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Probability And Statistics For Engineers And Scientists 3rd Edition Anthony Hayter - Solutions

DS 10.3.5 presents a data set compiled from a series of clinical trials to investigate the effectiveness of a certain medication in healing an infection. (a) Is it appropriate to say that there is a 50% chance that the infection is completely healed and a 10% chance that there is no change in the
A beverage company has three formulations of a soft drink product. DS 10.3.6 gives the results of some taste tests where participants are asked to declare which formulation they like best. Is it plausible that the three formulations are equally popular?
DS 10.3.7 gives a data set of the number of arrivals at a hospital emergency room during one-hour periods. Is there any evidence that it is not reasonable that the number of arrivals are modeled with a Poisson distribution with mean λ = 7?
DS 10.3.8 gives a data set of the number of radioactive particles emitted from a substance and passing through a counter in one-minute periods. Is there any evidence that it is not reasonable to model these with a Poisson distribution?
Oyster pearls are graded A. B. C. and D depending on whether their diameter is 9-10 mm, 7 9 mm. 5-7 mm, or 0-5 mm. respectively. DS 10.3.9 shows how a farmer's pearl crop was categorized. Is it plausible that the pearl oyster diameters have a uniform distribution between 0 and 10 mm?
A computer manufacturer has four suppliers of electrical circuit boards. Random samples of 200 circuit boards are taken from each of the suppliers and they are classified as being either acceptable or defective, as given in DS 10.4.1. Consider the problem of testing whether the defective rates are
An experiment was conducted to compare three types of asphalt. Samples of each type of asphalt were subjected to repeated loads at high temperatures, and the resulting cracking was analyzed. For type A. 57 samples were tested, of which 9 had severe cracking. 17 had medium cracking, ami 3 I had
Seedlings are grown without fertilizer or with one of two kinds of fertilizer. After a certain period of time a seedling's growth is classified into one of four categories, as given in DS 10.4.2. Test whether the seedlings' growth can be taken to be the same for all three sets of growing conditions.
DS 10.4.3 gives the results of a taste test in which a sample of 200 people in each of three age groups are asked which of three formulations of a soft drink they prefer. Test whether the preferences for the different formulations change with age.
A factory has five production lines that assemble electric motors. A random sample of ISO motors is taken from each production line and is given a quality examination. The results are given in DS 10.4.4. (a) Is there any evidence that the pass rates are any different for the five production lines?
An air-conditioner supplier employs four technicians who visit customers to install and provide maintenance for their air-conditioner units. In an after-visit questionnaire the customers are asked to rate their satisfaction with the technician. DS 10.4.5 gives the ratings obtained by the four
DS 10.4.6 presents a 2 × 2 contingency table that compares two drugs with respect to the speed with which a patient recovers from an ailment. Let ps be the probability that a patient recovers in less than one week if given the standard drug, and let pn be the probability that a patient recovers in
In the manufacture of semiconductors, reactive ion etching is a technique whereby the surface of the semiconductor is bombarded with ions to remove unwanted material and to leave the desired structure. In an experiment a set of semiconductors was produced by this technique, and each one was
A company's sales records show that an extended warranty was purchased on 38 out of 89 sales of a product of type A, an extended warranty was purchased on 62 out of 150 sales of a product of type B. and an extended warranty was purchased on 37 out of 111 sales of a product of type C. Do these data
Crystallization is an important step in the recovery and purification of biological molecules such as enzymes and proteins. The determination of conditions that produce suitable crystals is of great interest to molecular biologists. In one experiment, crystals had appeared within 24 hours in 27 out
A garage sells tires of types A. B. and C. and this year's and last year's sales of the three types are given in DS 10.6.5. Is there evidence of a change in the preferences for the three types of tire between the two years?
DS 10.6.6 gives the results of a set of clinical trials involving three different medications. Test whether the three medications are equally effective at treating the infection.
