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Questions and Answers of Statistics

Assume X is normally distributed with a mean of 10 and a standard deviation of 2. Determine the following:(a) P(X < 13) (b) P(X > 9)(c) P(6 < X < 14) (d) P(2 < X < 4)(e) P(-2 < X < 8)
Assume X is normally distributed with a mean of 10 and a standard deviation of 2. Determine the value for x that solves each of the following:(a) P(X > x) = 0.5(b) P(X > x) = 0.95(c) P(x < X < 10) =
Assume X is normally distributed with a mean of 5 and a standard deviation of 4. Determine the following: (a) P(X < 11) (b) P(X > 0)(c) P(3 < X < 7) (d) P(-2 < X < 9)(e) P(-2 < X < 8)
Assume X is normally distributed with a mean of 5 and a standard deviation of 4. Determine the value for x that solves each of the following:(a) P(X > x) = 0.5 (b) P(X > x) = 0.95(c) P(x < X < 9) =
The compressive strength of samples of cement can be modeled by a normal distribution with a mean of 6000 kilograms per square centimeter and a standard deviation of 100 kilograms per square
The tensile strength of paper is modeled by a normal distribution with a mean of 35 pounds per square inch and a standard deviation of 2 pounds per square inch.(a) What is the probability that the
The line width of for semiconductor manufacturing is assumed to be normally distributed with a mean of 0.5 micrometer and a standard deviation of 0.05 micrometer.(a) What is the probability that a
The fill volume of an automated filling machine used for filling cans of carbonated beverage is normally distributed with a mean of 12.4 fluid ounces and a standard deviation of 0.1 fluid ounce.(a)
The time it takes a cell to divide (called mitosis) is normally distributed with an average time of one hour and a standard deviation of 5 minutes.(a) What is the probability that a cell divides in
In the previous exercise, suppose that the mean of the filling operation can be adjusted easily, but the standard deviation remains at 0.1 ounce.(a) At what value should the mean be set so that 99.9%
The reaction time of a driver to visual stimulus is normally distributed with a mean of 0.4 seconds and a standard deviation of 0.05 seconds.(a) What is the probability that a reaction requires more
The speed of a file transfer from a server on campus to a personal computer at a student’s home on a weekday evening is normally distributed with a mean of 60 kilobits per second and a standard
The length of an injection-molded plastic case that holds magnetic tape is normally distributed with a length of 90.2 millimeters and a standard deviation of 0.1 millimeter.(a) What is the
In the previous exercise assume that the process is centered so that the mean is 90 millimeters and the standard deviation is 0.1 millimeter. Suppose that 10 cases are measured, and they are assumed
The sick-leave time of employees in a firm in a month is normally distributed with a mean of 100 hours and a standard deviation of 20 hours.(a) What is the probability that the sick-leave time for
The life of a semiconductor laser at a constant power is normally distributed with a mean of 7000 hours and a standard deviation of 600 hours.(a) What is the probability that a laser fails before
The diameter of the dot produced by a printer is normally distributed with a mean diameter of 0.002 inch and a standard deviation of 0.0004 inch.(a) What is the probability that the diameter of a dot
The weight of a sophisticated running shoe is normally distributed with a mean of 12 ounces and a standard deviation of 0.5 ounce.(a) What is the probability that a shoe weighs more than 13
Suppose that X is a binomial random variable with n = 200 and p = 0.4.(a) Approximate the probability that X is less than or equal to 70.(b) Approximate the probability that X is greater than 70 and
Suppose that X is a binomial random variable with n = 100 and p = 0.1.(a) Compute the exact probability that X is less than 4.(b) Approximate the probability that X is less than 4 and compare to the
The manufacturing of semiconductor chips produces 2% defective chips. Assume the chips are independent and that a lot contains 1000 chips.(a) Approximate the probability that more than 25 chips are
A supplier ships a lot of 1000 electrical connectors. A sample of 25 is selected at random, without replacement. Assume the lot contains 100 defective connectors.(a) Using a binomial approximation,
An electronic office product contains 5000 electronic components. Assume that the probability that each component operates without failure during the useful life of the product is 0.999, and assume
