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bayesian statistics an introduction

Probability & Statistics For Engineers And Scientists With R 1st Edition Michael Akritas - Solutions

The octane ratings of 12 gasoline blends were determined by two standard methods.The 12 differences Di are given in the first row of the table below.Is there evidence, at level α = 0.1, that the rating produced by the two methods differs?(a) Using hand calculations, carry out the test procedure
To assess the effectiveness of a political speech in changing public opinion on a proposed reform, a random sample of n = 300 voters was asked, both before and after the speech, if they support the reform. The before-after data are given in the table below:Was the political speech effective in
Changes in the turbidity of a body of water are used by environmental or soil engineers as an indication that surrounding land may be unstable, which allows sediments to be pulled into the water. Turbidity measurements using the Wagner test, from 10 locations around a lake taken both before and
Consider the study for comparing two methods for determining the iron content in ore samples described in Example 9.5-3. A total of 12 ore samples are analyzed by both methods, producing the paired data that yield D = −0.0167 and SD = 0.02645.Is there evidence that method B gives a higher average
Two different types of materials for making soles for children’s shoes are to be compared for durability. One way of designing this comparative experiment is to make n pairs of shoes where one (either the left or the right) is randomly selected to be made with material A, and the other
A certain lake has been designated for pollution clean-up. One way to assess the effectiveness of the clean-up measures is to randomly select a number n of locations from which water samples are taken and analyzed both before and after the cleanup.The n randomly chosen locations are the
3. Consider the information given in Exercise 3 in Section 9.2, and apply the F test to test for the equality of the two variances against the two-sided alternative at level α = 0.05. What assumptions are needed for the validity of this test?
2. Consider the information given in Exercise 1 in Section 9.2, and apply the F test to test for the equality of the two variances against the two-sided alternative at level α = 0.05. What assumptions are needed for the validity of this test?
1. Consider the information given in Exercise 3 in Section 9.3, and apply Levene’s test to test the equality of the two variances against the two-sided alternative at level α = 0.05.
Consider the data and testing problem described in Example 9.4-1, and assume that the underlying populations are normal.(a) Test H0 : σ2 1 = σ2 2 versus Ha : σ2 1 ̸= σ2 2 using the F test procedure (9.4.4).(b) Implement the F test procedure using R commands, and report the 95% CI for σ2 1
Numerous studies have shown that cigarette smokers have a lower plasma concentration of ascorbic acid (vitamin C) than nonsmokers. Given the health benefits of ascorbic acid, there is also interest in comparing the variability of the concentration in the two groups. The following data represent the
5. Consider the data set and testing problem given in Exercise 3 in Section 9.2, and compare the p-values obtained from the T test and the MWW rank-sum test.Next, compare the 90% CI obtained through the ranksum procedure and with the 90% T CI. Use R commands to obtain all p-values and CIs.
4. One of the variables measured during automobile driver-side crash tests with dummies is the left femur load (the femur is the largest and strongest bone in the human body, situated between the pelvis and the knee). The data file FemurLoads.txt gives left femur load measurements for vehicles of
3. An article reports on a study using high temperature strain gages to measure the total strain amplitude('εm/2) of different types of cast iron for use in disc brakes.6 The results for spheroidal graphite (SG) and compacted graphite (CG), multiplied by 10,000, are given below:Are the total
2. Six water samples taken from the eastern part of a lake and seven taken from the western part are subjected to a chemical analysis to determine the percent content of a certain pollutant. The data are as shown below:Is the pollutant concentration on the two sides of the lake significantly