DS 10.6.7 presents the results of an opinion poll conducted on students in the College of Engineering and students in the College of Arts and Sciences. Is there any evidence that opinions differ between students in these two colleges?
The dimensions of 3877 manufactured parts were examined and 445 were found to have a length outside a specified tolerance range. (a) Conduct a hypothesis test to investigate whether there is sufficient evidence to conclude that the probability of a part having a length outside the tolerance range
In a research report on the effect of moisture on a certain kind of composite material, it is reported that in 80' < of cases the effect is minimal, in 15% of cases the effect is strong, and in the remaining 5% of cases the effect is severe. An experimenter tested these claims by subjecting 800
An experiment is performed to investigate the best way to produce a chemical solution. Three different preparation methods are considered, and various trials are conducted with each method. The resulting solutions are classified as being either too weak, satisfactory, or too strong, as given in DS
Three types of a metal alloy were investigated to see how much damage they suffered when subjected to a high temperature. A total of 220 samples were obtained for each alloy, and after being subjected to the high temperature, the damage was classisfied as "none."' "slight," "medium," or "severe."
A company's orders are classified as coming from geographical area A, B, C, or D. Over a certain period, there were 119 orders from area A. 54 orders from area B. 367 orders from area C. and 115 orders from area D. (a) It is claimed that the probability that an order is from area A is 0.25, the
An experimenter obtained 84 samples of concrete. When the samples were each subjected to a load of size 115, a total of 17 of the samples broke while the other samples were unharmed. The remaining 67 samples were then subjected to a load of size 120, and 32 of them broke. Finally, the remaining 35
When a random sample of 64 male students was asked their opinion on a proposal, 28 of them expressed support. Also, when a random sample of 85 female students was asked their opinion on the proposal. 31 of them expressed support. (a) Use an appropriate hypothesis test to assess whether there is
A consumer watchdog organization takes a random sample of 500 bags of flour made by a company, weighs them, and discovers that 18 of them are underweight. Suppose that legal action can be taken if it can be demonstrated that the proportion of bags sold by the company that are underweight is more
Patients were diagnosed as being either condition A or condition B before undergoing a treatment. The treatment was successful for 56 out of 94 patients classified as condition A, and the treatment was successful for 64 out of 153 patients classified as condition B. (a) Perform a hypothesis test to
Are the following statements true or false? (a) Contingency tables can be used to summarize count frequencies for discrete data. (b) The degrees of freedom used in a chi-square goodness of fit test are related to the number of cells being examined. (c) Independence in a two-way contingency table
In a customer satisfaction survey a random sample of 635 customers were asked their opinion on the service they received. A total of 485 of these customers replied that they were very satisfied. (a) Construct a two-sided 05'/ confidence interval for the proportion of customers that are very
The records at the emergency rooms of five hospitals over a certain period of time were examined. Each patient was classified as either being "admitted to the hospital" or as "returned home." as given in DS 10.6.10. (a) Is there any evidence to support the claim that the hospital admission rates
After large Hoods and the consequent large river Hows, there can be a reduction in the level of the riverbed around the piers of bridges that is known as scouring. This can be a serious problem if the foundations of the piers become exposed. The amount of scouring can depend on the shape of the
A bank releases a new credit card that is to be targeted at a population group of about 1.000.000 customers. In a trial run the bank mails credit card applications to a random sample of 5000 customers within this group, and 384 of them request the credit card. If the bank goes ahead and mails
Upon checking hospital records, a hospital administrator notices that over a certain period of time. 443 out of 564 surgical operations performed in the morning turned out to be a "total success." whereas only 388 out of 545 surgical operations performed in the afternoon turned out to be a "total
Householders are polled on whether they support a tax increase to build more schools. The householders are also asked whether their annual household income is above or below $60,000. The results of the poll found 106 householders with an annual income above $60,000 of whom 32 support the tax
DS 10.6.1 contains a data set collected during an amateur archery contest. Calculate a X2 statistic to assess whether it is appropriate to model the probabilities of a bull's-eye and a missed target as both being 10%.
