to test the hypothesis at the 5% level of significance that the two temperatures result in the same yield.Temperature 1: 61 60 78 65 68 64
5. The following data give drying times (in hours) of two brands of oil-based paint.Use the Mann-Whitney test to test the hypothesis at the 5% level of significance that both paint brands dry in the same number of hours.Brand 1: 8.5 9.0 8.1 10.0 9.3 9.0 8.0 9.2 Brand 2: 10.3 8.5 8.9 10.2 9.8 8.5
4. A medical doctor wished to test the effect of a cholesterol-lowering drug when it is prescribed to children in one of two forms, tablet or suspension. The following data give the reduction in cholesterol levels (in mg/dL) after a full four weeks of treatment.Can we conclude at the 5% level of
3. The following data give the time (in eight-hour shifts) taken to complete a project by 10 technicians randomly selected from each of the two plants of a company. Based 608 14 Nonparametric Tests on these data, can we conclude at the 5% level of significance that the standards of hiring
2. A coordinator of an engineering program was interested in testing the effectiveness of onsite and online courses. Twenty-three students were selected to participate in this research project. Ten of them were assigned to attend the onsite class and the remaining 13 were assigned to attend the
1. A computer manufacturing company purchases memory chips from two different suppliers. The following data give the thickness of the coated film (coded data) on eight randomly selected chips received from two suppliers. Can we conclude at the 5% level of significance that thickness of coated films
9. The following data give “measured forced vital capacity” in eight asthma patients before and after a treatment. Use the Wilcoxon signed-rank test to test at the 5%level of significance that the treatment is effective.Before: 3878 4011 3685 3384 4091 3451 3898 3098 After: 4249 3569 4262 3839
8. Repeat Problem 7 using the Wilcoxon signed-rank test.
7. The following data give the total cholesterol level (HDL + LDL + 20% of triglycerides)of 20 Americans between the ages of 30 and 40 years. Use the sign test to test the hypothesis at the 5% level of significance that the median cholesterol of American males in that age group is 150 mg/dL.604 14
6. Reconsider the data in Problem 2 above. Use the Wilcoxon signed-rank test to test whether we can conclude that the median height of all the basketball players who were accepted with scholarships during that period is equal to 200 cm versus the hypothesis that the median height is greater than
5. Reconsider the data in Problem 3 above. Use the Wilcoxon signed-rank test to test the hypothesis that the median score in the placement test of all the applicants is greater than the required score of 115 points. Use α = 0.05.
4. The following data give the cholesterol levels before and after the administration of a cholesterol-lowering drug in 10 randomly selected patients. Can we conclude at the 5% level of significance that the drug is ineffective?Before: 183 174 145 140 177 153 175 173 140 152 After: 150 125 120 117
3. The following data give the placement test scores of a random sample of 20 candidates who have applied for undergraduate admission at a public university. Use the one-sample sign test to test that the median score in the placement test of all the applicants is greater than the required score of
2. The following data give the heights in centimeters (cm) of 12 basketball players who were accepted with scholarships during the past 20 years. Use a one-sample sign test to test whether we can conclude that the median height of all the basketball players who were accepted with scholarships
1. Two tests are used to determine the hardness of a metal used in SUV bumpers.Ten samples of the metal under investigation are used for these tests. The hardness indexes under a certain scale produced by the two tests are shown below. Use the two-sample sign test to see whether the two tests
24. A random sample of 500 teenagers is selected and classified according to age and the number of accidents he/she has had over a given period of time. The data are shown below. Test, at the 1% level of significance, the hypothesis that the age and number of accidents are independent.Accidents Age
23. A pharmaceutical company wants to study the form of a drug that is possibly more effective. Three different forms, namely tablet, suspension, and injection were prescribed randomly to 190 patients. After using that drug for four weeks, its effectiveness is observed and the data obtained are
22. To determine whether there is any relationship between the school attended to earn an engineering degree and success at a job, a sample of 300 engineers who earned their degree from different schools S1, S2, S3 is randomly selected, and then each engineer is classified according to his/her
21. A computer assembling company buys all its memory cards from three manufacturers M1,M2,M3. The quality control manager of the company decided to learn more about the quality of the memory cards purchased from the three manufacturers, and thus took a random sample of 500 cards from the shipments