1. An article reports on a cloud seeding experiment conducted to determine whether cloud seeding with silver nitrate increases rainfall.5 Out of 52 clouds, 26 were randomly selected for seeding, with the remaining 26 serving as controls. The rainfall measurements, in acre-feet, are given in
The sputum histamine levels (in μg/g) from a sample of size 9 allergic individuals and 13 non-allergic individuals4 are as follows:Is there a difference between the two populations? Test at level α = 0.01, and use R commands to construct a 95% CI for the median of the difference between the
10. In 85 10-mph crash tests with typeAcars, 19 sustained no visible damage. For type B cars, 22 out of 85 sustained no visible damage. Is this evidence sufficient to claim that type B cars do better in 10-mph crash tests than type A cars?(a) State the null and alternative hypotheses, carry out the
9. A tracking device, used for enabling robots to home in on a beacon that produces an audio signal, is said to be fine-tuned if the probability of correct identification of the direction of the beacon is the same for each side (left and right) of the tracking device. Out of 100 signals from the
8. An article3 reported results of arthroscopic meniscal repair with an absorbable screw. For tears greater than 25 millimeters, 10 of 18 repairs were successful, while for tears less than 25 millimeters, 22 of 30 were successful.(a) Is there evidence that the success rates for the two types of
7. Earnings management refers to a wide array of accounting techniques that can help earnings per share(EPS) meet analyst expectations. Because reported EPS in the United States are rounded to the nearest cent, earnings of 13.4 cents are rounded down to 13 cents while earnings of 13.5 cents are
6. A facility for bottling soft drinks uses two fill and seal machines. As part of quality control, data are periodically collected to test if the fill weight is the same for the two machines. A particular data collection of 12 fill weights from each machine yields sample mean and sample variance
5. Wrought aluminum alloy 7075-T6 is commonly used in applications such as ski poles, aircraft structures, and other highly stressed structural applications, where very high strength and good corrosion resistance are needed.A laboratory conducts an experiment to determine if the ultimate tensile
4. After curing for several days at 20oC, concrete specimens were exposed to temperatures of either −8oC or 15oC for 28 days, at which time their strengths were determined. The n1 = 9 strength measurements at −8oC resulted in X1 = 62.01 and S1 = 3.14, and the n2 = 9 strength measurements at
3. A company is investigating how long it takes its drivers to deliver goods from its factory to a port for export.Records reveal that with a standard driving route, the last 48 delivery times have sample mean 432.7 minutes and sample standard deviation 20.38 minutes. A new driving route is
2. To compare the corrosion-resistance properties of two types of material used in underground pipe lines, specimens of both types are buried in soil for a 2-year period and the maximum penetration (in mils) for each specimen is measured. A sample of size 42 specimens of material type A yielded X1
1. An article reports on a study regarding the effect of thickness in fatigue crack growth in aluminum alloy 2024-T351.1 Two groups of specimens were created, one with a thickness of 3 mm and the other with a thickness of 15 mm. Each specimen had an initial crack length of 15 mm. The same cyclic
Consider the manufacturing of tractors using two different assembly lines, as described in Example 9.2-4. Let p1 denote the proportion of tractors coming out of assembly line L1 that require adjustments, and let p2 be the corresponding proportion for assembly line L2. (a) Test H0 : p1 = p2 against
During an investigation into the flame-resistance properties of material types A and B, which are being considered for use in children’s pajamas, 85 randomly selected specimens of material type A and 100 randomly selected specimens of material type B were subjected to high temperatures.
Consider the data from the experiment of Example 9.2-2, which compares the current flow of two micro circuit designs. Thus, the n1 = 35 measurements using Design 1 give X1 = 24.2 amps, S1 = √10 amps, and the n2 = 40 measurements using Design 2 give X2 = 23.9 amps and S2 = √14.3 amps. Is the
