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mathematics
statistics

Questions and Answers of Statistics

Suppose X has a Poisson distribution with a mean 0.4. Determine the following probabilities: (a) P(X = 0) (b) P(X < 2) (c) P(X = 4) (d) P(X = 8)
Suppose that the number of customers that enter a bank in an hour is a Poisson random variable, and suppose that P(X = 0) = 0.05. Determine the mean variance of X.
The number of telephone calls that arrive at a exchange is often modeled as a Poisson random variable. Assume that on the average there are 10 calls per hour. (a) What is the probability that there
The number of flaws in bolts of cloth in textile manufacturing is assumed to be Poisson distributed with a mean flaw per square meter. (c) What is the probability that there are no flaws in 20
When a computer disk manufacturer tests a disk, it writes to the disk and then tests it using a certifier. The certifier counts the number of missing pulses or errors. The number of errors on a test
The number of cracks in a section of interstate highway that are significant enough to require repair is assumed to follow a Poisson distribution with a mean of two cracks per mile. (a) What is the
The number of failures for a cytogenics machine from contamination in biological samples is a Poisson random variable with a mean of 0.01 per 100 samples. (a) If the lab usually processes 500
The number of surface flaws in plastic panels used in the interior of automobiles has a Poisson distribution with a mean of 0.05 flaw per square foot of plastic panel. Assume an automobile interior
The number of failures of a testing instrument from contamination particles on the product is a Poisson random variable with a mean of 0.02 failure per hour. (a) What is the probability that the
A shipment of chemicals arrives in 15 totes. Three of the totes are selected at random, without replacement, for an inspection of purity. If two of the totes do not conform to purity requirements,
The probability that your call to a service line is answered in less than 30 seconds is 0.75. Assume that your calls are independent. (a) If you call 10 times, what is the probability that exactly 9
Continuation of Exercise 3-108. (a) What is the probability that you must call four times to obtain the first answer in less than 30 seconds? (b) What is the mean number of calls until you are
Continuation of Exercise 3-109. (a) What is the probability that you must call six times in order for two of your calls to be answered in less than 30 seconds? (b) What is the mean number of calls
The number of messages sent to a computer bulletin board is a Poisson random variable with a mean of 5 messages per hour. (a) What is the probability that 5 messages are received in 1 hour? (b)
A Web site is operated by four identical computer servers. Only one is used to operate the site; the others are spares that can be activated in case the active server fails. The probability that a
The number of errors in a textbook follow a Poisson distribution with a mean of 0.01 error per page. What is the probability that there are three or less errors in 100 pages?
The probability that an individual recovers from an illness in a one-week time period without treatment is 0.1. Suppose that 20 independent individuals suffering from this illness are treated with a
Patient response to a generic drug to control pain is scored on a 5-point scale, where a 5 indicates complete relief. Historically the distribution of scores is 1 2 3 4 5 0.05 0.1 0.2 0.25
In a manufacturing process that laminates several ceramic layers, 1% of the assemblies are defective. Assume that the assemblies are independent. (a) What is the mean number of assemblies that need
Continuation of Exercise 3-116. Determine the minimum number of assemblies that need to be checked so that the probability of at least one defective assembly exceeds 0.95.
Determine the constant c so that the following function is a probability mass function: for f(x) = cx for x = 1, 2, 3, 4.
A manufacturer of a consumer electronics product expects 2% of units to fail during the warranty period. A sample of 500 independent units is tracked for warranty performance. (a) What is the
Messages that arrive at a service center for an information systems manufacturer have been classified on the basis of the number of keywords (used to help route messages) and the type of message,
The random variable X has the following probability distribution: X 2 3 5 8 Probability 0.2 0.4 0.3 0.1 Determine the following: (a) P(X < 3) (b) P(X > 2.5) (c) P(2.7 < X < 5.1) (d)
Determine the probability mass function for the random variable with the following cumulative distribution function
Each main bearing cap in an engine contains four bolts. The bolts are selected at random, without replacement, from a parts bin that contains 30 bolts from one supplier and 70 bolts from
Assume the number of errors along a magnetic recording surface is a Poisson random variable with a mean of one error every 105 bits. A sector of data consists of 4096 eight-bit bytes. (a) What is the
An installation technician for a specialized communication system is dispatched to a city only when three or more orders have been placed. Suppose orders follow a Poisson distribution with a mean of
From 500 customers, a major appliance manufacturer will randomly select a sample without replacement. The company estimates that 25% of the customers will provide useful data. If this estimate is
It is suspected that some of the totes containing chemicals purchased from a supplier exceed the moisture content target. Samples from 30 totes are to be tested for moisture content. Assume that the
Messages arrive to a computer server according to a Poisson distribution with a mean rate of 10 per hour. Determine the length of an interval of time such that the probability that no messages arrive
Flaws occur in the interior of plastic used for automobiles according to a Poisson distribution with a mean of 0.02 flaw per panel. (a) If 50 panels are inspected, what is the probability that there
Derive the mean and variance of a hypergeometric random variable (difficult exercise).