Suppose that the number of asbestos particles in a sample of 1 squared centimeter of dust is a Poisson random variable with a mean of 1000. What is the probability that 10 squared centimeters of dust
A corporate Web site contains errors on 50 of 1000 pages. If 100 pages are sampled randomly, without replacement, approximate the probability that at least 1 of the pages in error are in the sample
Hits to a high-volume Web site are assumed to follow Poisson distribution with a mean of 10,000 per day. Approximate each of the following:(a) The probability of more than 20,000 hits in a day(b) The
Continuation of Exercise 4-68.(a) Approximate the expected number of days in a year (365 days) that exceed 10,200 hits.(b) Approximate the probability that over a year (365 days) more than 15 days
The percentage of people exposed to a bacteria who become ill is 20%. Assume that people are independent. Assume that 1000 people are exposed to the bacteria. Approximate each of the following:(a)
A high-volume printer produces minor print-quality errors on a test pattern of 1000 pages of text according to a Poisson distribution with a mean of 0.4 per page.(a) Why is the number of errors on
Suppose X has an exponential distribution with λ = 2. Determine the following: (a) P(x < 0) (b) P(X > 2) (c) P(X < 1) (d) P(1) < X < 2) (e) Find the value of x such that P(X < x) = 0.05.
Suppose X has an exponential distribution with mean equal to 10.Determine following:(a) P(X > 10)(b) P(X > 20)(c) P(X > 30)
Suppose the counts recorded by a Geiger counter follow a Poisson process with an average of two counts per minute. (a) What is the probability that there are no counts in a 30- second interval?(b)
Suppose that the logons to a computer network follow a Poisson process with an average of 3 counts per minute.(a) What is the mean time between counts?(b) What is the standard deviation of the time
The time to failure (in hours) for a laser in a cytometry machine is modeled by an exponential distribution with λ = 0.00004. (a) What is the probability that the laser will last at least
The time between calls to a plumbing supply business is exponentially distributed with a mean time between calls of 15 minutes.(a) What is the probability that there are no calls within a 30- minute
The life of automobile voltage regulators has an exponential distribution with a mean life of six years. You purchase an automobile that is six years old, with a working voltage regulator, and plan
The time to failure (in hours) of fans in a personal computer can be modeled by an exponential distribution with(a) What proportion of the fans will last at least 10,000 hours?(b) What proportion of
The time between the arrival of electronic messages at your computer is exponentially distributed with a mean of two hours.(a) What is the probability that you do not receive a message during a
The time between arrivals of taxis at a busy intersection is exponentially distributed with a mean of 10 minutes.(a) What is the probability that you wait longer than one hour for a taxi?(b) Suppose
Continuation of Exercise 4-81. (a) Determine x such that the probability that you wait more than x minutes is 0.10.(b) Determine x such that the probability that you wait less than x minutes is
The distance between major cracks in a highway follows an exponential distribution with a mean of 5 miles.(a) What is the probability that there are no major cracks in a 10-mile stretch of the
Continuation of Exercise 4-83.(a) What is the probability that the first major crack occurs between 12 and 15 miles of the start of inspection?(b) What is the probability that there are no major
The lifetime of a mechanical assembly in a vibration test is exponentially distributed with a mean of 400 hours.(a) What is the probability that an assembly on test fails in less than 100 hours?(b)
Continuation of Exercise 4-85.(a) If 10 assemblies are tested, what is the probability that at least one fails in less than 100 hours? Assume that the assemblies fail independently.(b) If 10
When a bus service reduces fares, a particular trip from New York City to Albany, New York, is very popular. A small bus can carry four passengers. The time between calls for tickets is exponentially
The time between arrivals of small aircraft at a county airport is exponentially distributed with a mean of one hour. What is the probability that more than three aircraft arrive within an hour?