Consider the data from the experiment of Example 9.2-3, which compares the strengths of two types of steel. Thus, the n1 = 32 cold-rolled steel observations yield X1 = 29.8, S1 = 4.0, and the n2 = 35 two-sided galvanized steel observations yield X2 = 34.7, S2 = 6.74. Are the mean strengths of the
A certain type of tractor is being assembled at two locations, L1 and L2. An investigation into the proportion of tractors requiring extensive adjustments after assembly finds that in random samples of 200 tractors from L1 and 400 from L2, the number requiring extensive adjustments were 16 and 14,
A random sample of n1 = 32 specimens of cold-rolled steel give average strength X1 = 29.80 ksi, and sample standard deviation of S1 = 4.00 ksi. A random sample of n2 = 35 specimens of two-sided galvanized steel give average strength X2 = 34.70 ksi, and S2 = 6.74 ksi. Construct a 99% CI for μ1 −
A manufacturer of video display units wants to compare two micro circuit designs in terms of the current flow they produce. Using Design 1, n1 = 35 current flow measurements give X1 = 24.2 and S1 = √10.0 amps. Using Design 2, n2 = 40 measurements give X2 = 23.9 and S2 = √14.3 amps. Construct a
To compare two catalysts in terms of the mean yield of a chemical process, n1 = 8 chemical processes are performed with catalyst A, and n2 = 8 are performed with catalyst B. From catalyst A we obtain X1 = 92.255, S1 = 2.39. From catalyst B we obtain X2 = 92.733, S2 = 2.98. Construct a 95% CI for
7. Ues R commands and the information given in Exercise 5 in Section 8.3, that is, n = 500, H0 : p = 0.05 vs Ha : p < 0.05, and α = 0.01, to complete the following.(a) Find the probability of type II error at pa = 0.04.(b) What sample size should be used to achieve power of 0.5 at pa = 0.04 while
6. Use R commands and the information given in Exercise 7 in Section 8.3, that is, n = 42, H0 : p = 0.2 vs Ha : p > 0.2, and α = 0.01, to complete the following.(a) Find the probability of type II error at pa = 0.25.(b) What sample size should be used to achieve power of 0.3 at pa = 0.25 while
5. Acid baths are used to clean impurities from the surfaces of metal bars used in laboratory experiments. For effective cleaning, the average acidity of the solution should be 8.5. Average acidity in excess of 8.65 may damage the plated surface of the bars. Before delivering a batch of acid bath
4. Use R commands and the information given in Exercise 2 in Section 8.3, that is, n = 36, S = 4.1, H0 : μ = 1 vs Ha : μ > 1, and α = 0.05, to complete the following.(a) Find the probability of type II error when the true concentration is 2 ppm.(b) OSHA would like the probability of type II
3. In 10-mph crash tests, 25% of a certain type of automobile sustain no visible damage. A modified bumper design has been proposed in an effort to increase this percentage.Let p denote the proportion of all cars with this new bumper that sustain no visible damage in 10-mph crash tests. The
2. To investigate the flame resistance of material type A, used in children’s pajamas, 85 specimens of the material were subjected to high temperatures and 28 of those specimens ignited. Consider testing H0 : pA = 0.3 vs Ha : pA ̸= 0.3, where pA is the probability that a specimen exposed to such
1. (a) When a null hypothesis is rejected, there is risk of committing which type of error?(b) When a null hypothesis is not rejected, there is risk of committing which type of error?
A proposed change to the tire design is justifiable only if the average life span of tires with the new design exceeds 20,000 miles. The life spans of a random sample of n = 16 tires with the new design will be used for the decision. It is known that the life spans are normally distributed with σ
A coalmining company suspects that certain detonators used with explosives do not meet the requirement that at least 90% will ignite. To test the hypothesis H0 : p = 0.9 vs Ha : p < 0.9, a random sample of n = 20 detonators is selected and tested. Let X denote the number of those that ignite. For
15. The standard deviation of a random sample of 36 chocolate chip cookie weights is 0.25 oz. Test the hypothesisH0: σ = 0.2 vsHa : σ ̸= 0.2 at α = 0.05, and report the p-value. State any assumptions that are needed to justify your results.
