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• 6 Hypothesis Testing -Two Populations7Exercise 1:A random sample of size n = 25 taken from a normal population with σ = 5.2 has a meanequals 81. A second random sample of size n = 36, taken from a different normal populationwith σ = 3.4, has a mean equals 76.(a) Do the data indicate that the true mean value µ1 and µ2 are different? Carry out a test at α = 0.01(b) Find 90% CI on the difference in mean strengthExercise 2:Two machines are used for filling plastic bottles with a net volume of 16.0 oz. The fillvolume can be assumed normal with, s1 = 0.02 and s2 = 0.025. A member of the qualityengineering staff suspects that both machines fill to the same mean net volume, whether ornot this volume is 16.0 oz. A random sample of 10 bottles is taken from the output of eachmachine with the following results:(a) Do you think the engineer is correct? Use the p – value approach.(b) Find a 95% CI on the difference in means. 1
• Probability and Statistics Work BookExercise 3: (Tutorial 6, No.1)Two machine are used to fill plastic bottles with dishwashing detergent. The standarddeviations of fill volume are known to be σ1= 0.01 and σ2 = 0.15 fluid ounce for twomachines, respectively. Two random samples of n1 = 12 bottles from machine 1 and n2=10bottles from machine 2 are selected, and the sample mean fill volumes are x 1 =30.61x 2 =30.24 fluid ounces. Assume normality.(i) Test the hypothesis that both machines fill to the same mean volume. Use the P- value approach;(ii) Construct a 90% two-sided CI on the mean difference in fill volume; and(iii) Construct a 95% two-sided CI on the mean difference in fill volume. Compare and comment on the width of this interval to the width of the interval in part (ii).Exercise 4:To find out whether a new serum will arrest leukemia, 9 mice, all with an advanced stage ofthe disease are selected. 5 mice receive the treatment and 4 do not. Survival, in years, fromthe time the experiment commenced are as follows: Treatment 2.1 5.3 1.4 4.6 0.9 No treatment 1.9 0.5 2.8 3.1At the 0.05 level of significance can the serum be said to be effective? Assume the twodistributions to be of equal variances. 2
• Probability and Statistics Work BookExercise 5: (Tutorial 6, No.2)A new policy regarding overtime pay was implemented. This policy decreased the pay factorfor overtime work. Neither the staffing pattern nor the work loads changed. To determine ifovertime loads changed under the policy, a random sample of employees was selected. Theirovertime hours for a randomly selected week before and for another randomly selected weekafter the policy change were recorded as follows: Employees: 1 2 3 4 5 6 7 8 9 10 11 12 Before: 5 4 2 8 10 4 9 3 6 0 1 5 After: 3 7 5 3 7 4 4 1 2 3 2 2Assume that the two population variances are equal and the underlying population isnormally distributed.(i) Is there any evidence to support the claim that the average number of hours worked as overtime per week changed after the policy went into effect. Use a P-value approach in arriving at this conclusion.(ii) Construct a 95% CI for the difference in mean before and after the policy change. Interpret this interval.Exercise 6:The diameter of steel rods manufactured on two different extrusion machines is beinginvestigated. Two random samples of sizes n1 = 15 and n2 = 17 are selected, andrespectively. s1 = 0.35 and x2 = drawn s2 = 0.40 x1 = 8.37, Assume that data are 8.68, normal distribution with equal variances. 2 2(a) Is there evidence to support the claim that the two machines produce rods with different mean diameters ? Use the p – value approach.(b) Construct a 95% CI on the difference in mean rod diameter. 3
• Probability and Statistics Work BookExercise 7:The following data represent the running times of films produced by 2 motion-picturecompanies. Test the hypothesis that the average running time of films produced by company2 exceeds the average running time of films produced by company 1 by 10 minutes againstthe one-sided alternative that the difference is less than 10 minutes? Use a = 0.01 and assumethe distributions of times to be approximately normal with unequal variances. Time Company X1 102 86 98 109 92 X2 81 165 97 134 92 87 114Exercise 8:Two companies manufacture a rubber material intended for use in an automotive application.25 samples of material from each company are tested, and the amount of wear after 1000cycles are observed. For company 1, the sample mean and standard deviation of wear arex1 = 20.12mg / 1000cycles and s1 = 1.9mg / 1000cyclesand for company 2, we obtain x2 = 11.64mg / 1000cycles and s2 = 7.9mg / 1000cycles(a) Do the sample data support the claim that the two companies produce material with different mean wear? Assume each population is normally distributed but unequal variances?(b) Construct a 95% CI for the difference in mean wear of these two companies. Interpret this interval.Exercise 9: (Tutorial 6, No.3) 4
