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

A new battery supposedly with a charge of 1.5 volts actually has a voltage with a uniform distribution between 1.43 and 1.60 volts. (a) What is the expectation of the voltage? (b) What is the standard deviation of the voltage? (c) What is the cumulative distribution function of the voltage? (d)
A computer random-number generator produces numbers that have a uniform distribution between 0 and 1. (a) If 20 random numbers are generated, what are the expectation and variance of the number of them that lie in each of the lour intervals [0.00, 0.30). [0.30, 0.50), 10.50.0.75). and [0.75.
The lengths in meters of pieces of scrap wood found on a building site are uniformly distributed between 0.0 and 2.5. (a) What are the expectation and variance of the lengths? (b) What is the probability that at least 20 out of 25 pieces of scrap wood are longer than I meter?
Suppose that a metal pin has a diameter that has a uniform distribution between 4.182 mm and 4.185 mm. (a)What is the probability that a pin will lit into a hole that has a diameter of 4.184 mm'.' (b) If a pin does lit into the hole, what is the probability that the difference between the diameter
When employees undergo an evaluation, their scores are independent and uniformly distributed between 60 and 100.(a) If six employees take the evaluation, what is the probability that half of them score more than 85 and half less'.'(b) If six employees take the evaluation, what is the probability
The lengths of telephone calls can be modeled by an exponential distribution with parameter λ = 0.3 per minute, with the call lengths being independent. What is the probability that out of 10 telephone calls, 2 will be shorter than 1 minute, 4 will last between 1 minute and 3 minutes, and the
Costomers arrive at a service window according to a poisson process with parameter λ = 0.2 per minute. (a) What is the probability that the time between two successive arrivals is less than 6 minutes? (b) What is the probability that there will be exactly three arrivals during a given 10-minute
Suppose that components have failure limes that are independent and that can be modeled with an exponential distribution with λ = 0.0065 per day. If a box contains 10 components, what is the probability that the box has at least 8 componenls that last longer than 150 days'.'
As a metal detector is passed over the ground, signals are received according to a Poisson process with λ. = 0.022 per meter. What is the probability that there is no more than one signal in a 100-meter stretch?
Ninety identical electrical circuits are monitored at an extreme temperature to see how long they last before failing. The 50th failure occurs after 263 minutes. If the failure times are modeled with an exponential distribution, when would you predict that the"80th failure will occur?
Suppose that you are waiting for a friend to call you and that the time you wait in minutes has an exponential distribution with parameter λ = 0.1.(a) What is the expectation of your waiting time?(b) What is the probability that you will wait longer than 10 minutes '(c) What is the probability
The time in days between breakdowns of a machine is exponentially distributed with λ = 0.2.(a) What is the expected time between machine breakdowns?(b) What is the standard deviation of the time between machine breakdowns?(c) What is the median time between machine breakdowns.'(d) What is the
A researcher plants 12 seeds whose germination times in days are independent exponential distributions with λ = 0.31.(a) What is the probability that a given seed germinates within five days?(b) What are the expectation and variance of the number of seeds germinating within live day s?(c) What is
A double exponential distribution, often called a Laplace distribution, has a probability density functionfor - oo (a) P(X (b) P(X > 1)
Imperfections in an optical fiber are distributed according to a Poisson process such that the distance between imperfections in meters has an exponential distribution with parameter λ. = 2m-1. (a) What is the expected distance between imperfections.' (b) What is the probability that the distance
The arrival times of workers at a factory first-aid room satisfy a Poisson process with an average of 1.8 per hour. (a) What is the value of the parameter A of the Poisson process? (b) What is the expectation of the time between two arrivals at the first-aid room? (c) What is the probability that
Engineers observe that about 90% of graphite samples fracture within five hours when subjected to a certain stress. (a) If the time to fracture is modeled with an exponential distribution, what would be a suitable value for the parameter A? (b) Use the model to estimate the probability that a
Consider a Poisson process with parameter A = 0.8. (a) What is the probability that the time between two adjacent events is longer than 1.5? (b) What is the probability that there will be at least three events in a period of length 2?