14. A tire manufacturer will adopt a new tire design unless there is evidence that the standard deviation of tread life span of the new tires is more than 2.5 thousand miles.(a) Specify the null and alternative hypotheses.(b) Use t=read.table(”TireLifeTimes.txt”, header=T);var(t$x) to compute
13. Sign test for percentiles. The sign test procedure can also be applied to test a null hypothesis H0 : xπ = η0 for the (1−π)100th percentile, xπ , by converting it to the null hypothesis H0 : p = π, where p is the probability that an observation is larger than η0. The alternative
12. Use rt=read.table(”RobotReactTime.txt”, header=T);r2=rt$Time[rt$Robot==2] to copy the reaction times of robot 2 into the R object r2.(a) It is desired to test either H0 : (μ = 28 vs Ha (μ > 28 or H0 : μ = 28 vs Ha μ > 28. Use a normal Q-Q plot to decide which of the two hypotheses tests
11. It is claimed that the median increase in home owners’taxes in a certain county is $300. A random sample of 20 home owners gives the following tax-increase data(arranged from smallest to largest): 137, 143, 176, 188, 195, 209, 211, 228, 233, 241, 260, 279, 285, 296, 312, 329, 342, 357, 412,
10. A historic (circa 1920) data set on the relationship between car speed (X) and stopping distance (Y) is given in the R data frame cars. Use the R output given below to answer the following questions(a) What is the sample size in this study?(b) Give the estimate of the standard deviation of the
9. In the context of a study on the relationship between exercise intensity and energy consumption, the percentage of maximal heart rate reserve (X) and the percentage of maximal oxygen consumption (Y) were measured for 26 male adults during steady states of exercise on a treadmill.5 Use the data
8. An experiment examined the effect of temperature on the strength of new concrete. After curing for several days at 20oC, specimens were exposed to temperatures of−10oC, −5oC, 0oC, 10oC, or 20oC for 28 days, at which time their strengths were determined. The data can be found in
7. A food processing company is considering themarketing of a new product. The marketing would be profitable if more than 20% of consumers would be willing to try this new product. Among 42 randomly chosen consumers, 9 said that they would purchase the new product and give it a try.(a) Set up the
6. KitchenAid will discontinue the bisque color for its dishwashers, due to reports suggesting it is not popular west of the Mississippi, unless more than 30% of its customers in states east of the Mississippi prefer it. As part of the decision process, a random sample of 500 customers east of the
5. In the airline executive traveler’s club setting of Exercise 5 in Section 8.2, a random sample of 500 customers yielded 40 who would qualify.(a) Test the hypothesis H0 : p = 0.05 vs Ha : p < 0.05 at α = 0.01, and state what action the airline should take.(b) Compute the p-value using either
4. Researchers are exploring alternative methods for preventing frost damage to orchards. It is known that the mean soil heat flux for plots covered only with grass isμ0 = 29 units. An alternative method is to use coal dust cover.(a) Heat flux measurements of 8 plots covered with coal dust yielded
3. To investigate the corrosion-resistance properties of a certain type of steel conduit, 16 specimens are buried in soil for a 2-year period. The maximum penetration(in mils) for each specimen is then measured, yielding a sample average penetration of X = 52.7 and a sample standard deviation of S
2. Studies have shown that people who work with benzene longer than five years have 20 times the incidence of leukemia than the general population. As a result, OSHA(Occupational Safety and Health Administration) has set a time-weighted average permissible exposure limit of 1 ppm. A steel
1. In the context of Exercise 4 in Section 8.2, suppose that the old machine achieves μ0 = 9.5 cuts per hour.The appliance manufacturer decides to test H0 : μ = 9.5 against Ha : μ > 9.5 at level α = 0.05. She gets the machine manufacturer to lend her a new machine and she measures the number of
An optical firm is considering the use of a particular type of glass for making lenses.As it is important that the various pieces of glass have nearly the same index of refraction, this type of glass will be used if there is evidence that the standard deviation of the refraction index is less than
Elevated blood pressure among infants is thought to be a risk factor for hypertensive disease later in life.4 However, because blood pressure is rarely measured on children under the age of three, there is little understanding of what blood pressure levels should be considered elevated. Systolic
The n = 59 measurements of temperature and dissolved oxygen of Example 8.3-3 yield MSE = 1.5329 and MSR = 87.924.(a) Using the given information, construct the ANOVA table by hand calculations and report the percent of the total variability explained by the regression model.(b) Conduct the ANOVA F
Measurements along the river Ijse3 on temperature (oC) and dissolved oxygen(mg/L), taken from March 1991 to December 1997, can be found in OxygenTempData.txt.(a) The question of scientific interest is whether temperature (X) can be used for predicting the amount of dissolved oxygen (Y).(i) Does the
It is thought that more than 70% of all faults in transmission lines are caused by lightning. In a random sample of 200 faults from a large data base, 151 are due to lightning. Does the data provide strong evidence in support of this contention? Test at level of significance α = 0.01, and report
The proposed federal health standard for the maximum acceptable level of exposure to microwave radiation in the US2 is an average of 10 W/cm2. It is suspected that a radar installation used for air traffic control may be pushing the average level of radiation above the safe limit. A random sample
11. In the context of Exercise 2, suppose it is decided to test H0 : p = 0.25 vs Ha : p > 0.25. In n = 50 independent crashes of car prototypes with the new bumper, X = 8 result in no visible damage. Let ZH0 be the test statistic given in (8.2.9).(a) Draw by hand a figure similar to those in Figure
10. Suppose a tire manufacturer wants to claim that the average tread life of a certain type of tire is more than 28,000 miles. To gain empirical evidence that this claim is true, a study is initiated to test H0 : μ = 28,000 vs Ha : μ > 28,000. The tread life spans of a random sample of n = 25
9. Working as in Example 8.2-3, give the CI-based rejection rule for testingH0 : θ =θ0 versus Ha : θ ̸=θ0 in terms of a test statistic when(a) θ stands for the regression slope β1.(b) θ stands for the expected response μY|X(x) at a given value x of the covariate.