DS 10.6.2 gives a data set of the number of car accidents occurring during evening rush-hour traffic in a certain city. Does it look like it's reasonable to model these with a Poisson distribution?
A random-number generator is supposed to provide numbers that are uniformly distributed between 0 and 1. A total of n = 10.000 simulations are obtained, and they are classified as falling into one of ten intervals of length 0.1. as given in DS 10.6.3. Is there any evidence that the random-number
Use Table IV to determine whether these probabilities are greater than 10%, between 5% and 10%, between 1% and 5%, or less than 1%. Then use a computer package to evaluate them exactly. (a) P(X > 4.2) where X has an F4,15 distribution (b) P(X > 2.3) where X has an F7,30 distribution (c) P(X > 31.7)
Consider the data set given in DS 11.1.1.(a) Calculate the sample averages 1,(b) Calculate the overall average ...(c) Calculate the treatment sum of squares SSTr.(e) Calculate the total sum of squares SST.(f) Calculate the error sum of squares SSE.(g) Complete the analysis of variance table. What
Complete parts (a)-(i) of Problem 11.1.11 for the data set in DS 11.1.2.(a) Calculate the sample averages 1,(b) Calculate the overall average ...(c) Calculate the treatment sum of squares SSTr.(e) Calculate the total sum of squares SST.(f) Calculate the error sum of squares SSE.(g) Complete the
A factory has three production lines producing glass sheets that are all supposed to be of the same thickness. A quality inspector takes a random sample of n = 30 sheets from each production line and measures their thicknesses. The glass sheets from the first production line have a sample average
A retail company is interested in examining whether the time taken for it to obtain approval for a credit card number by phone varies from one day of the week to another. The manager collects the following results. On Mondays, a sample of n1 = 20 cases had a sample average of 1 = 20.44 seconds
The data set in DS 11.1.3 concerns the infrared radiation readings from an energy source measured by a particular meter with various levels of background radiation levels. The meter's manufacturers are interested in how the readings are affected by the background radiation levels. What would you
DS 11.1.4 gives the times taken to perform a numerical task using three different kinds of keyboard layouts for the numerical keys. Perform hand calculations to investigate how the different layouts affect the time taken to perform the task.
A chemical engineer is interested in how polymethyl methacrylate (PMMA) particles can be prepared by dispersion polymerization in the presence of a stabilizer polyvinyl pyrrolidone (PVP). DS 11.1.5 gives the average particle diameters achieved in various experiments with three different amounts of
A computer manufacturer has production lines that utilize three different assembly methods. These methods differ in the order in which tasks are performed and the consequent layout of the production lines. A quality inspector randomly observes 30 computers assembled under each of the three methods
Three catalysts were compared to investigate how they affect the strength of a compound. When catalyst A was used 8 samples of the compound were obtained and the strengths of these 8 samples had a sample average of 42.91 and a sample standard deviation of 5.33. When catalyst B was used 11 samples
Find the missing values in the analysis of variance table shown in Figure 11.20. What is the p-value?In Figure 11.20
The data set in DS 11.1.7 is a one-way layout to compare the quality scores of a product made from five different treatment methods.(a) Construct the ANOVA table.(b) Construct pairwise comparisons of the treatment effects and make a graphical presentation of what you find. Which treatment mean is
In a one-way layout to compare five treatments an experiment gave n1 = 10, 1= 46.09, n2 = 9, 2 = 42.21, n3 = 9, 3= 55.32, n4 = 7, 4 = 54.62. n5 = 10, 5= 38.79, with Construct pairwise comparisons of the treatment means Î¼i, 1
An experiment was performed to compare four metal alloys. Various samples were made of each alloy, and their strengths were measured. The following data were obtained:Type A: sample size 8, sample mean 10.50, sample standard deviation 1.02.Type B: sample size 8, sample mean 9.22, sample standard
Thirty patients with a similar medical condition were randomly allocated among six physicians, so that each physician received exactly five patients. The times in minutes taken for the physicians to perform an investigatory surgical procedure on the patients were measured and the data set given in
Four samples of water were taken at five different locations along a river and measurements were obtained of the amount of E. Coli colonies within the water as given in DS 11.1.9. Investigate whether there is any evidence that the E. Coli pollution levels are different at the five locations. Make a
In a one-way layout, seven observations are taken from treatment I and their sum is 327.8, eight observations are taken from treatment 2 and their sum is 381.3, seven observations are taken from treatment 3 and their sum is 337.0, and six observations are taken from treatment 4 and their sum is
Find the missing values in the analysis of variance table shown in Figure 11.21. What is the p-value?In Figure 11.21
A balanced experimental design has a sample size of n = 12 observations at each of k = 7 factor levels. The total sum of squares is SST = 133.18, and the sample averages are 1 = 7.75, 2 = 8.41, 3 = 8.07. 4. = 8.30, 5 = 8.17, 6 = 8.81, and 7 = 8.32. Compute the analysis of variance table.