Use a computer package to (ind both the probability density function and cumulative distribution function at x = 3 and the median of gamma distributions with the following parameter values: (a) k = 3.2. λ = 0.8 (b) k = 7.5, λ = 5.3 (c) k = 4.0, λ = 1.4 In part (c), check the value of the
A day's sales in $1000 units at a gas station have a gamma distribution with parameters A = 5 and k = 0.9. (a) What is the expectation of a day's sales? (b) What is the standard deviation of a day's sales.' (c) What are the upper and lower quartiles of a day's sales'.' (d) What is the probability
Recall Problem 4.2.6 concerning imperfections in an optical fiber. Suppose that five adjacent imperfections are located on a fiber. (a) What is the distribution of the distance between the first imperfection and the fifth imperfection? (b) What is the expectation of the distance between the first
Recall Problem 4.2.7 concerning the arrivals at a factory first-aid room. (a) What is the distribution of the time between the first arrival of the day and the fourth arrival? (b) What is the expectation of this time? (c) What is the variance of this time? (d) By using (i) the gamma distribution
Suppose that the time in minutes taken by a worker on an assembly line to complete a particular task has a gamma distribution with parameters k = 44 and λ = 0.7. (a) What are the expectation and standard deviation of the time taken to complete the task'.' (b) Use a software package to find the
Suppose that the random variable X has a Weibull distribution with parameters a = 4.9 and λ. = 0.22. Find: (a) The median of the distribution (b) The upper and lower quartiles of the distribution (c) P(2 < X < 7)
Suppose that the random variable X has a Weibull distribution with parameters a = 2.3 and λ = 1.7. Find: a. The median of the distribution b. The upper anil lower quartiles ol" the distribution c. P(0.5 < X < 1.5)
The time to failure in hours of an electrical circuit subjected to a high temperature has a Weibull distribution with parameters a = 3 and λ = 0.5. (a) What is the median failure time of a circuit? (b) The circuit engineers can be 99% confident that a circuit will last as long as what time? (c)
A biologist models the time in minutes between the formation of a cell and the moment at which it splits into two new cells using a Weibull distribution with parameters a = 0.4 and λ = 0.5. (a) What is the median value of this distribution? (b) What are the upper and lower quartiles of this
The lifetime in minutes of a mechanical component has a Weibull distribution with parameters a = 1.5 and λ = 0.03.(a) What are the median, upper quartile. and 99th percentile of the lifetime of a component?(b) If 500 independent components are considered, what are the expectation and variance of
Suppose that the time in days taken for bacteria cultures to develop after they have been prepared can be modeled by a Weibull distribution with parameters A. = 0.3 and a = 0.6. A biologist prepares several sets of cultures at the same time, and after four days opens them one by one until five
A physician conducts a study to investigate the time taken to recover from an ailment under a certain treatment. A group of 82 patients with the ailment are given the treatment, and when they are checked 7 days later, it is found that 9 of them have recovered. The remaining 73 patients are checked
Consider the probability density function f(x) = Ax3 (I - x)2 for 0 < x < I and f(x) = 0 elsewhere. (a) Find the value of A by direct integration. (b) Find by direct integration the expectation and variance of this distribution. (c) What are the parameter values a and b of a beta distribution for
Consider the beta probability density function f(x) = Ax9 (1-x)3 for 0 < x < 1 and f(x) = 0 elsewhere. (a) What are the values of the parameters a and b! (b) Use the answer to part (a) to calculate the value of A. (c) What is the expectation of this distribution? (d) What is the standard deviation
Use a computer package to find the probability density function and cumulative distribution function at x = 0.5, and the upper quartile, of beta distributions with the following parameter values: (a) a = 3.3, b = 4.5 (b) a = 0.6,b = 1.5 (c) a = 2,b = b In part (c), check the value of the
Suppose thai the random variable A" has a beta distribution with parameters a = b = 2.1, and consider the random variable Y = 3 + 4X (a) What is the state space of the random variable Y? (b) What are the expectation and variance of the random variable Y? (c) What is P(Y < 5)?
The purity of a chemical batch, expressed as a percentage, is equal to 100x-, where the random variable X has a beta distribution with parameters a = 7.2 and b = 2.3. (a) What are the expectation and variance of the purity levels? (b) What is the probability that a chemical batch has a purity of at
The proportion of tin in a metal alloy has a beta distribution with parameters a = 8.2 and b = 1 1.7. (a) What is the expected proportion of tin in the alloy? (b) What is the standard deviation of the proportion of tin in the alloy? (c) What is the median proportion of tin in the alloy)?