• Probability and Statistics Work BookProfessor A claims that a probability and statistics student can increase his or her score ontests if the person is provided with a pre-test the week before the exam. To test her theory sheselected 16 probability and statistics students at random and gave these students a pre-test theweek before an exam. She also selected an independent random sample of 12 students whowere given the same exam but did not have access to the pre-test. The first group had a meanscore of 79.4 with standard deviation 8.8. The second group had sample mean score 71.2with standard deviation 7.9.(i) Do the data support Professor A claims that the mean score of students who get a pre-test are different from the mean score of those who do not get a pre test before an exam. Use the P-value approach and assume that their variances are not equal.(ii) Construct a 95% CI for the difference in mean score of students who get a pre-test and those who do not get a pre-test before an exam. Interpret this interval.Exercise 10:A vote is to be taken among residents of a town and the surrounding county to determinewhether a proposed chemical plant should be constructed. If 120 of 200 town voters favourthe proposal and 240 of 500 county residents favour it, would you agree that the proportionof town voters favouring the proposal is higher than the proportion of county voters? Use a =0.05 5
• Probability and Statistics Work BookExercise 11: (Tutorial 6, No.4)The rollover rate of sport utility vehicles is a transportation safety issue. Safety advocatesclaim that the manufacturer A’s vehicle has a higher rollover rate than that of manufacturerB. One hundreds crashes for each of this vehicles were examined. The rollover rates werepA=0.35 and pB=0.25.(i) By using the P-value approach, does manufacturer A’s vehicle has a higher rollover rate than manufacturer B’s?(ii) Construct a 95% CI on the difference in the two rollover rates of the vehicle. Interpret this interval.Exercise 12:Professor Rady gave 58 A’s and B’s to a class of 125 students in his section of English 101.The next term Professor Hady gave 45 A’s and B’s to a class of 115students in his section ofEnglish 101.(i) By using a 5% significance level, test the claim that Professor Rady gives a higher percentage of A’s and B’s in English 101 than Professor Hady does. What is comment?(ii) Construct a 95% CI on the difference in the percentage of A’s and B’s in English 101 given by this two professors. 6
• 7 Probability and Statistics Work Book Simple Linear Regression8Exercise 1:The manager of a car plant wishes to investigate how the plant’s electricity usage dependsupon the plant production. The data is given below Production 4.51 3.58 4.31 5.06 5.64 4.99 5.29 5.83 4.7 5.61 4.9 4.2 (RMmillion) (x) Electricity 2.48 2.26 2.47 2.77 2.99 3.05 3.18 3.46 3.03 3.26 2.67 2.53 Usage (y)(a) Estimate the linear regression equation Y = β 0 + β1 x(b) An estimate for the electricity usage when x = 5(c) Find a 90% Confidence Interval for the electricity usage. 7
• Probability and Statistics Work BookExercise 2:An experiment was set up to investigate the variation of the specific heat of a certainchemical with temperature. The data is given below Temperature oF 50 60 70 80 90 100 (x) Heat 1.60 1.63 1.67 1.70 1.71 1.71 (y) 1.64 1.65 1.67 1.72 1.72 1.74(a) Estimate the linear regression equation Y = β 0 + β1 x(b) Plot the results on a scatter diagram(c) An estimate for the specific heat when the temperature is 75oF(d) Find a 95% Confidence Interval for the specific heat.Exercise 3:An engineer at a semiconductor company wants to model the relationship between the deviceHFE (y) and the parameter Emitter - RS ( x1). Data for Emitter - RS was first collected anda statistical analysis is carried out and the output is displayed in the table given.Regression Analysis: y = 1075.2 – 63.87x1Predictor Coef SE Coef T P-valueConstant 1075.2 121.1 8.88 0.000 x1 -63.87 8.002 -7.98 0.000S = 19.4 R-Sq = 0.78Analysis of varianceSource DF SS MS FRegression 1 23965 23965 63.70Residual 18 6772 376Total 19 30737(a) Estimate HFE when the Emitter - RS is 14.5.(b) Obtain a 95 % confidence interval for the true slope β.(c) Test for significance of regression for a = 0.05. 8
