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applied statistics and probability for engineers

Questions and Answers of Applied Statistics And Probability For Engineers

=+c. For a random sample of 64 grills, calculate the approximate probability that the average number of flaws per grill exceeds 1.
=+b. What are the mean and standard deviation of the sampling distribution of the average number of flaws per grill in a random sample of 64 grills?
=+a. Calculate the mean and standard deviation of x.
=+56. The number of flaws x on an electroplated automobile grill is known to have the following probability mass function:x: 0 1 2 3 p(x): .8 .1 .05 .05
=+b. How large a sample would be required to ensure that the first probability in part (a) is at least .99?
=+a. If a random sample of 25 such specimens is selected, what is the probability that the sample average sediment density is at most 3.00? Between 2.65 and 3.00?
=+of .85 (“Modeling Sediment and Water Column Interactions for Hydrophobic Pollutants,” Water Research, 1984: 169–174).
=+55. Suppose that the sediment density (in g@cm3) of specimens from a certain region is normally distributed with a mean of 2.65 and a standard deviation
=+b. What is the approximate probability that the average hardness in a random sample of 40 pins is at least 52?
=+a. If the distribution of all such pin hardness measurements is known to be normal, what is the probability that the average hardness for a random sample of 9 pins is at least 52?
=+54. The Rockwell hardness of certain metal pins is known to have a mean of 50 and a standard deviation of 1.5.
=+of T0 hours will be replaced free of charge to the customer. If it costs the manufacturer $3.00 to replace a package of batteries (materials plus mailing to customer), calculate the expected
=+d. Refer to your answer to part (c). Suppose the battery manufacturer guarantees that any package of batteries that does not yield a total lifetime
=+c. If T denotes the total lifetime of the four batteries in a randomly selected package, find the numerical value of T0 for which P1T $ T02 5 .95.
=+b. What is the probability that the total lifetime of the batteries will exceed 36 hours?
=+a. What is the probability that the average lifetime of the four batteries exceeds 9 hours?
=+53. The lifetime of a certain battery is normally distributed with a mean value of 8 hours and a standard deviation of 1 hour. There are four such batteries in a package.
=+b. If the expected weight per bag is 49.8 lb rather than 50 lb (so that, on average, the bags are underfilled), calculate P149.75 # x # 50.252.
=+a. If the expected weight of each bag is 50 lb and the standard deviation of bag weights is known to be 1 lb, calculate the approximate value of P149.75 # x # 50.252 by relying on the Central Limit
=+52. Let x1, x2, x3,. .., x100 denote the actual weights of 100 randomly selected bags of fertilizer.
=+b. Calculate the probability P111.99 # x # 12.012 when n 5 64.
=+a. Calculate the probability P111.99 # x # 12.012 when n 5 16.
=+51. Refer to Exercise 48. Assume that the distribution of piston diameters is known to be normal.
=+50. A random sample of size 25 is selected from a large batch of electronic components, and the proportion of defective items in the sample is recorded. The proportion of defective items in the
=+c. Calculate the mean and standard error of the sampling distribution of the proportion of engineers in samples of size 100 who favor the proposed changes.
=+b. Calculate the mean and standard error of the sampling distribution of the proportion of engineers in samples of size 25 who do not favor the proposed changes.
=+a. Calculate the mean and standard error of the sampling distribution of the proportion of engineers in samples of size 25 who favor the proposed changes.
=+49. A survey of the members of a large professional engineering society is conducted to determine their views on proposed changes to an ASTM measurement standard. Suppose that 80% of the entire
=+c. Which is more likely to lie within .01 cm of 12 cm, the mean of a random sample of size 16 or the mean of a random sample of size 64?
=+b. Answer the questions in part (a) for sample means based on samples of size 64.
=+a. Where is the sampling distribution of x centered? What is the standard deviation of the sampling distribution of x?
=+48. The inside diameter of a randomly selected piston ring is a random variable with a mean of 12 cm and a standard deviation of .04 cm.