A dial is spun and an angle 9 is measured, which can be taken to be uniformly distributed between 0 and 360. If 0 < θ < 90, a player wins nothing; if 90 < θ < 270, then a player wins $(20 - 180); and if 270 < G < 360, then a player wins $(02 - 72,540). Draw the cumulative distribution function of
The strength of a chemical solution is measured on a scale between 0 and 1, with values smaller than 0.5 being too weak, values between 0.5 and 0.8 being satisfactory, and values larger than 0.8 being too strong. If chemical batches have strengths that are independently distributed according to a
Suppose that visits to a website can be modeled by a Poisson process with parameter λ. = 4 per hour. (a) What is the probability that there are exactly 10 visits within a given 2-hour interval? (b) A supervisor starts to monitor the website from the start of a new shift. What is the distribution
A hole is drilled into the Antarctic ice shelf and a core is extracted that provides information on the climate when the ice was formed at different times in the past. Suppose that a researcher is interested in high-temperature years, and that the places in the core corresponding to
Are the following statements true or false? (a) If a Beta distribution has the parameter a larger than the parameter b. then its expectation is smaller than 1/2. (b) The uniform distribution is a symmetric distribution. (c) In a Poisson process the distances between events are identically
Suppose that after operation, the electrical charge remaining on a circuit component has a uniform distribution between 50 and 100. and that these charges are independent of each other for different operations. II the machine is operated five times, what is the probability that the residual charge
Consider a Poisson process with parameter λ = 8.(a) Consider an interval of length 0.5. What is the probability of obtaining exactly four events within this interval?(b) What is the probability that the interval between two adjacent events is shorter than 0.2?
Suppose that customer waiting times are independent and can be modeled by a Weibull distribution with a = 2.3 and A = 0.09 per minute. What is the probability that out of 10 customers, exactly 3 wait less than 8 minutes, exactly 4 wait between 8 and 12 minutes, and exactly 3 wait more than 12
A commercial bleach eventually becomes ineffective because the chlorine in it becomes attached to other molecules. The company that manufactures the bleach estimates that the median time for this to happen is about one and a half years. (a) If an exponential distribution is used to model the time
A ship navigating through the southern regions of the North Atlantic ice flows encounters icebergs according to a Poisson process. The distances between icebergs in nautical miles are exponentially distributed with a parameter λ = 0.7. (a) What is the expected distance between iceberg
Calls arriving at a switchboard follow a Poisson process with parameter λ = 5.2 per minute. (a) What is the expected waiting time between the arrivals of two calls? (b) What is the probability that the waiting time between the arrivals of 2 calls is less than 10 seconds? (c) What is the
Figure 4.25 shows the probability density function of a triangle distribution T(a,b) with endpoints a and (a) What is the height of the probability density function at (a + b)/2? (b) If the random variable X has a T(a, b) distribution, what is P(X < a/4 + 3b/4)? (c) What is the variance of a T(a,
The fermentation time in weeks required by a brewery for a particular kind of beer has a Weibull distribution with parameters a = 4 and Î» = 0.2.(a) What are the median, upper quartile. and 95th percentile of the fermentation times?(b) What are the expectation and variance of the
The proportion of a day that a tiger spends hunting for food has a beta distribution with parameters a = 2.7 and b = 2.9. (a) What is the expected amount of time per day that the tiger spends hunting for food? (b) What is the standard deviation of the amount of time per day that the tiger spends