An experiment to compare k = 4 factor levels has n1 = 12 and = 1 = 16.09, n2 = 8 and 2 = 21.55, n3 = 13 and 3 = 16.72. and n4 = 11 and 4 = 17.57. The total sum of squares is SST = 485.53. Compute the analysis of variance table. What is the p-value?
An experiment to compare k = 3 factor levels has n1 = 17 and n2 = 32.30. n2 = 20 and 2 = 34.06, and n3 = 18 and 3 = 32.02. Also,Compute the analysis of variance table. What is the p-value?
A balanced experimental design has a sample size of n = 14 observations at each of k = 4 factor levels. The averages and variance estimates at each of the four factor levels are 1 = 0.705 and s21 = 0.766 × 10-3, 2 = 0.715 and s22 = 1.276 - 10-3, 3 = 0.684 and s23 = 2.608 × 10-3 and 4 = 0.692
A balanced experimental design has a sample size of n = 11 observations at each of k = 3 factor levels. The sample averages are 1 = 48.05, 2 = 44.74, and 3 = 49.11, and MSE = 4.96. (a) Calculate pairwise confidence intervals for the factor level means with an overall confidence level of 95%. (b)
A balanced experimental design has a sample size of n = 6 observations at each of k = 6 factor levels. The sample averages are 1 = 136.3, 2 = 152.1, 3 = 125.7, 4 = 130.2, 5 = 142.3, and 6 = 128.0, and MSE = 15.95. (a) Calculate pairwise confidence intervals for the factor level means with an
Find the missing values in the analysis of variance table shown in Figure 11.37. What are the p-values?In Figure 11.37
A randomized block design is used to compare k = 3 radar systems for detecting airborne objects. A total of b = 8 objects are flown toward the radar stations and the distances at detection by the radar systems are recorded in DS 11.2.4. Perform hand calculations to investigate whether there is
A total of b = 9 golfers are each asked to hit a golf ball with each of k = 4 types of driver. DS 11.2.5 gives the straight line distances traveled by the balls. Are some drivers better than others? Perform hand calculations.
A manufacturing company wants to investigate which of three possible assembly methods for an electric motor is the most efficient. Since there is a fair amount of variability in the expertise of the workforce, it is decided to take workers as a blocking variable in the experiment. A random
A Realtor's office has live agents, and the manager wants to determine who is the best agent. A data set, given in DS 11.2.7, is collected that consists of the agents' commissions in each month of the previous year. Since house sales vary from one month to the next, the manager decides that it is
Find the missing values in the analysis of variance table shown in Figure 11.38. What are the p-values?In Figure 11.38
A randomized block design has k = 4 factor levels and b = 10 blocks. The total sum of squares is SST = 3736.64 and the block sum of squares is SSB1 = 2839.97. The factor level sample averages are 1 = 45.12, 2 = 42.58, 3 = 43.10, and 4 = 41.86. Compute the analysis of variance table. What are
A randomized block design has k = 5 factor levels and b = 15 blocks. The total sum of squares is SST = 1947.89 and the block sum of squares is SSB1 = 1527.12. The factor level sample averages are 1 = 21.50, 2 = 22.51, 3 = 26.08, 4 = 21.03, and 5 = 23.43. Compute the analysis of variance table.