20. An experiment of tossing a certain coin four times is repeated 100 times and the number of heads appearing in each experiment is recorded and shown below. Test at the 5% level of significance the hypothesis that the coin used is unbiased.Numberofheads 0 1 2 3 4 Frequency 8 18 35 25 14
19. The data below gives the time (in minutes) elapsed between the admission of patients to a certain hospital. Can we conclude, at the 5% level of significance, that these data follow an exponential distribution?15 18 78 5 16 58 78 60 40 60 55 49 28 30 39 18 10 12 17 21 24 23 27 14 11 8 28 36 69
18. The data below provide the number of customers entering in a bank during morning hours (9 am–12 noon) on each day of a given week. Do these data provide sufficient evidence that the number of customers entering in the bank is not the same on the different working days of the week? Use α =
17. The data given below provide the frequency distribution of daily traffic violation tickets issued by the city police over a period of eight weeks. Test at the 5% level of significance the hypothesis that the number of traffic tickets is the same for each of the past eight weeks.Week 1 2 3 4 5 6
16. The data below give the frequency distribution of ages for 100 students selected randomly from a large university. Test at the 5% level of significance that the data in the table follow a multinomial distribution, with θ1 = 0.35, θ2 = 0.25, θ3 = 0.15, θ4 =0.10, θ5 = 0.10, θ6 = 0.05.Class
15. The following data set gives the scores of 50 students of an engineering exam. The results are shown below. Test at the 1% level of significance that these data follow a normal distribution.79 81 82 74 86 92 95 87 70 78 79 88 89 85 87 92 96 91 83 83 77 76 84 87 88 91 90 92 94 96 94 91 90 77 81
14. The direct investment by US companies in millions of dollars in five different countries(A, B, C, D, and E) during 1995, 2000, and 2005 are as shown below. Do these data provide sufficient evidence to indicate that the US investments over time in these countries differ significantly? Use α =
13. A random sample of bulbs is taken from lots manufactured in the United States, Canada, and Mexico. Then, from each sample, the defectives and nondefective bulbs are separated. The results are shown below. Test at the 1% level of significance the hypothesis that the quality of bulbs in all three
12. The data given below show the classification of 357 persons by race who visit a physician’s office a certain number of times over a period of one year. The results are shown below. Test, at the 5% level of significance, the hypothesis that the four races are homogeneous with respect to the
11. A certain strain of guinea pig is such that 75% of its progeny are born with white eyes and 75% of its progeny are born with webbed feet. A sample of 160 piglets from newly born litters is classified as shown below. Test whether the modes of classification are independent. Use α = 0.05.Webbed
10. The number of contaminated tablets were counted for 720 samples with each sample consisting of 100 tablets. The results are shown below. Fit a Poisson distribution to this data and test for goodness of fit. Specify α.Contaminated tablets (Xi) in a sample 0 1 2 3 4 5 6 7 8 9 10 Observed sample
9. A study was performed on the effect of time of work on quality of work in a certain plant. It was the practice in the plant for a crew to change shifts once a month. A study of three months of operations by one crew that remained intact for the entire period showed the numbers of defective and
8. The lateral deflection and range in yards obtained in firing 75 rockets are shown below(from Crow et al. (1955)]). Test at the 5% level of significance the hypothesis that lateral deflection and range are independent.Lateral deflection (yards)Range(yards) −250 to −51 −50 to +49 50 to 199
7. In field tests of mine fuses, 216 of each of the two types of fuses A and B, chosen at random from large lots, were buried, and then simulated tanks ran over them. The number of “hits” and “not hits” was recorded for each type of fuse, with the results shown below (from Ordnance Corps
6. Pieces of vulcanite were examined according to porosity and dimensional defects, and the results are shown below (data from Hald, 1952). Test the hypothesis that the two criteria of classification are independent. Specify α.Porous Nonporous With defective dimensions 142 331 Without defective
5. A small roulette wheel was spun 380 times and yielded frequencies shown below for the 38 roulette numbers (taken in pairs). Test the hypothesis that the wheel is true, using the chi-square test at the 1% level of significance.586 13 Analysis of Categorical Data Spins Frequency Spins Frequency
4. Houses with and without air conditioners on nine different streets in a certain city are shown below (from Brownlee, 1957). Test (at the 5% level) the null hypothesis that having an air conditioner is independent of street locations of the houses.Street With A/C Without A/C 1 5 18 2 8 35 3 18 25
3. Using the method of Example 13.2.4, fit a normal distribution to the data of Problem 14 of Review Practice Problems in Chapter 2 and test the null hypothesis at the 5% level of significance that the data behave as a sample from a normal population.Find the observed level of significance.