Continuation of Exercise 4-88.(a) If 30 separate one-hour intervals are chosen, what is the probability that no interval contains more than three arrivals? (b) Determine the length of an interval of
The time between calls to a corporate office is exponentially distributed with a mean of 10 minutes.(a) What is the probability that there are more than three calls in one-half hour?(b) What is the
Continuation of Exercise 4-90.(a) What is the probability that there are no calls within a two-hour interval?(b) If four non-overlapping one-half hour intervals are selected, what is the probability
If the random variable X has and exponential distribution with mean θ determine the following: (a) P(X > θ) (b) P(X > 2θ) (c) P(X > 3θ) (d) How do the results depend on θ?
Assume that the flaws along a magnetic tape follow a Poisson distribution with a mean of 0.2 flaws per meter. Let X denotes the distance between two successive flaws.(a) What is the mean of X?(b)
Continuation of Exercise 4-93. (More difficult questions.)(a) What is the probability that the first time the distance between two flaws exceeds 8 meters is at the fifth flaw?(b) What is the mean
Derive the formula for the mean and variance of an exponential random variable.
Calls to a telephone system follow a Poisson distribution with a mean of five calls per minute.(a) What is the name applied to the distribution and parameter values of the time until the tenth
Continuation of Exercise 4-96.(a) What is the probability that exactly four calls occur within one minute?(b) If 10 separate one-minute intervals are chosen, what is the probability that all
Raw materials are studied for contamination. Suppose that the number of particles of contamination per pound of material is a Poisson random variable with a mean of 0.01 particle per pound.(a) What
The time between failures of a laser in a cytogenesis machine is exponentially distributed with a mean of 25,000 hours.(a) What is the expected time until the second failure?(b) What is the
In a data communication system, several messages that arrive at a node are bundled into a packet before they are transmitted over the network. Assume the messages arrive at the node according to a
Errors caused by contamination on optical disks occur at the rate of one error every 105 bits. Assume the errors follow a Poisson distribution.(a) What is the mean number of bits until five errors
Calls to the help line of a large computer distributor follow a Passion distribution with a mean of 20 calls per minute. (a) What is the mean time until the one-hundredth calls?(b) What is the mean
The time between arrivals of customers at an automatic teller machine is an exponential random variable with a mean of 5 minutes.(a) What is the probability that more than three customers arrive in
The time between process problems in a manufacturing line is exponentially distributed with a mean of 30 days. (a) What is the expected time until the fourth problem?(b) What is the probability that
Use the properties of the gamma function to evaluate the following: (a) Г(6) (b) Г(5/2) (c) Г(9/2)
Use integration by parts to show that Г(r) = (r – 1) Г(r – 1).
Show that the gamma density function integrates to f(x, λ, r) integrates to 1.
Use the result for the gamma distribution to determine the mean and variance of a chi-square distribution with r = 7/2.
Suppose that X has a Weibull distribution with β = 0.2 and δ = 100 hours. Determine the mean and variance of X.
Suppose that X has a Weibull distribution β = 0.2 and δ = 100 hours. Determine that following: (a) P(X < 10,000) (b) P(X > 5000)
Assume that the life of a roller bearing follows a Weibull distribution with parameters β = 2 and δ = 10,000 hours. (a) Determine the probability that a bearing lasts at least 8000
The life (in hours) of a computer processing unit (CPU) is modeled by a Weibull distribution with parameters β = 3 and δ = 900 hours. (a) Determine the mean life of the CPU. (b) Determine
Assume the life of a packaged magnetic disk exposed to corrosive gases has a Weibull distribution with β = 0.5 and the mean life is 600 hours. (a) Determine the probability that a packaged disk
The life of a recirculating pump follows a Weibull distribution with parameters β = 2, and δ = 700 hours. (a) Determine the mean life of a pump. (b) Determine the variance of the life of
The life (in hours) of a magnetic resonance imagining machine (MRI) is modeled by a Weibull distribution with parameters β = 2 and δ = 500 hours. (a) Determine the mean life of the MRI.
If X is a Weibull random variable with β = 1, and δ = 1000, what is another name for the distribution of X and what is the mean of X?