• Probability and Statistics Work BookExercise 4:An chemical engineer wants to model the relationship between the purity of oxygen (y)produced in a chemical distillation process and the percentage of hydrocarbons (x ) that arepresent in the main condenser of the distillation unit. A statistical analysis is carried out andthe output is displayed in the table given.Regression Analysis: y = 74.3 + 14.9xPredictor Coef SE Coef T P-valueConstant 74.283 1.593 46.62 0.000 x1 14.947 1.317 11.35 0.000S = 1.087 R-Sq = 87.7%Analysis of varianceSource DF SS MS FRegression 1 152.13 152.13 12.86Residual 18 21.25 1.18Total 19 173.38(a) Estimate the purity of oxygen when the percentage of hydrocarbon 1%.(b) Obtain a 95 % confidence interval for the true slope β.(c) Test for significance of regression for a = 0.05. 9
• Probability and Statistics Work BookExercise 5: (Tutorial 7, No.1)Regression methods were used to analyze the data from a study investigating the relationshipbetween roadway surface temperature (x) and pavement deflection (y). The data follow. Temperature Deflection Temperature Deflection x y x y 70.0 0.621 72.7 0.637 77.0 0.657 67.8 0.627 72.1 0.640 76.6 0.652 72.8 0.623 73.4 0.630 78.3 0.661 70.5 0.627 74.5 0.641 72.1 0.631 74.0 0.637 71.2 0.641 72.4 0.630 73.0 0.631 75.2 0.644 72.7 0.634 76.0 0.639 71.4 0.638(a) Estimate the intercept and slope regression coefficients. Write the estimated regression line.(b) Compute SSE and estimate the variance.(c) Find the standard error of the slope and intercept coefficients.(d) Show that(e) Compute the coefficient of determination, R2. Comment on the value.(f) Use a t-test to test for significance of the intercept and slope coefficients at . Give the P-values of each and comment on your results.(g) Construct the ANOVA table and test for significance of regression using the P-value. Comment on your results and their relationship to your results in part (f).(h) Construct 95% CIs on the intercept and slope. Comment on the relationship of these CIs and your findings in parts (f) and (g). 10
• Probability and Statistics Work BookExercise 6: (Tutorial 7, No.2)The designers of a database information system that allows its users to search backwards forseveral days wanted to develop a formula to predict the time it would be take to search.Actually elapsed time was measured for several different values of days. The measured datais shown in the following table: Number of Days 1 2 4 8 16 25 Elapsed Time 0.6 0.79 1.36 2.26 3.59 5.39 5(i) Estimate the intercept and slope regression coefficients. Write the estimated regression line.(ii) Compute SSE and estimate the variance.(iii) Find the standard error of the slope and intercept coefficients.(iv) Show that(v) Compute the coefficient of determination, R2. Comment on the value.(vi) Use a t-test to test for significance of the intercept and slope coefficients at . Give the P-values of each and comment on your results.(vii) Construct the ANOVA table and test for significance of regression using the P-value. Comment on your results and their relationship to your results in part (vi).(viii)Construct 95% CIs on the intercept and slope. Comment on the relationship of these CIs and your findings in parts (vi) and (vii). 11
• Probability and Statistics Work Book8 Multiple Linear Regressions9Exercise 1:Given the data: Test Number y x1 x2 1 1.6 1 1 2 2.1 1 2 3 2.4 2 1 4 2.8 2 2 5 3.6 2 3 6 3.8 3 2 7 4.3 2 4 8 4.9 4 2 9 5.7 4 3 10 5 3 4(a) Fit a multiple linear regression model to these data. 12
• Probability and Statistics Work BookExercise 2:Given the data:Observation Number Pull Strength y Wire Length x1 Die Height x2 1 9.95 2 50 2 24.45 8 110 3 31.75 11 120 4 35.00 10 550 5 25.02 8 295 6 16.86 4 200 7 14.38 2 375 8 9.60 2 52 9 24.35 9 100 10 27.50 8 300 11 17.08 4 412 12 37.00 11 400 13 41.95 12 500 14 11.66 2 360 15 21.65 4 205 16 17.89 4 400 17 69.00 20 600 18 10.30 1 585 19 34.93 10 540 20 46.59 15 250 21 44.88 15 290 22 54.12 16 510 23 56.63 17 590 24 22.13 6 100 25 21.15 5 400(b) Fit a multiple linear regression model to these data. 13
