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probability statistics
Probability And Statistics For Engineers And Scientists 9th Edition Ronald E. Walpole, Raymond H. Myers, Sharon L. Myers, Keying E. Ye - Solutions
If the standard deviation of the mean for the sampling distribution of random samples of size 49, from large or infinite population, is 3, how large must the sample size become if the standard deviation is to be reduced to 1?
If all possible samples of size 25 are drawn from a normal population with a mean of 60 and standard deviation of 9, what is the probability that a sample mean, , will fall in the interval between and ? Assume that the sample means can be measured to any degree of accuracy.
In the 2014–15 cricket season, the captain of a university cricket team scored the following runs in 12 different one-day matches: 84, 25, 74, 53, 40, 31, 64, 71, 18, 63, 88, and 49. Find 21. the mean runs;2. the median runs.
Calculate the variance of the sample, 8, 10, 16, 18, 24, and 25. Use this answer, along with the results of Exercise 8.14, to find 1. the variance of the sample 16, 20, 32, 36, 48 and 50;2. the variance of the sample 12, 14, 20, 22, 28, and 29.
1. Show that the sample variance is unchanged if a constant c is added to or subtracted from each value in the sample.2. Show that the sample variance becomes c times its original value if each observation in the sample is multiplied by c.
The heights, in meters, of 20 randomly selected college seniors are as follows:Calculate the standard deviation.
The average contents of saturated fat in eight bars of a certain brand of low-fat cereal selected at random are measured as follows: 0.65, 0.72, 0.45, 0.55, 0.58, 0.39, 0.68, and 0.52 grams. Calculate 1. the mean;2. the variance.
For the data of Exercise 8.5, calculate the variance using the formula 1. of form (8.2.1);2. in Theorem 8.1.
For the sample of reaction times in Exercise 8.3, calculate 1. the range;2. the variance, using the formula of form (8.2.1).
Consider the data in Exercise 8.2, find 1. the range;2. the standard deviation.
According to ecology writer Jacqueline Killeen, phosphates contained in household detergents pass right through our sewer systems, causing lakes to turn into swamps that eventually dry up into deserts. The following data show the amount of phosphates per load of laundry, in grams, for a random
A random sample of students from a city school scored the following marks for a paper in the annual examination: 45, 57, 68, 34, 50, 32, 89, 47, 97, 67, 79, 84, 43, 35, 68, 55, 72, 63, 68, and 49. Calculate 1. the mean;2. the mode.
Find the mean, median, and mode for a sample whose observations, 5, 12, 3, 7, 11, 1, 85, 5, 9, and 5, represent the number of medical leaves claimed by 10 employees of a company in a year. Which of the values appears to be the best measure of the center of the data? State reasons for your
The numbers of incorrect answers on a true-false competency test for a random sample of 15 students were recorded as follows: 2, 1, 3, 0, 1, 3, 6, 0, 3, 3, 5, 2, 1, 4, and 2. Find 1. the mean;2. the median;3. the mode.
The number of tickets issued for traffic violations by 8 state troopers during the Memorial Day weekend are 5, 4, 7, 7, 6, 3, 8, and 6.1. If these values represent the number of tickets issued by a random sample of 8 state troopers from Montgomery County in Virginia, define a suitable population.2.
The reaction times for a random sample of 9 subjects to a stimulant were recorded as 2.5, 3.6, 3.1, 4.5, 2.9, 2.3, 2.6, 4.1, and 3.4 seconds. Calculate 1. the mean;2. the median.
The lengths of time, in minutes, that 10 patients waited in a doctor’s office before receiving treatment were recorded as follows: 5, 11, 9, 5, 10, 15, 6, 15, 5, and 12. Treating the data as a random sample, find 1. the mean;2. the median;3. the mode.
Define suitable populations from which the following samples are selected:1. A car manufacturing company calls customers for feedback after servicing their vehicles at the company’s authorized service center.2. Four out of 10 randomly selected college students are girls.3. The total marks
By expanding e in a Maclaurin series and integrating term by term, show that Mx (t) = [*e* f(x) dr 1+t+ +
Both X and Y independently follow a geometric distribution with a probability mass function of P(X = r) = P(Y =r) = q p, r = 0, 1, 2, …, where, p and q are positive numbers such that p + q = 1. Find Xr 1. the probability distribution of U = X + Y;2. the conditional distribution of X/(X + Y) = u.
Using the moment-generating function of Exercise 7.21, show that the mean and variance of the chisquared distribution with v degrees of freedom are, respectively, v and 2v.
Show that the moment-generating function of the random variable X having a chi-squared distribution with v degrees of freedom is Mx(t) (1-2)/2.
The moment-generating function of a certain Poisson random variable X is given byFind P(μ – σ ≤ X ≤ μ + σ). Mx (t) = (-1)
A random variable X has the Poisson distribution p(x; μ) =e μ /x! for x = 0, 1, 2, …. Show that the moment-generating function of X is x – 1 X–μ x Using M (t), find the mean and variance of the Poisson distribution.
A random variable X has the geometric distribution g(x; p)= pq for x = 1, 2, 3, …. Show that the moment-generating function of X is and then use M (t) to find the mean and variance of the geometric distribution.
Show that the rth moment about the origin of the gamma distribution is 22 2[Hint: Substitute y = x/β in the integral defining and then use the gamma function to evaluate the integral.]
Let X have the probability distribution Find the probability distribution of the random variable Y = X .
Let X be a random variable with probability distribution Find the probability distribution of the random variable Y = X .
A current of I amperes flowing through a resistance of R ohms varies according to the probability distribution 1 2 1 2 1 1 2 2 1 1 2 If the resistance varies independently of the current according to the probability distribution find the probability distribution for the power W = I R watts.
