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Solution Manual for Probability and Statistics with R for Engineers and Scientists, 1st Edition

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    S OLUTIONS M ANUAL S ONGFENG Z HENG Missouri State University P ROBABILITY & S TATISTICS WITH R FOR E NGINEERS AND S CIENTISTS Michael Akritas The Pennsylvania State University

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    Contents 1 Basic Statistical Concepts 1 1.2 Populations and Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.3 Some Sampling Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.4 Random Variables and Statistical Populations . . . . . . . . . . . . . . . 3 1.5 Basic Graphics for Data Visualization . . . . . . . . . . . . . . . . . . . . 4 1.6 Proportions, Averages, and Variances . . . . . . . . . . . . . . . . . . . . 20 1.7 Medians, Percentiles, and Boxplots . . . . . . . . . . . . . . . . . . . . . 23 1.8 Comparative Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2 Introduction to Probability 35 2.2 Sample Spaces, Events, and Set Operations . . . . . . . . . . . . . . . . . 35 2.3 Experiments with Equally Likely Outcomes . . . . . . . . . . . . . . . . 39 2.4 Axioms and Properties of Probabilities . . . . . . . . . . . . . . . . . . . 44 2.5 Conditional Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.6 Independent Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3 Random Variables and Their Distributions 53 3.2 Describing a Probability Distribution . . . . . . . . . . . . . . . . . . . . 53 3.3 Parameters of Probability Distributions . . . . . . . . . . . . . . . . . . . 59 3.4 Models for Discrete Random Variables . . . . . . . . . . . . . . . . . . . 63 3.5 Models for Continuous Random Variables . . . . . . . . . . . . . . . . . 68 4 Jointly Distributed Random Variables 77 4.2 Describing Joint Probability Distributions . . . . . . . . . . . . . . . . . 77 4.3 Conditional Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 4.4 Mean Value of Functions of Random Variables . . . . . . . . . . . . . . . 87 4.5 Quantifying Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 4.6 Models for Joint Distributions . . . . . . . . . . . . . . . . . . . . . . . . 97 5 Some Approximation Results 103 5.2 The LLN and the Consistency of Averages . . . . . . . . . . . . . . . . . 103 5.3 Convolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 5.4 The Central Limit Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 106

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    CONTENTS 6 Fitting Models to Data 111 6.2 Some Estimation Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . 111 6.3 Methods for Fitting Models to Data . . . . . . . . . . . . . . . . . . . . . 113 6.4 Comparing Estimators: The MSE Criterion . . . . . . . . . . . . . . . . 119 7 Confidence and Prediction Intervals 121 7.3 Type of Confidence Intervals . . . . . . . . . . . . . . . . . . . . . . . . . 121 7.4 The Issue of Precision . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 7.5 Prediction Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 8 Testing of Hypotheses 133 8.2 Setting Up a Test Procedure . . . . . . . . . . . . . . . . . . . . . . . . . 133 8.3 Types of Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 8.4 Precision in Hypothesis Testing . . . . . . . . . . . . . . . . . . . . . . . 144 9 Comparing Two Populations 147 9.2 Two-Sample Tests and CIs for Means . . . . . . . . . . . . . . . . . . . . 147 9.3 The Rank-Sum Test Procedure . . . . . . . . . . . . . . . . . . . . . . . 155 9.4 Comparing Two Variances . . . . . . . . . . . . . . . . . . . . . . . . . . 158 9.5 Paired Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 10 Comparing k > 2 Populations 163 10.2 Types of k -Sample Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 10.3 Simultaneous CIs and Multiple Comparisons . . . . . . . . . . . . . . . . 173 10.4 Randomized Block Designs . . . . . . . . . . . . . . . . . . . . . . . . . . 181 11 Multifactor Experiments 187 11.2 Two-Factor Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 11.3 Three-Factor Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 11.4 2 r Factorial Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 12 Polynomial and Multiple Regression 209 12.2 The Multiple Linear Regression Model . . . . . . . . . . . . . . . . . . . 209 12.3 Estimating, Testing, and Prediction . . . . . . . . . . . . . . . . . . . . . 212 12.4 Additional Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 13 Statistical Process Control 237 13.2 The ¯ X Chart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 13.3 The S and R Charts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 13.4 The p and c Charts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 13.5 CUSUM and EWMA Charts . . . . . . . . . . . . . . . . . . . . . . . . . 249

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    1 Chapter 1 Basic Statistical Concepts 1.2 Populations and Samples 1. (a) The population consists of the customers who bought a car during the previous year. (b) The population is not hypothetical. 2. (a) There are three populations, one for each variety of corn. Each variety of corn that has been and will be planted on all kinds of plots make up the population. (b) The characteristic of interest is the yield of each variety of corn at the time of harvest. (c) There are three samples, one for each variety of corn. Each variety of corn that was planted on the 10 randomly selected plots make up the sample. 3. (a) There are two populations, one for each shift. The cars that have been and will be produced on each shift make up the population. (b) The populations are hypothetical. (c) The characteristic of interest is the number of nonconformances per car. 4. (a) The population consists of the all domestic flights, past or future. (b) The sample consists of the 175 domestic flights. (c) The characteristic of interest is the air quality, quantified by the degree of staleness. 5. (a) There are two populations, one for each teaching method. (b) The population consists of all students who took or will take a statistics course for engineering using one of each teaching methods. (c) The populations are hypothetical. (d) The samples consist of the students whose scores will be recorded at the end of the semester.