Show that the function f (x) in Example 3-5 satisfies the properties of a probability mass function by summing the infinite series.
Derive the formula for the mean and standard deviation of a discrete uniform random variable over the range of integers.
A company performs inspection on shipments from suppliers in order to defect nonconforming products. Assume a lot contains 1000 items and 1% are nonconforming. What sample size is needed so that the
A company performs inspection on shipments from suppliers in order to detect nonconforming products. The company’s policy is to use a sample size that is always 10% of the lot size. Comment on the
Surface flaws in automobile exterior panels follow a Poisson distribution with a mean of 0.1 flaw per panel. If 100 panels are checked, what is the probability that fewer than five panels have any
A large bakery can produce rolls in lots of either 0, 1000, 2000, or 3000 per day. The production cost per item is $0.10. The demand varies randomly according to the following distribution: demand
A manufacturer stocks components obtained from a supplier. Suppose that 2% of the components are defective and that the defective components occur independently. How many components must the
Each of three machined parts is classified as either above or below the target specification for the part.
Each of four transmitted bits is classified as either in error or not in error.
In the final inspection of electronic power supplies, three types of nonconformities might occur: functional, minor, or cosmetic. Power supplies that are defective are further classified as to type
In the manufacturing of digital recording tape, each of 24 tracks is classified as containing or not containing one or more bits in error.
An ammeter that displays three digits is used to measure current in milliamperes.
The following two questions appear on an employee survey questionnaire. Each answer is chosen from the five point scale 1 (never), 2, 3, 4, 5 (always).
An order for an automobile can specify either an automatic or a standard transmission, either with or without air-conditioning, and any one of the four colors red, blue, black or white. Describe the
A sampled injection-molded part could have been produced in either one of two presses and in any one of the eight cavities in each press.
An order for a computer system can specify memory of 4, 8, or 12 gigabytes, and disk storage of 200, 300, or 400 gigabytes. Describe the set of possible orders.
Three events are shown on the Venn diagram in the following figure:Reproduce the figure and shade the region that corresponds to each of the following events.(a) A’(b) A ∩ B(c) (A ∩ B) U C(d)
Three events are shown on the Venn diagram in the following figure:
A digital scale is used that provides weights to the nearest gram. (a) What is the sample space for this experiment? Let A denote the event that a weight exceeds 11 grams, let B denote the event
In an injection-molding operation, the length and width, denoted as X and Y, respectively, of each molded part are evaluated. Let A denote the event of 48 = X = 52 centimeters B denote the event
Four bits are transmitted over a digital communications channel. Each bit is either distorted or received without distortion. Let Ai denote the event that the ith bit is distorted, i = 1,.., 4. (a)
A sample of three calculators is selected from a manufacturing line, and each calculator is classified as either defective or acceptable. Let A, B, and C denote the events that the first, second, and
Samples of a cast aluminum part are classified on the basis of surface finish (in micro inches) and edge finish. The results of 100 parts are summarized as follows: Edge finish Excellent
The rise time of a reactor is measured in minutes (and fractions of minutes). Let the sample space be positive, real numbers. Define the events A and B as follows: A = {x | x < 72.5} And B = {x |
A sample of two items is selected without replacement from a batch. Describe the (ordered) sample space for each of the following batches: (a) The batch contains the items {a, b, c, d}. (b) The
A sample of two printed circuit boards is selected without replacement from a batch. Describe the (ordered) sample space for each of the following batches: (a) The batch contains 90 boards that are
Counts of the Web pages provided by each of two computer servers in a selected hour of the day are recorded. Let A denote the event that at least 10 pages are provided by server 1 and let B denote
Suppose that f(x) = e-x for 0 < x. Determine the following probabilities:(a) P(1 < X) (b) P(1 < X < 2.5)(c) P(X = 3)(d) P(X < 4)(e) P(3 < X
Suppose that for f(x) = e-x for 0 < x (a) Determine x such that P(x < X) = 0.10.(b) Determine x such that P(X < x) = 0.10.
Suppose that f(x) = e/8 for 3 < x < 5 for Determine the following probabilities:(a) P(X < 4)(b) P(X > 3.5)(c) P(4 < X < 5) (d) P(X < 4.5)(e) P(X < 3.5 or X > 4.5)
Suppose that Determine the f(x) = e-(x-4) for 4 < x. following probabilities:(a) P(1 < X)(b) P(2 > X < 5)(c) P(5 < X)(d) P(8 < X < 12)(e) Determine x such that P(X _ x) = 0.90.