• Probability and Statistics Work BookExercise 3:A study was performed to investigate the shear strength of soil (y) as it related to depth inmeter (x1) and percentage moisture content (x2). Ten observations were collected and thefollowing summary quantities obtained: n = 10, ∑x i1 = 223, ∑x i2 = 553,∑y i = 1,916, ∑x 2 i1 = 5,200.9, ∑ x = 31,729, 2 i2 ∑x x = 12,352,i1 i 2 ∑x i1 i y = 43,550.8, ∑ x y = 104,736.8, ∑ y = 371,595.6 i2 i 2 i(a) Estimate the parameters to fit the multiple regression models for these data.(b) What is the predicted strength when x1=18meter and x2= 43%. 14
• Probability and Statistics Work BookExercise 4:A set of experimental runs were made to determine a way of predicting cooking time y atvarious levels of oven width x1, and temperature x2. The data were recorded as follows: y x1 x2 6.4 1.32 1.15 15.05 2.69 3.4 18.75 3.56 4.1 30.25 4.41 8.75 44.86 5.35 14.82 48.94 6.3 15.15 51.55 7.12 15.32 61.5 8.87 18.18 100.44 9.8 35.19 111.42 10.65 40.4(a) Fit a multiple linear regression model to these data.(b) Estimate and the standard errors of the regression coefficients.(c) Test for significance of and .(d) Predict the useful range when brightness = 80 and contrast = 75. Construct a 95% PI.(e) Compute the mean response of the useful range when brightness = 80 and contrast = 75. Compute a 95% CI.(f) Interpret parts (d) and (e) and comment on the comparison between the 95% PI and 95% CI. 15
• Probability and Statistics Work BookExercise 5: (Tutorial 8, No.1)An article in Optical Engineering (“Operating Curve Extraction of a Correlators Filter,” Vol.43, 2004, pp. 2775–2779) reported the use of an optical correlator to perform an experimentby varying brightness and contrast. The resulting modulation is characterized by the usefulrange of gray levels. The data are shown Brightness (%): 5 6 6 10 10 10 50 57 54 4 1 5 0 0 0 Contrast (%): 5 8 7 50 65 80 25 35 26 6 0 0 Useful range (ng): 9 5 5 11 96 80 15 14 25 6 0 0 2 5 4 5(a) Fit a multiple linear regression model to these data.(b) Estimate and the standard errors of the regression coefficients.(c) Test for significance of and .(d) Predict the useful range when brightness = 80 and contrast = 75. Construct a 95% PI.(e) Compute the mean response of the useful range when brightness = 80 and contrast = 75. Compute a 95% CI.(f) Interpret parts (d) and (e) and comment on the comparison between the 95% PI and 95% CI. 16
• Probability and Statistics Work BookExercise 6: (Tutorial 8, No.2)A study was performed on wear of a bearing y and its relationship to x1 = oil viscosity andx2 = load. The following data were obtained: x 1.6 15.5 22.0 43.0 33.0 40.0 1 x 85 816 1058 120 135 111 2 1 1 7 5 y 29 230 172 91 113 125 3(a) Fir a multiple regression model to these data.(b) Estimate σ2 and the standard errors of the regression coefficients.(c) Use the model to predict wear when x1 = 25 and x2 = 1000.(d) Fit a multiple regression model with an interaction term to these data.(e) Estimate σ2 and se(βj) for this new model. How did these quantities change? Does this tell you anything about the value of adding the interaction term to the model?(f) Use the model in (d), to predict when x1=25 and x2=1000. Compare this prediction with the predicted value from part (c) above. 17
• Probability and Statistics Work Book9 Factorial Experiments – 22 Factorial designExercise 1:An engineer is investigating the thickness of epitaxial layer which will be subject to twovariations in A, deposition time (+ for short time, and – for long time) and two levels of B,arsenic flow rate (- for 55% and + for 59%). The engineer conduct 22 factorial design with n= 4 replicates. The data are as follow: 18
• Probability and Statistics Work Book Arsenic Level B– B+ (Low - 55%) (High – 59%) Deposition Time 14.037 13.880 14.165 13.860 A - (Long) 13.972 14.032 13.907 13.914 14.821 14.888 14.757 14.921 A + (Short) 14.843 14.415 14.878 14.932 a) Construct the 2 X 2 factorial design table. b) Find the estimate of all effects and interaction. c) Construct the ANOVA table for each effect, test the null hypothesis that the effect is equal to 0.Exercise 2: (Tutorial 9, No.1)A two factor experimental design was conducted to investigate the lifetime of a componentbeing manufactured. The two factors are A (design) and B (cost of material). Two levels ((+)and (-)) of each factor are considered. Three components are manufactured with eachcombination of design and material, and the total lifetime measured (in hours) is as shown intable below 19