=+c. Will the variance of the sampling distribution of M for samples of size n52 be the same or different from the variance of the sampling distribution of M for samples of size n5100? Give a simple
=+b. Which do you expect to be larger, the mean of the sampling distribution of M for samples of size n52 or the mean of the sampling distribution of M for samples of size n5100? Use the definition
=+a uniform distribution on the interval [2, 4]. For a sample of size n 5 2, do you expect the mean of the sampling distribution of M to be closer to 2 or 4?
=+a. Suppose that the actual distribution of forces needed to disconnect tubes can be described by
=+several tests are made. For a random sample of n connections, the forces x1, x2, x3, . . . , xn required to disconnect the tubes are recorded and the maximum, M, of the n readings is used to
=+connecting intravenous tubes used on hospital patients. To be comparable to an already-existing product, the force required to disconnect two tubes joined by the new device must not exceed 5 lb.
=+47. The Food and Drug Administration (FDA) oversees the approval of both medical devices and new drugs.To gain FDA approval, a new device must be shown to perform at least as well, and hopefully
=+d. What will happen to the variance of the sampling distribution of R as the sample size n increases? Give a simple justification for your answer based on the definition of the range.
=+c. Will the variance of the sampling distribution of R for samples of size n52 be the same or different from the variance of the sampling distribution of R for samples of size n5100? Give a
=+b. For samples of size n5100, what do you predict the mean of the sampling distribution of R will be?
=+a. For samples of size n52, what do you predict the mean of the sampling distribution of R will be?
=+46. Random samples of size n are selected from a population that is uniformly distributed over the interval[10, 20]. Without sampling or performing any calculations, describe what you expect the
=+c. Answer the question in part (a) for samples of size n 5 100. Hint: Use the normal approximation to the binomial distribution.
=+b. Answer the question in part (a) for samples of size n 5 25.
=+a. Suppose that 5% of the items in a particular lot are defective and that a random sample of size n 5 5 is to be taken from the lot. Calculate the probability that the sample proportion p falls
=+sample. The result is then reported in terms of the proportion p 5 xyn of defective items in the sample.Assume that the binomial distribution can be used to describe the behavior of the random
=+45. Refer to Exercise 37. Suppose that a large lot of items is inspected by taking a random sample of size n and determining the number x of defective items in the
=+44. What primary purpose do sampling distributions serve in statistical inference?
=+d. Are x and y independent?
=+c. Find the probability that x 1 y # 21.
=+b. Find the probability mass function of y.
=+a. Find the probability mass function of x.
=+43. Let x be the cost ($) of an appetizer and y be the cost of a main course at a certain restaurant for a customer who orders both courses. Suppose that x and y have the following joint
=+d. What is the expected time for completing both tasks?
=+c. What is the probability that the total time to complete both tasks is less than 35 days?
=+b. Find the probability distribution of the total time for completing both tasks (assume that the framing and wiring tasks are independent).
=+a. Calculate the expected completion time for each task.
=+42. On a construction site, subcontractor A is responsible for completing the structural frame of a building. When this task is complete, subcontractor B then begins the task of installing
=+c. From your answer to part (a), find a general formula (for any value of MTBF) for expressing the median time to failure in terms of the mean time before failure.
=+b. Is the median time to failure from part (a) larger or smaller than the mean time before failure(MTBF)?
=+a. Suppose the lifetime x of an electronic assembly follows an exponential distribution with an MTBF of 500 hours (see Example 5.12 for the definition of MTBF). Find the median of this
=+41. The concept of the median of a set of data can also be applied to the probability distribution of a random variable. If x is a random variable with density function f(x), then the median of
=+40. When used to model lifetimes of components, a probability distribution is said to be “memoryless”if, for a component that has already lasted (without failure) for t hours, the probability
=+c. The exam administrators want to make sure that there is a very small chance, say, 1%, that a person who is guessing will pass the test. What minimum passing score should they allow on the exam
=+b. For the 25-question test, what are the mean and standard deviation of x?
=+a. What type of probability distribution does x have?