2. Five thumbtacks of a certain type were thrown 200 times and the number of tacks of the five falling point up in each throw was counted. The experimental results are shown below.Tacks falling point up, X Frequency 0 5 1 27 2 41 3 67 4 43 5 17 Total 200(a) Estimate the probability θ that a tack
1. Suppose that a coin is tossed 1000 times with the result that 462 heads and 538 tails are obtained. Are these results consistent with the hypothesis that the coin is true at the 5% level of significance? (Use the chi-square test.)
6. The following data give the income level for 400 married couples and the number of children that they have. Test the hypothesis at the 5% level of significance that the number of children is independent of the level of income.Number of children 0–2 3 4 5 or more Less than $75K 32 30 16
5. A social worker conducted an experiment to study if the level of education among immigrants to the United States is dependent on the region of migration. The following data give the level of education of 815 immigrants. Test the hypothesis at the 1% level of significance that the level of
4. A medical team conducted an experiment to observe if hypertension is dependent on drinking habits. The data below give information on 250 individuals. Test the hypothesis at the 5% level of significance that having or not having hypertension is independent of drinking habits.Hypertension
3. A social worker was interested in finding the effect of education among women on their married life. The data given below are the levels of their education and the number of years they remained married to their first husband. Test at the 5% level of significance the null hypothesis that the
2. Recently, a new strain of virus that causes influenza was detected. A new vaccine to counter this strain was developed and tested using two selected groups. One group was inoculated with the new vaccine, while the other group did not receive any vaccination. The results showing how many people
1. A random sample of 106 engineers was asked their starting salaries, and whether they graduated from a public or a private school. The results are given below:School $40,000–$49,999 $50,000–$59,999 $60,000–$69,999 Over $70,000 Total Public 12 18 8 6 44 Private 8 24 14 16 62 Total 20 42 22
10. The data below give the number of flights arriving late at a regional airport during the last 10 weeks. Test the hypothesis, at the 5% level of significance, that the number of flights arriving late is the same for each of the past 10 weeks.Week 1 2 3 4 5 6 7 8 9 10 Frequency 12 8 14 16 8 7 12
9. The following data give the scores made by a basketball player in 40 consecutive games. Test at the 1% level of significance that these data follow a normal distribution.34 36 34 34 36 30 31 35 31 36 34 35 36 25 36 25 30 33 27 36 31 26 24 25 33 36 33 34 35 26 25 33 29 27 33 31 33 24 35 25
8. The following data give the grades in a required engineering course for a given semester. Test at the 5% level of significance that the data follow a multinomial distribution, where θ1 = 0.22, θ2 = 0.28, θ3 = 0.30, θ4 = 0.08, θ5 = 0.07, θ6 = 0.05.Use α = 0.05.Grade A B C D F Incomplete
7. It is believed that the number of cars passing through a toll booth in a given interval of time is a random variable having Poisson distribution with unknown parameter λ.In an experiment, the number of cars passing through the toll booth is determined for each of 100 time intervals of equal
6. In 100 throws of a single die, the data obtained are shown below. Test at the 5%level of significance the null hypothesis that the die is a fair die.Number of points 1 2 3 4 5 6 Number of throws 10 25 20 16 19 10
5. It is believed that when a certain type of uranium is placed in a radioactive counter for a given interval of time, the number X of gamma particles emitted during the interval behaves as a random variable having the Poisson distribution. In an experiment, the number of emissions from a piece of
4. Fit a normal distribution to the data of Problem 22 in Review Practice Problems of Chapter 2 and test the null hypothesis at 5% level of significance that the data behave like a sample from a normal population. The data of Problem 22 is reproduced below. Find the observed level of