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    2 Chapter 1 Basic Statistical Concepts 1.3 Some Sampling Concepts 1. The second choice provides a closer approximation to simple random sample. 2. (a) It is not a simple random sample. (b) In (a), each member of the population does not have equal chance to be selected, thus it is not a simple random sample. Instead, the method described in (a) is a stratified sampling. 3. (a) The population includes all the drivers in the university town. (b) The student’s classmates do not constitute a simple random sample. (c) It is a convenient sample. (d) Young college students are not experienced drivers, thus they tend to use seat belts less. Consequently, the sample in this problem will underestimate the proportion. 4. We identify each person with a number from 1 to 70. Then we write each number from 1 to 70 on separate, identical slips of paper, put all 70 slips of paper in a box, and mix them thoroughly. Finally, we select 15 slips from the box, one at a time, without replacement. The 15 selected numbers specify the desired sample of size n = 15 from the 70 iPhones. The R command is y = sample(seq(1,70), size=15) A sample set is 52 8 14 48 62 6 70 35 18 20 3 41 50 27 40. 5. We identify each pipe with a number from 1 to 90. Then we write each number from 1 to 90 on separate, identical slips of paper, put all 90 slips of paper in a box, and mix them thoroughly. Finally, we select 5 slips from the box, one at a time, without replacement. The 5 selected numbers specify the desired sample of size n = 5 from the 90 drain pipes. The R command is y = sample(seq(1,90), size=5) , A sample set is 7 38 65 71 57. 6. (a) We identify each client with a number from 1 to 1000. Then we write each number from 1 to 1000 on separate, identical slips of paper, put all 1000 slips of paper in a box, and mix them thoroughly. Finally, we select 100 slips from the box, one at a time, without replacement. The 100 selected numbers specify the desired sample of size n = 100 from the 1000 clients.

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    1.4 Random Variables and Statistical Populations 3 (b) Using stratified sampling: Get a simple random sample of size 80 from the sub-population of Caucasian-Americans, a simple random sample of size 15 from the sub-population of African-Americans, and a simple random sample of size 5 from the sub-population of Hispanic-Americans. Then combine the three subsamples together. (c) The R command for part (a) is y = sample(seq(1,1000), size=100) and the R command for part (b) is y1 = sample(seq(1,800), size=80) y2 = sample(seq(801,950), size=15) y3 = sample(seq(951,1000), size=5) y = c(y1, y2, y3) 7. One method is to take a simple random sample of size n from the population of N customers (of all dealerships of that car manufacturer) who bought a car the previous year. The second method is to divide the population of the previous year’s customers into three strata according to the type of car each customer bought and perform stratified sampling with proportional allocation of sample sizes. That is, if N 1 , N 2 , N 3 denote the sizes of the three strata, take simple random samples of approximate sizes (due to round-off) n 1 = n ( N 1 /N ), n 2 = n ( N 2 /N ), n 3 = n ( N 3 /N ), respectively, from each of the three strata. Stratified sampling assures that the sample representation of the three strata equals their population representation. 8. It is not a simple random sample because products from facility B have a smaller chance to be selected than products from facility A. 9. No, because the method excludes samples consisting of n 1 cars from the first shift and n 2 = 9 − n 1 from the second shift for any ( n 1 , n 2 ) different from (6 , 3). 1.4 Random Variables and Statistical Populations 1. (a) The variable of interest is the number of scratches in each plate. The statistical population consists of 500 numbers, 190 zeros, 160 ones, and 150 twos. (b) The variable of interest is quantitative. (c) The variable of interest is univariate. 2. (a) Statistical population: If there are N undergraduate students enrolled at PSU, the statistical population is a list of length N and the i -th element in the list is the major of the i -th student. The variable of interest is qualitative. Another possible variable: gender.