A randomized block design has k = 3 factor levels and b = 8 blocks. The block sum of squares is SSBI = 50.19, and the factor level sample averages are 1 = 5.93, 2 = 4.62, and 3 = 4.78. Also,(a) Compute the analysis of variance table. What is the p-value?(b) Calculate pairwise confidence intervals
Consider the data set given in DS 11.2.1.(a) Calculate the factor level sample averages i(b) Calculate the block sample averages j(c) Calculate the overall average ..(d) Calculate the treatment sum of squares SSTr.(e) Calculate the block sum of squares SSBI.(g) Calculate the total sum of squares
Complete parts (a)-(k) of Problem 1 1.2.7 for the data set in DS 11.2.2. If the experimenter wants conlidence intervals for the pairwise differences of the factor level means with an overall confidence level of 95% that are no longer than 4.0. how much additional sampling would you recommend?
DS 1 1.2.3 gives a data set of brightness measurements for b = 7 batches of kaolin processed through k = 3 calciners. The calciner operators are interested in whether their calciners are operating at different efficiencies. What would you advise them? Perform hand calculations.
An experimenter measured the Young's modulus values of silicon carbide-silica biaxial nanowires. These synthesized nanowires are potentially useful for high-strength composites in which mechanical properties are critical. DS 11.4.1 gives a completely randomized design consisting of the Young's
The data set in DS 11.4.9 is a one-way layout to compare the hardness measurements of three different alloys.(a) Construct the ANOVA table. What conclusion can you draw from the ANOVA table about the differences between the alloys?(b) Construct pairwise comparisons of the alloys and make a
A randomized block design has four factor levels with sample averages 1 = 11.43. 2 = 12.03, 3 = 14.88, and 4 = 11.76. There are 9 blocks and SSBI = 53.28.1 Also. SSE = 14.12. Compute the ANOVA table. What do the p-values tell you? Construct a graphical representation of the differences between
An investigation was conducted of the radon levels in five different rivers. Six samples of water were obtained from each river and the radon levels given in DS 11.4.10 were obtained. Construct an appropriate ANOVA table. What does the p-value in the ANOVA table tell you? Construct an appropriate
A completely randomized design to compare k = 4 factor levels has n1 = 11 and 1 = 213.7, n2 = 8 and 2 = 206.3. n3 = 13 and 3. = 205.7, and n4 = 12 and 4 = 215.8. The mean square error is MSE = 11.23. Calculate pairwise confidence intervals for the factor level means with an overall confidence
A courier service buys four new cars of the same make and model, and they are driven under similar city and suburban conditions. Each time the cars are filled with gasoline, the car's mileage and amount of gasoline required to top off the tank are noted. At the end of a year, the company manager
An experiment is conducted to assess the effect of the temperature during curing of a certain kind of cement. Six batches of the cement are prepared, and each batch is separated into five parts that are then cured at the five different temperature levels under consideration. The strengths of the
A comparison of k = 5 fertilizers is made in a randomized block design that incorporates b = 10 fields of com. Each field is partitioned into five equal areas, and the five fertilizers are randomly allocated to these areas. The resulting corn yields are given in DS 11.4.4. Which fertilizer is best?