3. The Occupational Safety and Health Administration revealed the data below on safety violations per week by a manufacturing company over a period of 80 weeks.Fit a Poisson distribution to these data and test the null hypothesis at the 1% level of significance that the data behave like a sample
2. An engineering society is interested in finding if males and females have the same interest in graduate work in engineering. The society surveyed 100 graduate programs in engineering each of which had admitted seven PhD students. The data below give the distribution of males and females among
1. The table below gives the month of birth of a sample of 756 artists. Test, at the 5%level of significance, the null hypothesis that there is no seasonal variation in the months of the year in which artists are born.Birth month Jan Feb Mar Apr May June July Aug Sept Oct Nov Dec Total Frequency 68
21. Refer to the data in “Review Problem 21” on the website: www.wiley.com/college/gupta/statistics2e.(a) Using the covariance structures listed in Table 12.6.1 select the appropriate number of clusters for this data.(b) Based on the BIC values which covariance structure seems reasonable for
20. Refer to the data in Review Problem 17.(a) Use the BIC option in ‘mclust’ package in R to select the appropriate number of model-based clusters and the covariance structure for this data. Note that by the default ‘Mclust()’ function produces 14 different models.(b) Explain basic
19. Refer to the data in Review Problem 17. Use the Density-Based method to cluster US states (Hint: use eps = 20 and minPts = 3 in your DBSCAN algorithm).Review Practice Problems 557
18. Refer to the data in Review Problem 17. Use the K-means method to cluster US states into six distinct clusters.
17. The following data contains US arrest data published in McNeil (1977). The table shows arrests per 100,000 residents for “assault,” “murder,” and “rape” in each of the 50 US states in 1973. The variable “UP” indicates the percent of the urban population living in urban areas.(a)
16. Consider the dichotomous coded variables in Review Problem 5.(a) Use the single, complete, and the average linkage methods to construct the dendrograms.(b) Compare your results in part (a).556 12 Cluster Analysis
15. Consider the following data set.Object A B C D E F G H I J X 0 1 2 2 4 5 6 6 6 5 Y 1 0 0 6 8 6 2 1 1 8(a) Use the K-means algorithm to split the above 10 objects into three distinct clusters by considering objects A, B, and C as their initial cluster centroids.(b) Using part (a), plot the
14. Refer to the data in Review Problem 7. Use the Density-Based method to cluster students into distinct clusters (Hint: use eps = 0.3 and minPts = 2 in your DBSCAN algorithm).
13. Refer to the data in Review Problem 7 and the K-mediods discussion in Review Problem 12.(a) Use the K-mediods method to cluster students into four distinct clusters using the city-block distance. Report your cluster results. (Hint: you may use ‘kmed’ library in R for this purpose).(b)
12. The K-medoids algorithm can also perform effective clustering. That is, in the K-means algorithm, we replace mean centroid with median centroid and perform rest of the calculations accordingly. Illustrate the strength and weakness of the K-means in comparison with the K-medoids algorithm (see
11. Refer to the data in Review Problem 7.(a) Use the K-means method to cluster students into four distinct clusters. Report your resulting clusters and their centroids.(b) Identify the common features of the resulting clusters in part (a).
10. Refer to the data in Review Problem 7.(a) Use the average linkage method to cluster students based on their high school and college GPA values. Report your dendrogram.(b) Using part (a), identify the resulting clusters at distance = 0.5.
9. Refer to the data in Review Problem 7.(a) Use the complete linkage method to cluster students based on their high school and college GPA values. Report your dendrogram.(b) Using part (a), identify the resulting clusters at distance = 0.5.