The starting time of a class is uniformly distributed between 10:00 and 10:05. If a student arrives early and has to wait t minutes for the class to start, then the student incurs a penalty of A1t, which accounts for the waste in the student's time. On the other hand, if the student arrives t
An herbalist finds that about 25% of plants sprout within 35 days, and that about 75% of plants sprout within 65 days. (a) If the time of sprouting is modeled with a Weibull distribution, what parameter values would be appropriate? (b) Use this model to estimate the time by which 90% of plants
Suppose that Z ~ N(0, 1). Find: (a) P(Z ≤ 1.34) (b) P(Z ≥ -0.22) (c) P(-2.19 ≤ Z ≤ 0.43) (d) P(0.09 ≤ Z ≤ 1.76) (e) P(|Z| ≤ 0.38) (f) The value of x for which P(Z ≤ x) = 0.55 (g) The value of x for which P(Z ≥ x) = 0.72 (h) The value of x for which P(|Z| ≤ x) = 0.31
The amount of sugar contained in 1-kg packets is actually normally distributed with a mean of μ = 1.03 kg and a standard deviation of σ = 0.014 kg. (a) What proportion of sugar packets are underweight? (b) If an alternative package-filling machine is used for which the weights of the packets are
The thicknesses of metal plates made by a particular machine are normally distributed with a mean of 4.3 mm and a standard deviation of 0.12 mm. (a) What are the upper and lower quartiles of the metal plate thicknesses? (b) What is the value of c for which there is 80% probability that a metal
The density of a chemical solution is normally distributed with mean 0.0046 and variance 9.6 × 10-8. (a) What is the probability that the density is less than 0.005? (b) What is the probability that the density is between 0.004 and 0.005? (c) What is the 10th percentile of the density level? (d)
The resistance of one meter of copper cable at a certain temperature is normally distributed with mean μ = 23.8 and variance σ2 = 1.28. (a) What is the probability that a one-meter segment of copper cable has a resistance less than 23.0? (b) What is the probability that a one-meter segment of
The weights of bags filled by a machine are normally distributed with a standard deviation of 0.05 kg and a mean that can be set by the operator. At what level should the mean be set if it is required that only 1% of the bags weigh less than 10 kg?
Suppose a certain mechanical component produced by a company has a width that is normally distributed with a mean μ = 2600 and a standard deviation σ = 0.6.(a) What proportion of the components have a width outside the range 2599 to 2601?(b) If the company needs to be able to guarantee to its
Bricks have weights that are independently distributed with a normal distribution that has a mean 1320 and a standard deviation of 15. A set of 10 bricks is chosen at random. What is the probability that exactly 3 bricks will weigh less than 1300, exactly 4 bricks will weigh between 1300 and 1330,
Manufactured items have a strength that has a normal distribution with a standard deviation of 4.2. The mean strength can be altered by the operator. At what value should the mean strength be set so that exactly 95% of the items have a strength less than 100?
Suppose that Z ~ N(0, 1). Find: (a) P(Z ≤ -0.77) (b) P(Z ≥ 0.32) (c) P(-3.09 ≤ Z ≤ -1.59) (d) P(-0.82 ≤ Z ≤ 1.80) (e) P(|Z| ≥ 0.91) (f) The value of .v for which P(Z ≤ x) = 0.23 (g) The value of x for which P(Z ≥x)= 0.51 (h) The value of x for which P(|Z| ≥ x) = 0.42
Suppose that X ~ N(10, 2). Find: (a) P(X ≤ 10.34) (b) P(X ≥ 11.98) (c) P(7.67 ≤ X ≤ 9.90) (d) P(10.88 ≤ X ≤ 13.22) (e) P(|X - 10| ≤ 3) (f) The value of x for which P(X ≤ x) = 0.81 (g) The value of x for which P(X ≥ x) = 0.04 (h) The value of x for which P(|X - 10| ≥ x) = 0.63
Suppose that X ~ N(-7, 14). Find. (a) P(X ≤ 0) (b) P(X ≥ -10) (c) P(-15 ≤ X ≤ -1) (d) P(-5 ≤ X ≤ 2) (e) P(|X + 7| ≥ 8) (f) The value of X for which P(X ≤ x) = 0.75 (g) The value of x for which P(X ≥ x) = 0.27 (h) The value of x for which P(|X + 7| ≤ x) = 0.44
Suppose that X ~ N(μ, σ2) and that P(X ≤ 5) = 0.8 and P(X ≥ 0) = 0.6 What are the values of μ and σ2?
Suppose that X ~ N(μ, σ2) and that P(X ≤ 10) = 0.55 and P(X ≤ 0) = 0.40 What are the values of μ and σ2?