A hospital has k = 4 clinics where it can send blood samples for adhesion measurements of sickle red blood cells to endothelium layers. A hospital administrator decides to perform a randomized block experiment to investigate the consistency between the four clinics. A set of b = 12 blood samples is
The insertion gain of a hearing aid is defined to be the difference between the sound pressure level at the eardrum with and without the hearing aid in place. These sound pressure measurements can be made by a probe microphone placed in the ear canal of a subject. An experiment was conducted to
The reduction of drag from air resistance for large road vehicles is an important component of attempts to improve fuel efficiency. Wind tunnel tests were performed on models of four different vehicle designs, and the data set given in DS 11.4.7 was obtained. What conclusions can you draw from this
When leather is used to produce furniture or clothes, its shrinkage is an important consideration. An experiment was conducted to discover the differences between four different preparation methods in terms of the resulting shrinkages of the leather. Seven batches of leather were split into four
Are the following statements true or false? (a) A one-way layout with only k = 2 treatments is equivalent to an independent samples design for comparing two means. (b) In an ANOVA table the sum of squares for treatments can be negative if the only difference between the treatment levels can be
Suppose that a dependent variable y is related to an explanatory variable x through a linear regression model with β0 = 4.2, β1 = 1.7. and with normally distributed error terms with a zero mean and a standard deviation of σ = 3? (a) What is the expected value of the dependent variable when x =
Suppose that the purity of a chemical solution y is related to the amount of a catalyst x through a linear regression model with β0 = 123.0. β1 = -2.16. and with an error standard deviation σ = 4.1. (a) What is the expected value of the purity when the catalyst level is 20? (b) How much does the
Consider the linear regression model y = 5 + 0.9x, with σ = 1.4, which relates the air content of a concrete sample x to the porosity of the sample y. (a) What is the expected value of the porosity for a concrete sample with an air content of 20? (b) How much does the expected value of the
Suppose that you wish to use simple linear regression to investigate the relationship between two variables. Why does the model you obtain depend on which variable is chosen as the y variable? How should you decide which of your two variables should be the y variable?
Consider the linear regression model y = 675.30 - 5.87x with σ = 7.32. which relates the viscosity y of a substance to the temperature x. If a sample of the substance is prepared at temperature 80, what is the probability that its viscosity is less than 220?
DS 12.11.1 gives the times taken by a class of n = 14 students to finish a math test together with the math SAT scores of the students. Perform the following linear regression analysis with time as the dependent variable and SAT as the explanatory variable.(a) Calculate the point estimates β^0,
A simple linear regression analysis has n = 20 and R2 = 0.853. What can you say about the p-value for the two-sided hypothesis test of H0: β1 =0?
Consider the data set in DS 12.11.9 from the mining industry that shows the amount of ore processed in various locations together with the resulting amount of gold that was obtained. A linear regression model is used to analyze the data with the amount of ore processed as the input variable and
Are the following statements true or false? (a) In simple linear regression, the choice of which variable should be designated as the dependent variable and which variable should be designated as the explanatory variable is determined by considering the correlation between the variables. (b) In
The data set in DS 12.11.10 shows the size of some files and the times that they took to download from a website. A linear regression model is used to analyze the data with size as the input variable and time as the output variable.(a) Find β^0,β^1, and σ^2(b) Perform the hypothesis test to
Pressure changes in seawater affect the speed at which sound is transmitted, and the data set in DS 12.11.11 shows how the underwater speed of sound relates to the depth at which it is measured. Use a linear regression model to analyze the data with the depth as the input variable and with the
The data used in Example 43 arc taken from the data set given in DS 12.11.2, which relates the variable % scrap to the number of times that the metal plate is passed through the rollers. It is useful to be able to model the amount of discarded metal. 95 scrap, as a function of the number of passes
DS 12.11.3 contains data from a set of automobile engines concerning the diameter of the bearing and the friction power loss that occurs in the bearing of the engine. Perform a linear regression analysis with power loss as the dependent variable and bearing diameter as the explanatory variable. Can
A double pipe heat exchanger is a device that allows energy to be exchanged between a hot liquid and a cold liquid. It consists of two concentric pipes of different diameter. The hot liquid flows through the inner pipe, and the cold liquid How in the gap between the two pipes, usually in the

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