• Probability and Statistics Work Book Total lifetime of 3 Design Material AB components Treatment A B (in hours) Combination (1) - - + 122 a + - - 60 b - + - 120 ab + + + 118(a) Perform a two way analysis of variance to estimate the effects of design and materialexpense on the component life time if the sum squares of total are 1050.(b) Based on your results in part (a), what conclusions can you draw from the factorial experiment?(c) Indicate which effects are significant to the lifetime of a component.(d) Write the least square fitted model using only the significant sources.Exercise 3:An engineer suspects that the surface finish of metal parts is influenced by the type of paintused and the drying time. He selected three drying times – 20, 25, and 30 minutes and usedtwo types of paint. Three parts are tested with each combination of paint typoe and dryingtime. The data are as follow: Drying Time (min) Paint 20min 25min 30min ICI 74 73 78 64 61 85 50 44 92 NIPPON 92 98 66 20 86 73 45 68 88 85
• Probability and Statistics Work Book(a) Compute the estimates of the effects and their standard errors for this design.(b) Construct two-factor interaction plots and comment on the interaction of the factors.(c) Use the t ratio to determine the significance of each effect with .Comment on your findings.(d) Compute an approximate 95% CI for each effect. Compare your results with those in part (c) and comment.(e) Perform an analysis of variance of the appropriate regression model for this design. Include in your analysis hypothesis tests for each coefficient, as well as residualExercise 4: (Tutorial 9, No.2)An experiment involves a storage battery used in the launching mechanism of a shoulder-fired ground-to-air missile. Two material types can be used to make the battery plates. Theobjective is to design a battery that is relatively unaffected by the ambient temperature. Theoutput response from the battery is effective life in hours. Two temperature levels areselected, and a factorial experiment with four replicates is run. The data are as follows: 21
• Probability and Statistics Work Book Temperature (°F) Material Low High 1 13 15 2 70 0 5 0 74 18 8 58 0 2 2 13 11 9 10 8 0 6 4(a) Compute the estimates 16 16 8 60 of the effects and their 8 0 2 standard errors for this design.(b) Construct two-factor interaction plots and comment on the interaction of the factors.(c) Use the t ratio to determine the significance of each effect with .Comment on your findings.(d) Compute an approximate 95% CI for each effect. Compare your results with those in part (c) and comment.(e) Perform an analysis of variance of the appropriate regression model for this design. Include in your analysis hypothesis tests for each coefficient, as well as residual analysis. State your final conclusions about the adequacy of the model. Compare your results to part (c) and comment.Exercise 5: 22
• Probability and Statistics Work BookAn article in the IEEE Transactions on Semiconductor Manufacturing (Vol. 5, 1992, pp.214-222) describes an experiment to investigate the surface charge on a silicon wafer. Thefactors thought to influence induced surface charge are cleaning method (spin rinse dry orSRD and spin dry or SD and the position on the wafer where the charge was measured. Thesurface charge ( X1011 q/cm3) response data are shown. Test Position L R 1.66 1.84 Cleaning SD 1.90 1.84 Method 1.92 1.62 -4.21 -7.58 SRD -1.35 -2.20 -2.08 -5.36(a) Compute the estimates of the effects and their standard errors for this design.(b) Construct two-factor interaction plots and comment on the interaction of the factors.(c) Use the t ratio to determine the significance of each effect with .Comment on your findings.(d) Compute an approximate 95% CI for each effect. Compare your results with those in part (c) and comment.(e) Perform an analysis of variance of the appropriate regression model for this design. Include in your analysis hypothesis tests for each coefficient, as well as residual analysis. State your final conclusions about the adequacy of the model. Compare your results to part (c) and comment. 23