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    4 Chapter 1 Basic Statistical Concepts (b) Statistical population: If there are N restaurants on campus, the statistical population consists of a list of N numbers, and the i -th element is the capacity of the i -th restaurant. The variable of interest is quantitative. Another possible variable: food type. (c) Statistical population: If there are N books in Penn State libraries, the sta- tistical population consists of a list of N numbers, and the i -th element is the check-out frequency of the i -th book in the library. The variable of interest is quantitative. Another possible variable: pages of the book. (d) Statistical population: If there are N steel cylinders made in the given month, the population consists of a list of N numbers, and the i -th element is the diameter of the i -th steel cylinder made in the given month. The variable of interest is quantitative. Another possible variable: weight. 3. (a) The variable of interest is univariate. (b) The variable of interest is quantitative. (c) If N is the number cars of available for inspection, the statistical population consists of N numbers, { v 1 , · · · , v N } , where v i is the total number of engine and transmission nonconformances of the i th car. (d) If the number of nonconformances in the engine and transmission are recorded separately for each car, the new variable would be bivariate. 4. (a) The variable of interest is the degree of staleness. Statistical population consists of a list of 175 numbers, and the i -th number is the degree of staleness of the air in the i -th domestic flight. (b) The variable of interest is quantitative. (c) The variable of interest is univariate. 5. (a) The variable of interest is the type of car a customer bought and his/her satisfaction level. Statistical population: If there are N customers who bought a new car in the previous year, the statistical population is a list of N elements, and the i -th element is the car type the i -th customer bought along with his/her satisfaction level, which is a number between 1 to 6. (b) The variable of interest is bivariate. (c) The variable of interest has two components. The first is qualitative and the second is quantitative. 1.5 Basic Graphics for Data Visualization 1. The histogram produced by the commands is shown as following:

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    1.5 Basic Graphics for Data Visualization 5 Histogram of Str Str Density 42 44 46 48 50 0.00 0.10 0.20 The stem and leaf plot is as following: The decimal point is at the | 41 | 5 42 | 39 43 | 1445788 44 | 122357 45 | 1446 46 | 00246 47 | 3577 48 | 36 49 | 3 2. The histogram on the waiting time is as following

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    6 Chapter 1 Basic Statistical Concepts Histogram of waiting waiting Frequency 40 50 60 70 80 90 100 0 10 20 30 40 50 The corresponding stem and leaf plot is given below. It is clear that the shape of the stem and leaf plot is similar to that of the histogram. The decimal point is 1 digit(s) to the right of the | 4 | 3 4 | 55566666777788899999 5 | 00000111111222223333333444444444 5 | 555555666677788889999999 6 | 00000022223334444 6 | 555667899 7 | 00001111123333333444444 7 | 555555556666666667777777777778888888888888889999999999 8 | 000000001111111111111222222222222333333333333334444444444 8 | 55555566666677888888999 9 | 00000012334 9 | 6 The histogram with title and the colored smooth curve superimposed is shown as

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    1.5 Basic Graphics for Data Visualization 7 Waiting times before Eruption the Old Faithful Geyser Density 40 50 60 70 80 90 100 0.00 0.01 0.02 0.03 0.04 3. The scatterplot is shown below. From the scatter plot, it seems that if the waiting time before eruption is longer, the duration is also longer. 50 60 70 80 90 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 waiting eruptions 4. (a) The scatterplot matrix is given below. From the figure, it seems that the lati- tude is a better predictor of the temperature because as the latitude changes, the temperature shows a clear pattern, while there is no pattern as the longi- tude changes.

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    8 Chapter 1 Basic Statistical Concepts JanTemp 25 30 35 40 45 0 10 30 50 25 30 35 40 45 Lat 0 10 30 50 70 80 90 100 120 70 80 90 100 120 Long (b) The following figure gives the 3D scatter plot. The 3D scatter plot also shows that the latitude is a better predictor for the temperature. 25 30 35 40 45 50 0 10 20 30 40 50 60 70 70 80 90 100 110 120 130 Lat Long JanTemp

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    1.5 Basic Graphics for Data Visualization 9 5. The 3D scatterplot is shown below 10 15 20 25 30 35 0 100 200 300 400 500 10 20 30 40 50 60 Neck.G Chest.G Weight 6. The scatterplot is shown below. From the scatterplot, it is clear that in general, if the speed is high, the breaking distance is larger. 5 10 15 20 25 0 20 40 60 80 100 120 speed dist 7. The required graph is given below:

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    10 Chapter 1 Basic Statistical Concepts 20 40 60 80 100 0 100 200 300 400 500 600 Speed StopDist Cars Trucks 8. The resulting graph is given below. The figure shows that for SMaple and WOak, the growing speed in terms of the diameter of the tree is constant, while for ShHick- ory, when the tree gets older, it grows faster. 40 60 80 100 100 150 200 250 300 diam age SMaple ShHickory WOak 9. (a) The basic histogram with smooth curve superimposed:

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    1.5 Basic Graphics for Data Visualization 11 Histogram of t1 t1 Density 28 29 30 31 32 33 0.00 0.05 0.10 0.15 0.20 0.25 0.30 (b) The stem and leaf plot for the reaction time of Robot 1 is given below. The decimal point is at the | 28 | 4 29 | 0133688 30 | 03388 31 | 0234669 32 | 47 10. The produced basic scatter plot is given below. It seems that the surface conduc- tivity can be used for predicting sediment conductivity.

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    12 Chapter 1 Basic Statistical Concepts 200 300 400 500 600 700 800 400 500 600 700 800 900 X Y 11. The produced basic scatter plot is given below. It seems that the rainfall volume is useful for predicting the runoff volume. 20 40 60 80 100 120 20 40 60 80 100 X Y 12. The produced scatterplot matrix is as following, and it seems that the variable temperature is a better single predictor for the amount of electricity consumed.

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