Suppose that f(x) = 1.5x2 for – 1 < x < 1. Determine the following probabilities:(a) P(0 < X)(b) P(0.5 < X)(c) P( - 0.5 < X < 0.5)(d) P(X < - 2)(e) P(X < 0 or X) – 0.5)(f) Determine x such that
The probability density function of the time to failure of an electronic component in a copier (in hours) is f(x) =e-x/100/1000 for x > 0 Determine the probability that(a) A component lasts more than
The probability density function of the net weight in pounds of a packaged chemical herbicide is f(x) = 2.0 for 49.75 < x < 50.25 pounds.(a) Determine the probability that a package weighs more than
The probability density function of the length of a hinge for fastening a door is f(x) = 1.25 for 74.6 < x < 75.4 millimeters. Determine the following:(a) P(X < 74.8)(b) P(X < 74.8 or X > 75.2)(c) If
The probability density function of the length of a metal rod is f(x) =2 for 2.3 < x < 2.8 meters.(a) If the specifications for this process are from 2.25 to 2.75 meters, what proportion of the bars
If X is a continuous random variable, argue that P(x1 < X < x2) – P(x1 M X < x2) – P(x1 < X < x2) = P(x1 < X < x2).
Suppose the cumulative distribution function of the random variable X isDetermine the following:(a) P(X (b) P(X > 1.5)(c) P(X M €“ 2)(d) P(X > 6)
Suppose the cumulative distribution function of the random variable X isDetermine the following:(a) P(X (b) P(X > - 1.5)(c) P(X (d) P(- 1
Determine that cumulative distribution function for the distribution. Exercise 4-1,
Determine the cumulative distribution function for the distribution in Exercise 4-3.
Determine the cumulative distribution function for the distribution in Exercise 4-4.
Determine the cumulative distribution function for the distribution in Exercise 4-6. Use the cumulative distribution function to determine the probability that a component lasts more than 3000 hours
Determine the cumulative distribution function for the distribution in Exercise 4-8. Use the cumulative distribution function to determine the probability that a length exceeds 75 millimeters.
F(x) = 1 – e-2x x > 0
The gap width is an important property of a magnetic recording head. In coded units, if the width is a continuous random variable over the range from 0 < x < 2 with f(x) = 0.5x, determine the
Suppose f(x) = 0.25 for 0 < x < 4. Determine the mean and variance of X.
Suppose f(x) = 0.125x for 0 < x < 4. Determine the mean and variance of X.
Suppose f(x) = 1.5x2 for -1 < x < 1. Determine the mean and variance of X.
Suppose that f(x) – x/8 for 3 < x < 5. Determine the mean and variance for x.
Determine the mean and variance of the weight of packages in Exercise 4.7.
The thickness of a conductive coating in micrometers has a density function of 600x-2 for 100 μm < x
Suppose that contamination particle size (in micrometers) can be modeled as for Determine the mean of X.
Integration by parts is required. The probability density function for the diameter of a drilled hole in millimeters is for mm. Although the target diameter is 5 millimeters, vibrations, tool wear,
Suppose the probability density function of the length of computer cables is f (x) = 0.1 from 1200 to 1210 millimeters.(a) Determine the mean and standard deviation of the cable length.(b) If the
Suppose X has a continuous uniform distribution over the interval [1.5, 5.5].(a) Determine the mean, variance, and standard deviation of X.(b) What is P(X < 2.5)?
Suppose X has a continuous uniform distribution over the interval [-1, 1]. (a) Determine the mean, variance, and standard deviation of X.(b) Determine the value for x such that P(-x < X < x) = 0.90.
The net weight in pounds of a packaged chemical herbicide is uniform for pounds.(a) Determine the mean and variance of the weight of packages.(b) Determine the cumulative distribution function of the
The thickness of a flange on an aircraft component is uniformly distributed between 0.95 and 1.05 millimeters.(a) Determine the cumulative distribution function of flange thickness.(b) Determine the
Suppose the time it takes a data collection operator to fill out an electronic form for a database is uniformly between 1.5 and 2.2 minutes.(a) What is the mean and variance of the time it takes an
The probability density function of the time it takes a hematology cell counter to complete a test on a blood sample is f(x) = 0.2 for 50 < x < 75 seconds.(a) What percentage of tests require more
The thickness of photo resist applied to wafers in semiconductor manufacturing at a particular location on the wafer is uniformly distributed between 0.2050 and 0.2150 micrometers.(a) Determine the
The probability density function of the time required to complete an assembly operation is for seconds.(a) Determine the proportion of assemblies that requires more than 35 seconds to complete.(b)
Use Appendix Table II to determine the following probabilities for the standard normal random variable Z:(a) P(Z < 1.32) (b) P(Z < 3.0)(c) P(Z > 1.45) (d) P(Z > - 2.15)(e) P(- 2.34 < Z < 1.76)
Use Appendix Table II to determine the following probabilities for the standard normal random variable Z:(a) P(-1 < Z < 1) (b) P(- 2 < Z < 2) (c) P(- 3 < Z < 3) (d) P(Z > 3)(e) P(0 < Z < 1)
Assume Z has a standard normal distribution. Use Appendix Table II to determine the value for z that solves each of the following:(a) P(Z < z) = 0.9(b) P(Z < z) = 0.5(c) P( Z > z) = 0.1 (d) P(Z > z)
Assume Z has a standard normal distribution. Use Appendix Table II to determine the value for z that solves each of the following:(a) P(-z < Z < z) = 0.95 (b) P(-z < Z < z) = 0.99(c) P(-z < Z < z) =

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