=+Let x denote the number of correct answers given by a person who is guessing each answer on a 25-question exam, with each question having five possible answers (for each question, assume only one
=+39. Qualification exams for becoming a state-certified welding inspector are based on multiple-choice tests. As in any multiple-choice test, there is a possibility that someone who is simply
=+c. For what value of is the probability of accepting a shipment about .05?
=+b. Plot the OC curve for the zero acceptance plan that uses sample sizes of n 5 10.
=+a. Let denote the proportion of defective items in a shipment. Develop a general formula for the probability of accepting a shipment having 3 100% defective items.
=+against accepting shipments of inferior quality, they also tend to reject many shipments of good quality.
=+38. Refer to Exercise 37. Acceptance sampling plans that use an acceptance number of c 5 0 are given the name zero acceptance plans. Zero acceptance plans are not frequently used because, although
=+Connect the points on the graph with a smooth curve. The resulting curve is called the operating characteristic (OC) curve of the sampling plan. It gives a visual summary of how the plan performs
=+in a given shipment. Use your answers to parts(a)2(c) to plot the probability of accepting a shipment (on the vertical axis) against 5 0,.05, .10, .20, and .50 (on the horizontal axis).
=+d. Let denote the proportion of defective items
=+c. Rework part (a) for shipments that are 5%, 20%, and 50% defective.
=+b. Suppose that a certain shipment contains no defective items. What is the probability that the shipment will be accepted by the sampling plan in part (a)?
=+a. Suppose a company uses samples of size n 5 10 and an acceptance number of c 5 1 to evaluate shipments. If 10% of the items in a certain shipment are defective, what is the probability that this
=+denoted by x. As long as x does not exceed a prespecified integerc, called the acceptance number, then the entire shipment is accepted for use. If x exceeds c, then the shipment is returned to the
=+37. Acceptance sampling is a method that uses small random samples from incoming shipments of products to assess the quality of the entire shipment. Typically, a random sample of size n is
=+b. Suppose the measuring instrument in part (a)is replaced with a more precise measuring instrument having a standard deviation of .5 mm.What is the probability that a measurement from the new
=+a. If the measurements of the length of an object have a normal probability distribution with a standard deviation of 1 mm, what is the probability that a single measurement will lie within 2 mm
=+University Science Books, Sausalito, CA, 1997). In this context, a measured quantity x is assumed to have a normal distribution whose mean is assumed to be the “true” value of the object being
=+36. The normal distribution is commonly used to model the variability expected when making measurements(Taylor, J. R., An Introduction to Error Analysis: The Study of Uncertainties in Physical
=+b. What is the probability that 60 or more sheets will have a breakthrough?
=+a. What is the expected number of sheets that will experience a breakthrough?
=+1989: 81–89). The height from which the ball is dropped is determined so that there is a 50% chance of breaking through the glass. A breakthrough is considered to be a failure, whereas a ball
=+35. A standard procedure for testing safety glass is to drop a 1/2-lb iron ball onto a 12-in. square of glass supported on a frame (“Statistical Methods in Plastics Research and Development,”
=+c. After a certain period of time, all of the test tubes are examined, and it is found that 40%of the tubes contain at least one bacterial cell.Use your answer from part (b) to estimate , the mean
=+b. In terms of , what is the probability that a test tube contains at least one bacterial cell?
=+a. Express the probability that a particular test tube contains no bacteria, in terms of .
=+Suppose a dilute suspension of bacteria is divided into several different test tubes. The number of bacteria x in a test tube has a Poisson mass function with a parameter that represents the
=+34. The Poisson mass function is often used in biology to model the number of bacteria in a solution.
=+c. What is the probability that a given PCB has two or more defective solder joints?
=+b. What proportion of all PCBs are defect-free?
=+a. What are the mean and standard deviation of the number of defective solder joints on a PCB?
=+33. A printed circuit board (PCB) has 285 small holes, called “joints,” into which are inserted the thin leads or “pins” emanating from electronic components soldered to the PCB (see

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