8. Refer to the data in Review Problem 7.(a) Use the single linkage method to cluster students based on their high school and college GPA values. Report your dendrogram.(b) Using part (a), identify the resulting clusters at distance = 0.4.Review Practice Problems 555
7. Consider the following GPA data set for 10 selected freshman from a certain university.The data consist of students from high school and current college GPA values.ID High school GPA Current college GPA S1 3.0 3.2 S2 4.0 3.9 S3 2.8 3.1 S4 3.1 3.2 S5 3.7 3.1 S6 3.8 2.9 S7 3.6 3.0 S8 2.9 3.5 S9
6. Consider the dichotomous coded variables in Review Problem 5.(a) Obtain the city-block distances between all the patients.(b) Graph the distance matrix using an appropriate graphing tool.(c) How do you compare your results with results from Review Problem 5?
5. Consider the following patient data set. As in Example 12.2.2, introduce appropriate binary variables and calculate the Match-Mismatch coefficient for all the patients.Explain your results.Patient ID Gender Age (yr) Blood type Blood pressure(mm Hg)Glucose level (mg/dl)P011 F 37 A 120 111 P012 F
4. Consider the following data set and calculate (a), (b), and (c) below:A 1 0 1 1 0 B 1 1 0 1 0 C 0 0 1 1 1 D 0 1 0 1 0 E 1 0 1 0 1(a) What are the SMCs for all the objects?(b) What are the Jaccard coefficients for all the objects?(c) Compare your results in part (a) and part (b).554 12 Cluster
3. For the data in Review Problem 1,(a) Compute the city-block distance matrix for all the items.(b) Graph the distance matrix using an appropriate graphing tool.(c) Does the city-block distance matrix convey the same information as of the Euclidean distance matrix?
2. For the data in Review Problem 1,(a) Compute the Euclidean distance matrix for all the items.(b) Graph the distance matrix using an appropriate graphing tool.(c) Describe the main features of the distance matrix.
1. Consider the following data set and calculate (a), (b), and (c) below:Items X Y Z W A 13.2 236 58 21.2 B 10.0 263 48 44.5 C 8.1 294 80 31.0 D 8.8 190 50 19.5 E 9.0 276 91 40.6 F 7.9 204 78 38.7 G 3.3 110 77 11.1 H 5.9 238 72 15.8(a) Euclidean distance between items A and B?(b) The city-block
19. Use the following testing data to predict outcomes of the classification tree derived in Review Problem 18. Discuss the accuracy of the fitted model, using an appropriate measure.Q1 Q2 Q3 Q4 Q5 H1 H2 H3 H4 H5 T1 T2 Final Grade 10.0 10.0 10.0 8.0 8.0 10.0 12.0 9.5 10.0 9.5 75 69 76 C 10.0 9.0
18. A professor in a certain college is interested in predicting final class grades of his students using some data prior to the exam. He recorded the following data that include students quiz grades (Q1–Q5), homework grades (H1–H7), mid-term exam scores (T1 & T2), final exam score (Final), and
17. Split the data in Review Problem 15 into training and testing sets, using a 80:20 ratio. That is, use observations 1–202 as training data to build a regression tree using all the variables except “Index” to predict the “Percent of body fat index,” and use observations 203–252 as
16. Split the data in Review Problem 15 into training and testing sets, using a 80:20 ratio.That is, use observations 1–202 as training data to build a classification tree using all the variables except “Percent of body fat index” to predict the “Index,” and use observations 203–252 as
15. The data listed in “Review Problem 15” (see website: www.wiley.com/college/gupta/statistics2e) contain data from 252 people reported in “Generalized body composition prediction equation for men using simple measurement techniques” by Penrose et al.(1985). The variables listed in the
14. Consider the data given in Review Problem 4.(a) Use the logistic regression concepts discussed in Section 11.6 (see Section 16.8 for more details) to predict the outcome (i.e., “Survival”) using the variables “Parch,”“Fare,” “Embarked.” Please disregard the observations with
13. Investigate the relationship between cutoff values (0–1) and misclassification error rates for the predicted probabilities using the logistic model used in Review Problem 12.(a) Plot both class 0 and class 1 classification errors against the range of cutoff values(from 0 through 1).(b) What
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