• 10Probability and Statistics Work Book Concept of Probability00Learning Outcome:The students should be able to understand the basic concept of probability, sample space,probability of events, counting rule; conditional probability; multiplication rule and BayestheoremExercise 1:Each message in a digital communication system is classified as to whether it is receivedwithin the time specified by the system design. If 3 messages are classified, what is anappropriate sample space for this experiment?Exercise 2:A digital scale is used that provide weights to the nearest gram. Let event A: a weightexceeds 11 grams, B: a weight is less than or equal to 15 grams, C: a weight is greater than orequal to 8 grams and less than 12 grams.What is the sample space for this experiment? and find(a) A U B (b) A’ (c) A ∩ B(d) (A U C)’ (e) A ∩ B ∩ C (f) B’ ∩ C 24
• Probability and Statistics Work BookExercise 3:Samples of building materials from three suppliers are classified for conformance to air-quality specifications. The results from 100 samples are summarized as follows: Conforms Yes No R 30 10 Supplier S 22 8 T 25 5Let A denote the event that a sample is from supplier R, and B denote the event that a sampleconforms to the specifications. If sample is selected at random, determine the followingprobabilities:(a) P(A) (b) P(B) (c) P(B’)(d) P(AUB) (e) P(A ∩ B) (f) P(AUB’)(g) P( A B) (h) P( B A) 25
• Probability and Statistics Work BookExercise 4: (Tutorial 10, No.1)The compact discs from a certain supplier are analyzed for scratch and shock resistance. Theresults from 100 discs tested are summarized as follows: Scratch Resistance High Low High 30 10 Shock Medium 22 8 Resistance Low 25 5Let A denote the event that a disc has high shock resistance, and B denote the event that adisc has high scratch resistance. If sample is selected at random, determine the followingprobabilities:(a) P(A) (b) P(B) (c) P(B’)(d) P(AUB) (e) P(A ∩ B) (f) P(AUB’)(g) P( A B) (h) P( B A)Exercise 5:The reaction times ( in minutes) of a reactor for two batches are measured in an experiment.a) Define the sample space of the experiment.b) Define event A where the reaction time of the first batch is less than 45 minutes and event B is the reaction time of the second batch is greater than 75 minutes.c) Find A U B, A ∩ B and A’d) Verify whether events A and B are mutually exclusive. 26
• Probability and Statistics Work BookExercise 6: (Tutorial 10, No.2)When a die is rolled and a coin is tossed, use a tree diagram to describe the set of possibleoutcomes and find the probability that the die shows an odd number and the coin shows ahead.Exercise 7: (Tutorial 10, No.3)A bag contains 3 black and 4 while balls. Two balls are drawn at random one at a timewithout replacement.(i) What is the probability that a second ball drawn is black?(ii) What is the conditional probability that first ball drawn is black if the second ball is known to be black?Exercise 8:An oil-prospecting firm plans to drill two exploratory wells. Past evidence is used to assessthe possible outcomes listed in the following table: Event Description Probability A Neither well produces oil or gas 0.80 B Exactly one well produces oil or gas 0.18 C Both wells produce oil or gas 0.02Find and give description for 27
• Probability and Statistics Work Book P ( A ∪ B ), P ( B ∪ C ) and P ( B )Exercise 9:In a residential suburb, 60% of all households subscribe to the metro newspaper published ina nearby city, 80% subscribe to the local paper, and 50% of all households subscribe to bothpapers. Draw a Venn diagram for this problem.If a household is selected at random, what is the probability that it subscribes toa) at least one of the two newspapersb) exactly one of the two newspapersExercise 10:In a student organization election, we want to elect one president from five candidates, onevice president from six candidates, and one secretary from three candidates. How manypossible outcomes?Exercise 11:Suppose each student is assigned a 5 digit number. How many different numbers can becreated?Exercise 12:A chemical engineer wishes to conduct an experiment to determine how these four factorsaffect the quality of the coating. She is interested in comparing three charge levels, five 28