8. A coal mining company suspects that certain detonators used with explosives do not meet the requirement that at least 90% will ignite. To investigate this suspicion, a random sample of n detonators is selected and tested.Let X denote the number of those that ignite.(a) State the null and
7. In a simple linear regression context, consider testing a hypothesis regarding the expected response at a value X = x of the covariate.(a) If the hypothesis is of the form H0 : μY|X(x) = μY|X(x)0 versus Ha : μY|X(x) > μY|X(x)0, whereμY|X(x)0 is a specified value, state the form of the
6. Consider the hypothesis-testing problem in part (b) of Example 8.2-1. Thus, the null hypothesis is H0 : μ ≤ 28, 000. Let X denote the average tread life span of a random sample of n tires.(a) The rule for rejecting this null hypothesis should be of the form X ≥ C, for some constant C >
5. In making plans for an executive traveler’s club, an airline would like to estimate the proportion of its current customers who would qualify for membership. A random sample of 500 customers yielded 40 who would qualify.(a) The airline wants to proceed with the establishment of the executive
4. An appliance manufacturer is considering the purchase of a new machine for cutting sheet metal parts. Ifμ0 is the average number of metal parts cut per hour by her old machine and μ is the corresponding average for the new machine, the manufacturer wants to test the null hypothesis H0 : μ =
3. The CEO of a car manufacturer is considering the adoption of a new type of grille guard for the upcoming line of SUVs. If μ0 is the average protection index of the current grille guard and μ is the corresponding average for the new grille guard, the manufacturer wants to test the null
2. In 10-mph crash tests, 25% of a certain type of automobile sustain no visible damage. A modified bumper design has been proposed in an effort to increase this percentage. Let p denote the probability that a car with the modified bumper design sustains no visible damage in a 10-mph crash test.
1. Researchers are exploring alternative methods for preventing frost damage to orchards. It is known that the mean soil heat flux for plots covered only with grass is 29 units. An alternative method is to use coal dust cover. Due to the additional cost of covering plots with coal dust, this method
Consider testing the hypothesis H0 : μ = 16 vs Ha : μ ̸= 16 using the sample of 15 observations given in Example 8.2-4. Find the p-value.
It is suspected that a machine, used for filling plastic bottles with a net volume of 16.0 oz, on average, does not perform according to specifications. An engineer will collect 15 measurements and will reset the machine if there is evidence that the mean fill volume is different from 16 oz. The
Let X1, . . . ,Xn be a simple random sample from a normal(μ, σ2) population, and consider testing the hypothesis H0 : μ = μ0 versus Ha : μ ̸= μ0 by rejecting H0 ifμ0 does not lie inside the (1−α)100% T CI for μ. Express this RR in terms of a test statistic, and show that the level of
Consider the two testing problems in parts (a) and (b) of Example 8.2-1, and suppose that the tread life spans are approximately normally distributed and the population variance σ2 is known. Let X ≤ C1 and X ≥ C2 be the rejection regions for the testing problems in parts (a) and (b),
Designate H0 and Ha for each of the following testing situations.(a) A trucking firm suspects that a tire manufacturer’s claim that certain tires last at least 28,000 miles, on average, is faulty. The firm intends to initiate a study, involving data collection and hypothesis testing, to provide
5. Fifty black bears, captured in the period September to November, were anesthetized and their bodies were measured and weighed.7 Use bd=read.table(”BearsData.txt”, header=T); x=bd$Chest.G; y=bd$Weight to import the data into the R data frame bd and to copy the chest girth and weight
4. The article Effects of Bike Lanes on Driver and Bicyclist Behavior reports data from a study on X = distance between a cyclist and the roadway center line, and Y = the separation distance between the cyclist and a passing car (both determined by photography).6 The data, in feet, from n = 10
3. Use cs=read.table(”Concr.Strength.1s.Data.txt”, header=T) to read into the R data frame cs data on 28-day compressive strength measurements of concrete cylinders using water/cement ratio 0.4,5 and the command y=cs$Str to copy the n = 32 data points into the R object y. Use R commands to