Suppose that X ~ N(μ, σ2). Show that P(X ≤ μ + σzα) = 1 - α and that P(μ - σzα/2 ≤ X ≤ μ + σza/2) = 1 - α
What are the upper and lower quartiles of a N(0, 1) distribution? What is the interquartile range? What is the interquartile range of a N(μ, σ2) distribution?
The thicknesses of glass sheets produced by a certain process are normally distributed with a mean of μ = 3.00 mm and a standard deviation of σ = 0.12 mm. (a) What is the probability that a glass sheet is thicker than 3.2 mm? (b) What is the probability that a glass sheet is thinner than 2.7
Suppose that X ~ N(3.2, 6.5), Y ~ N(-2.1, 3.5), and Z ~ N (12.0, 7.5) are independent random variables. Find the probability that (a) X + Y ≥ 0 (b) X + Y - 2Z ≤ -20 (c) 3X + 5Y ≥ 1 (d) 4X - 4Y + 2Z ≤ 25 (e) |X + 6Y + Z| ≥ 2 (f) |2X - Y - 6| ≤ 1
Recall Problem 5.1.15, where mechanical components have a width that is normally distributed with a mean u = 2600 and a standard deviation σ = 0.6. In an assembly procedure, four of these components need to be fitted side by side into a slot in another part. (a) Suppose that the slots have a width
Let X1,..., X15 be independent identically distributed N(4.5, 0.88) random variables, with an average . (a) Calculate P(4.2 ≤ ≤ 4.9). (b) Find the value of c for which P(4.5 - c ≤ ≤ 4.5 + c) = 0.99.
Five students are waiting to talk to the TA when office hours begin. The TA talks to the students one at a time, starting with the first student and ending with the fifth student, with no breaks between students. Suppose that the time taken by the TA to talk to a student has a normal distribution
Components of type A have heights that are independently distributed as a normal distribution with a mean 190 and a standard deviation of 10. Components of type B have heights that are independently distributed as a normal distribution with a mean 150 and a standard deviation of 8. What is the
The times taken for worker 1 to perform a task are independently distributed as a normal distribution with mean 13 minutes and standard deviation 0.5 minutes. The times taken for worker 2 to perform a task are independently distributed as a normal distribution with mean 17 minutes and standard
Bricks' weights are independently distributed as a normal distribution with mean 110 and standard deviation 2. What is the smallest value of n such that there is a probability of at least 99% that the average weight of n randomly selected bricks is less than 111?
A piece of wire is cut, and the length of the wire has a normal distribution with a mean 7.2 m and a standard deviation 0.11 m. If the piece of wire is then cut exactly in half, what are the mean and the standard deviation of the lengths of the two pieces?
The amount of timber available from a certain type of fully grown tree has a mean of 63400 with a standard deviation of 2500. (a) What are the mean and the standard deviation of the total amount of timber available from 20 trees? (b) What are the mean and the standard deviation of the average
A chemist can set the target value for the elasticity of a polymer compound. The resulting elasticity is normally distributed with a mean equal to the target value and a standard deviation of 47. (a) What target value should be set if it is required that there is only a 10% probability that the
Suppose that X ~ N(-1.9, 2.2), Y ~ N(3.3, 1.7), and Z ~ N(0.8, 0.2) are independent random variables. Find the probability that (a) X - Y ≥ -3 (b) 2X + 3Y + 4Z ≤ 10 (c) 3Y - Z ≤ 8 (d) 2X - 2Y + 3Z ≤ -6 (e) |X + Y - Z| ≥ 1.5 (f) |4X - Y + 10| ≤ 0.5
Consider a sequence of independent random variables Xi, each with a standard normal distribution. (a) What is P(|Xi| ≤ 0.5)? (b) If is the average of eight of these random variables, what is P(|| ≤ 0.5)? (c) In general, if is the average of n of these random variables, what is the smallest
Recall Problem 5.1.11 where metal plate thicknesses are normally distributed with a mean of 4.3 mm and a standard deviation of 0.12 mm. (a) If one metal plate is placed on top of another, what is the distribution of their combined thickness? (b) What is the distribution of the average thickness of
A machine part is assembled by fastening two components of type A and three components of type B end to end. The lengths of components of type A in mm are independent N(37.0, 0.49) random variables, and the lengths of components of type B in mm are independent N(24.0, 0.09) random variables. What
(a) Suppose that X1 ~ N(Î¼1, Ïƒ12) and X2 ~ N(Î¼2, Ïƒ22) are independently distributed. What is the variance ofY = pX1 + (1 - p)X2?Show that the variance is minimized whenWhat is the variance of Y in this case? (b) More generally suppose that Xi ~
If $x is invested in mutual fund I, its worth after one year is distributed X1 ~ N(1.05x, 0.0002x2) and if $x is invested in mutual fund II, its worth after one year is distributed X11 ~ N(1.05x, 0.0003x2) Suppose that you have $1000 to invest and that you place $y in mutual fund I and $(1000 - y)
Recall Problem 5.1.9 where glass sheets have a N(3.00, 0.122) distribution. (a) What is the probability that three glass sheets placed one on top of another have a total thickness greater than 9.50 mm? (b) What is the probability that seven glass sheets have an average thickness less than 3.10 mm?