• Probability and Statistics Work Bookdensity levels, four temperature levels, and three speed levels. How many experimentalconditions are possible?Exercise 13: (Tutorial 10, No.4)A menu has five appetizers, three soup, seven main course, six salad dressings and eightdesserts. In how many ways cana) A full meal be chosen?b) A meal be chosen if either and appetizer or a soup is ordered, but not both?Exercise 14:Ten teaching assistants are available to grade a test of four questions. Wish to select adifferent assistant to grade each question (only one assistant per question). How manypossible ways can the assistant be chosen for grading?Exercise 15:Participant samples 8 products and is asked to pick the best, the second best, and the thirdbest. How many possible ways?Exercise 16:Suppose that in the taste test, each participant samples eight products and is asked to selectthe three best products. What is the number of possible outcomes? 29
• Probability and Statistics Work BookExercise 17:A contractor has 8 suppliers from which to purchase electrical supplies. He will select 3 ofthese at random and ask each supplier to submit a project bid. In how many ways can theselection of bidders be made?Exercise 18:Twenty players compete in a tournament. In how may ways cana) rankings be assigned to the top five competitors?b) the best five competitors be randomly chosen?Exercise 19:Three balls are selected at random without replacement from the jar below. Find theprobability that one ball is red and two are black.Exercise 20: 30
• Probability and Statistics Work BookA university warehouse has received shipment of 25 printers, of which 10 are laser printersand 15 are inkjet models. If 6 of these 25 are selected at random by a technician, what is theprobability that exactly 3 of those selected are laser printers?Exercise 21:There are 17 broken light bulbs in a box of 100 light bulbs. A random sample of 3 light bulbsis chosen without replacement.a) How many ways are there to choose the sample?b) How many samples contain no broken light bulbs?c) What is the probability that the sample contains no broken light bulbs?d) How many ways to choose a sample that contains exactly 1 broken light bulb?e) What is the probability that the sample contains no more than 1 broken light bulb?Exercise 22: (Tutorial 10, No.5)An agricultural research establishment grows vegetables and grades each one as either goodor bad for taste, good or bad for its size, and good or bad for its appearance. Overall, 78% ofthe vegetables have a good taste. However, only 69% of the vegetables have both a goodtaste and a good size. Also, 5% of the vegetables have a good taste and a good appearance,but a bad size. Finally, 84% of the vegetables have either a good size or a good appearance. a) if a vegetable has a good taste, what is the probability that it also has a good size? b) if a vegetable has a bad size and a bad appearance, what is the probability that it has a good taste? 31
• Probability and Statistics Work BookExercise 23:A local library displays three types of books entitled “Science” (S),“Arts” (A), and “Novels” (N). Reading habits of randomly selectedreader with respect to these types of books areRead regularly S A N S∩A S∩N A∩N S∩A∩N Probability 0.14 0.23 0.37 0.08 0.09 0.13 0.05Find the following probabilities and interpret a) P( S | A ) b) P( S | A U N ) c) P( S | reads at least one ) d) P( S U A | N)Exercise 24: (Tutorial 10, No.6)A batch of 500 containers for frozen orange juice contains 5 that are defective. Two areselected at random, without replacement, from the batch. Let A and B denote that the firstand second selected is defective respectivea) Are A and B independent events?b) If the sampling were done with replacement, would A and B be independent? 32
• Probability and Statistics Work BookExercise 25:Everyday (Mon to Fri) a batch of components sent by a first supplier arrives at certaininspection facility. Two days a week, a batch also arrives from a second supplier. Eightypercent of all batches from supplier 1 pass inspection, and 90% batches of supplier 2 passinspection. On a randomly selected day, what is the probability that two batches passinspection?Exercise 26:The probability is 1% that an electrical connector that is kept dry fails during the warrantyperiod of a portable computer. If the connector is ever wet, the probability of a failure duringthe warranty period is 5%. If 90% of the connectors are kept dry and 10% are wet, whatproportion of connectors fail during the warranty period? 33