2. Soil heat flux is used extensively in agrometeorological studies, as it relates to the amount of energy stored in the soil as a function of time. A particular application is in the prevention of frost damage to orchards. Heat flux measurements of eight plots covered with coal dust yielded X =
1. A random sample of 16 chocolate chip cookies made by a machine yields an average weight of 3.1 oz and a standard deviation of 0.3 oz. Construct an interval that contains the weight of the next chocolate chip cookie to be sampled with confidence 90%, and state any assumptions that are needed for
Data on rainfall volume (X) and soil runoff volume (Y) can be found in SoilRunOffData.txt.4 Make a prediction for the volume of soil runoff at the next rainfall of volume X = 62, and construct a 95% PI for the volume of soil runoff at X = 62. What assumptions are needed for the validity of the
The following is fat content measurements (in g) from a sample of size n = 10 hot dogs of a certain type:y 24.21, 20.15, 14.70, 24.38, 17.02, 25.03, 26.47, 20.74, 26.92, 19.38 Use this data to construct a 95% PI for the fat content of the next hot dog to be sampled. What assumptions are needed for
4. A pilot study of the electro-mechanical protection devices used in electrical power systems showed that of 193 devices that failed when tested, 75 failed due to mechanical parts failures. Use either hand calculations or R commands to answer the following.(a) How large a sample is required to
3. A food processing company, considering the marketing of a new product, is interested in the proportion p of consumers that would try the new product. In a pilot sample of 40 randomly chosen consumers, 9 said that they would purchase the new product and give it a try. Use either hand calculations
2. A pilot sample of 50 measurements of the breaking strength of cotton threads gave Spr = 18 grams. Use either hand calculations or an R command to determine the sample size needed to obtain a 90% CI of length 4.
1. The estimation of the average shrinkage percentage of plastic clay should have an error bound of 0.2 with 98% confidence. A pilot sample of npr = 50 gave Spr = 1.2. Use either hand calculations or an R command to determine the sample size that should be used.
A new method of pre-coating fittings used in oil, brake, and other fluid systems in heavy-duty trucks is being studied for possible adoption. In this context, the proportion of such fittings that leak must be determined to within 0.01 with 95%confidence.(a) What sample size is needed if a
The estimation of a new operating system’s mean response time to an editing command should have an error bound of 5 milliseconds with 95% confidence.Experience with other operating systems suggests that Spr = 25 is a reasonable approximation to the population standard deviation. What sample size
20. Use rt=read.table(”RobotReactTime.txt”, header=T);t2=rt$Time[rt$Robot==2] to import the robot reaction times data set into the R data frame rt and to copy the 22 reaction times of Robot 2 into the R object t2. Construct a 95%CI for the population variance of reaction times of Robot 2, using
19. Kingsford’s regular charcoal with hickory is available in 15.7-lb bags. Long-standing uniformity standards require the standard deviation of weights not to exceed 0.1 lb. The quality control team uses daily samples of 35 bags to see if there is evidence that the standard deviation is within
18. An important quality characteristic of the lapping process that is used to grind certain silicon wafers is the population standard deviation, σ, of the thickness of die pieces sliced out from the wafers. If the thickness of 15 dice cut from such wafers have sample standard deviation of 0.64
17. Use formula (7.3.16) to confirm that in a sample of size n = 16 from a continuous population, a = 4 and a = 5 yield confidence levels as reported in Example 7.3-7.
16. Let X1, . . . ,X30 be a random sample from a continuous population. It has been decided to use (X(10),X(21))as a CI for the median. Find the confidence level of this CI.
15. Copy the n = 153 daily measurements of temperature(oF) and wind speed (mph), taken in New York from May to September 1973 and included in the R data set airquality, by x=airquality$Temp; y=airquality$Wind, and use R commands to complete the following parts.(a) Contruct a scatterplot of the data

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