Suppose that X has a lognormal distribution with parameters θ = 5 and w2 = 9. Determine the following: (a) P(X < 13, 300) (b) The value for x such that (c) The mean and variance of X
Suppose that X has a lognormal distribution with parameters θ = - 2 and w2. Determine the following: (a) P(500 < X < 1000) (b) The value for x such that (c) The mean and variance of X
Suppose that X has a lognormal distribution with parameters θ = 2 and w2. Determine the following: (a) P(X < 500) (b) The conditional probability that given that (c) What does the
The length of time (in seconds) that a user views a page on a Web site before moving to another page is a lognormal random variable with parameters θ = 2.5 and w2 = 1. (a) What is the
Suppose that X has a lognormal distribution and that the mean and variance of X are 100 and 85,000, respectively. Determine the parameters and of the lognormal distribution. (Hint: define x = exp
The lifetime of a semiconductor laser has a lognormal distribution, and it is known that the mean and standard deviation of lifetime are 10,000 and 20,000, respectively.(a) Calculate the parameters
Derive the probability density function of a lognormal random variable from the derivative of the cumulative distribution function.
Suppose that f(x) = 0.5 – 1 for 2 < x < 4. Determine the following:(a) P(X < 2.5)(b) P(X > 3)(c) P(2.5 < X < 3.5)
Continuation of Exercise 4-124. Determine the cumulative distribution function of the random variable.
Continuation of Exercise 4-124. Determine the mean and variance of the random variable.
The time between calls is exponentially distributed with a mean time between calls of 10 minutes.(a) What is the probability that the time until the first call is less than 5 minutes?(b) What is the
Continuation of Exercise 4-127.(a) If there has not been a call in 10 minutes, what is the probability that the time until the next call is less than 5 minutes?(b) What is the probability that there
Continuation of Exercise 4-127.(a) What is the probability that the time until the third call is greater than 30 minutes?(b) What is the mean time until the fifth call?
The CPU of a personal computer has a lifetime that is exponentially distributed with a mean lifetime of six years. You have owned this CPU for three years. What is the probability that the CPU fails
Continuation of Exercise 4-130. Assume that your corporation has owned 10 CPUs for three years, and assume that the CPUs fail independently. What is the probability that at least one fails within the
Suppose that X has a lognormal distribution with parameters θ = 0 and w2 = 4. Determine the following: (a) P(10 < X < 50) (b) The value for x such that (c) The mean and variance of X
Suppose that X has a lognormal distribution and that the mean and variance of X are 50 and 4000, respectively. Determine the following:(a) The parameters and of the lognormal distribution(b) The
Asbestos fibers in a dust sample are identified by an electron microscope after sample preparation. Suppose that the number of fibers is a Poisson random variable and the mean number of fibers per
Without an automated irrigation system, the height of plants two weeks after germination is normally distributed with a mean of 2.5 centimeters and a standard deviation of 0.5 centimeters.(a) What is
Continuation of Exercise 4-135. With an automated irrigation system, a plant grows to a height of 3.5 centimeters two weeks after germination.(a) What is the probability of obtaining a plant of this
The thickness of a laminated covering for a wood surface is normally distributed with a mean of 5 millimeters and a standard deviation of 0.2 millimeter.(a) What is the probability that a covering
The diameter of the dot produced by a printer is normally distributed with a mean diameter of 0.002 inch. Suppose that the specifications require the dot diameter to be between 0.0014 and 0.0026
Continuation of Exercise 4-138. Assume that the standard deviation of the size of a dot is 0.0004 inch. If the probability that a dot meets specifications is to be 0.9973, what specifications are
The life of a semiconductor laser at a constant power is normally distributed with a mean of 7000 hours and a standard deviation of 600 hours.(a) What is the probability that a laser fails before
Continuation of Exercise 4-140. What should the mean life equal in order for 99% of the lasers to exceed 10,000 hours before failure?
Continuation of Exercise 4-140. A product contains three lasers, and the product fails if any of the lasers fails. Assume the lasers fail independently. What should the mean life equal in order for

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