• Probability and Statistics Work BookExercise 27:Computer keyboard failures are due to faulty electrical connects (12%) or mechanical defects(88%). Mechanical defects are related to loose keys (27%) or improper assembly (73%).Electrical connect defects are caused by defective wires (35%), improper connections (13%)or poorly welded wires (52%). Find the probability that a failure is due toi. loose keysii. improperly connected or poorly welded wires.Exercise 28:During a space shot, the primary computer system is backed up by two secondary systems.They operate independently of one another, and each is 90% reliable. What is the probabilitythat all three systems will be operable at the time of the launch? 34
• Probability and Statistics Work BookExercise 29:A store stocks light bulbs from three suppliers. Suppliers A, B, and C supply 10%, 20%, and70% of the bulbs respectively. It has been determined that company A’s bulbs are 1%defective while company B’s are 3% defective and company C’s are 4% defective. If a bulbis selected at random and found to be defective, what is the probability that it came fromsupplier B?Exercise 30:A particular city has three airports. Airport A handles 50% of all airline traffic, while airportsB and C handle 30% and 20%, respectively. The rates of losing a baggage in airport A, B andC are 0.3, 0.15 and 0.14 respectively. If a passenger arrives in the city and losses a baggage,what is the probability that the passenger arrives at airport A? 35
• Probability and Statistics Work BookExercise 31:A company rated 75% of its employees as satisfactory and 25% unsatisfactory. Of thesatisfactory ones 80% had experience, of the unsatisfactory only 40%. If a person withexperience is hired, what is the probability that (s)he will be satisfactory?Exercise 32:In a certain assembly plant, three machines, B1, B2, B3, make 30%, 45% and 25%,respectively, of the products. It is known from past experience that 2%,3% and 2% of theproducts made by each machine, respectively, are defective. Now, suppose that a finishedproduct is randomly selected.i. What is the probability that it is defective? 36
• Probability and Statistics Work Bookii. If a product was chosen randomly and found to be defective, what is the probability that it was produced by machine B3?Exercise 33: (Tutorial 10, No.7)Three machines A, B and C produce identical items of their respective output 5%, 4% and3% of the items are faulty. On a certain day A has produced 25%, B has produced 30% andC has produced 45% of the total output. An item selected at random is found to be faulty.What are the chances that it was produced by C?Exercise 34: (Tutorial 10, No.8)Suppose that a test for Influenza A, H1N1 disease has a very high success rate: if a testedpatient has the disease, the test accurately reports this, a ’positive’, 99% of the time, and if atested patient does not have the disease, the test accurately reports that, a ’negative’, 95% ofthe time. Suppose also, however, that only 0.1% of the population have that disease. 37
• Probability and Statistics Work Book(i) What is the probability that the test returns a positive result?(ii) If the patient has a positive, what is the probability that he has the disease?(iii) What is the probability of a false positive?Exercise 35:An insurance company charges younger drivers a higher premium than it does older driversbecause younger drivers as a group tend to have more accidents. The company has 3 agegroups: Group A includes those less than 25 years old, have a 22% of all its policyholders.Group B includes those 25-39 years old, have a 43% of all its policyholders, Group Cincludes those 40 years old and older, have 35% of all its policyholders. Company recordsshow that in any given one-year period, 11% of its Group A policyholders have an accident.The percentages for groups B and C are 3% and 2%, respectively.(a) What is the probability that the company’s policyholders are expected to have an accident during the next 12 months?(b) Suppose Mr. Chong has just had a car accident. If he is one of the company’s policyholders, what is the probability that he is under 25? 38