Recall Problem 5.1.10 where sugar packets have weights with N(1.03, 0.0142) distributions. A box contains 22 sugar packets. (a) What is the distribution of the total weight of sugar in a box? (b) What are the upper and lower quartiles of the total weight of sugar in a box?
Calculate the following probabilities both exactly and by using a normal approximation: (a) P(X ≥ 8) where X ~ B(10, 0.7) (b) P(2 ≤ X ≤ 7) where X ~ B(15, 0.3) (c) P(X ≤ 4) where X ~ B(9, 0.4) (d) P(8 ≤ X ≤ 11) where X ~ B(14, 0.6)
Recall Problem 3.1.9, where a company receives 60% of its orders over the Internet. Estimate the probability that at least 925 of the company's next 1500 orders will be received over the Internet.
Consider again Supplementary Problem 4.7.10, where the strength of a chemical solution has a beta distribution with parameters a = 18 and b = 11. Estimate the probability that the average strength of 20 independently produced chemical solutions is between 0.60 and 0.65.
Suppose that a course has a capacity of at most 240 people, but that 1550 invitations are sent out. If each person who receives an invitation has a probability of 0.135 of attending the course, independently of everybody else, what is the probability that the number of people attending the course
The lifetimes of batteries are independent with an exponential distribution with a mean of 84 days. Consider a random selection of 350 batteries. What is the probability that at least 55 of the batteries have lifetimes between 60 and 100 days?
The time to failure of an electrical component has a Weibull distribution with parameters λ = 0.056 and a = 2.5. A random collection of 500 components is obtained. Estimate the probability that at least 125 of the 500 components will have failure times larger than 20.
Suppose that components have weights that are independent and uniformally distributed between 890 and 892. (a) Suppose that components are weighed one by one. What is the probability that the sixth component weighed is the third component that weighs more than 891.2? (b) If a box contains 200
Suppose that the time taken for food to spoil using a certain packaging method has an exponential distribution with a mean of 8 days. If a random sample of 100 packets are tested after 10 days, what are the expectation and the standard deviation of the number of packets that will be found with
Calculate the following probabilities both exactly and by using a normal approximation: (a) P(X ≥ 7) where X ~ B(10, 0.3) (b) P(9 ≤ X ≤ 12) where X ~ B(21, 0.5) (c) P(X ≤ 3) where X ~ B(7, 0.2) (d) P(9 ≤ X ≤ 11) where X ~ B(12, 0.65)
Suppose that a fair coin is tossed n times. Estimate the probability that the proportion of heads obtained lies between 0.49 and 0.51 for n = 100, 200, 500, 1000, and 2000.
Suppose that a fair die is rolled 1000 times. (a) Estimate the probability that the number of 6s is between 150 and 180. (b) What is the smallest value of n for which there is a probability of at least 99% of obtaining at least 50 6s in n rolls of a fair die?
The number of cracks in a ceramic tile has a Poisson distribution with parameter λ = 2.4. (a) How would you approximate the distribution of the total number of cracks in 500 ceramic tiles? (b) Estimate the probability that there are more than 1250 cracks in 500 ceramic tiles.
In a test for a particular illness, a false-positive result is obtained about 1 in 125 times the test is administered. If the test is administered to 15,000 people, estimate the probability of there being more than 135 false-positive results.

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