Let X and X be independent random variables each having the probability distribution Show that the random variables Y and Y are independent when Y = X + X and Y = X /(X + X ).
The random variables X and Y, representing the weights of creams and toffees, respectively, in 1-kilogram boxes of chocolates containing a mixture of creams, toffees, and cordials, have the joint density function 1. Find the probability density function of the random variable Z = X + Y.2. Using the
The hospital period, in days, for patients following treatment for a certain type of kidney disorder is a random variable Y = X + 4, where X has the density function 1. Find the probability density function of the random variable Y.2. Using the density function of Y, find the probability that the
A dealer’s profit, in units of $5000, on a new automobile is given by Y = X , where X is a random variable having the density function 1. Find the probability density function of the random variable Y.2. Using the density function of Y, find the probability that the profit on the next new
The speed of a molecule in a uniform gas at equilibrium is a random variable V whose probability distribution is given by where k is an appropriate constant and b depends on the absolute temperature and mass of the molecule. Find the probability distribution of the kinetic energy of the molecule W,
Given the random variable X with probability distribution 1 2 1 2 1 1 2 2 1 2 efind the probability distribution function of Y = – 3X + 5.
If the variable X has the probability distribution Show that the random variable Y = –4 log X has an exponential distribution with β = 4.
Let X and Y be two discrete random variables with a joint probability distribution of Find the probability distribution of the random variable Z = X +Y.
Let X and X be discrete random variables with the joint multinomial distribution 21 2 for x = 0, 1, 2; x = 0, 1, 2; x + x ≤ 2; and zero elsewhere. Find the joint probability distribution of Y = X + X and Y = X – X .
Let X be a binomial random variable with probability distribution Find the probability distribution of the random variable Y = X .
The probability distribution of the number X, when an unbiased die is tossed is Find the probability distribution of the random variable Y = 2X+ 1.
Group Project: Have groups of students observe the number of people who enter a specific coffee shop or fast food restaurant over the course of an hour, beginning at the same time every day, for two weeks. The hour should be a time of peak traffic at the shop or restaurant. The data collected will
The length of time, in seconds, that a computer user takes to read his or her e-mail is distributed as a lognormal random variable with μ = 1.8 and σ = 4.0.1. What is the probability that a user reads e-mail for more than 20 seconds? More than a minute?2. What is the probability that a user reads
From the relationship between the chi-squared random variable and the gamma random variable, prove that the mean of the chi-squared random variable is v and the variance is 2v.
Explain why the nature of the scenario in Review Exercise 6.82 would likely not lend itself to the exponential distribution.
Derive the cdf for the Weibull distribution. [Hint: In the definition of a cdf, make the transformation z = y .]
The length of life, in hours, of a drill bit in a mechanical operation has a Weibull distribution with α = 2 and β = 50. Find the probability that the bit will fail before 10 hours of usage.
The length of time between breakdowns of an essential piece of equipment is important in the decision of the use of auxiliary equipment. An engineer thinks that the best model for time between breakdowns of a generator is the exponential distribution with a mean of 15 days.1. If the generator has
In a human factor experimental project, it has been determined that the reaction time of a pilot to a visual stimulus is normally distributed with a mean of 1/2 second and standard deviation of 2/5 second.1. What is the probability that a reaction from the pilot takes more than 0.3 second?2. What
Consider Review Exercise 6.78. Given the assumption of the exponential distribution, what is the mean number of calls per hour? What is the variance in the number of calls per hour?
Consider now Review Exercise 3.74 on page 128. The density function of the time Z in minutes between calls to an electrical supply store is given by 1. What is the mean time between calls?2. What is the variance in the time between calls?3. What is the probability that the time between calls
The beta distribution has considerable application in reliability problems in which the basic random variable is a proportion, as in the practical scenario illustrated in Exercise 6.50 on page 226. In that regard, consider Review Exercise 3.73 on page 128. Impurities in batches of product of a
In Exercise 6.54 on page 226, the lifetime of a transistor is assumed to have a gamma distribution with mean 10 weeks and standard deviation weeks. Suppose that the gamma distribution assumption is incorrect. Assume that the distribution is normal.1. What is the probability that a transistor will
For Review Exercise 6.74, what is the mean of the average water usage per hour in thousands of gallons?
The average rate of water usage (thousands of gallons per hour) by a certain community is known to involve the lognormal distribution with parameters μ = 5 and σ = 2. It is important for planning purposes to get a sense of periods of high usage. What is the probability that, for any given hour,
For Review Exercise 6.72, what are the mean and variance of the time that elapses before 2 failures occur?
Consider the information in Review Exercise 6.66. What is the probability that less than 200 hours will elapse before 2 failures occur?
A technician plans to test a certain type of resin developed in the laboratory to determine the nature of the time required before bonding takes place. It is known that the mean time to bonding is 3 hours and the standard deviation is 0.5 hour. It will be considered an undesirable product if the
The time taken by all the contestants in a food festival to prepare a dish is normally distributed, with a mean of 25 minutes and standard deviation of 7 minutes. Find the probability that 1. it takes more than 30 minutes to prepare the item;2. the item is prepared within 20 minutes;3. the item is
The average consultation time for a patient in an ENT department of a hospital is 15 minutes.1. Assuming that the consultation times for the patients are independent and exponential, what is the expected consultation time for two random patients?2. What is the probability that the total
In a chemical processing plant, it is important that the yield of a certain type of batch product stay above 80%. If it stays below 80% for an extended period of time, the company loses money. Occasional defective batches are of little concern.But if several batches per day are defective, the plant
A certain type of device has an advertised failure rate of 0.01 per hour. The failure rate is constant and the exponential distribution applies.1. What is the mean time to failure?2. What is the probability that 200 hours will pass before a failure is observed?
A survey on gender status reveals that 56% of the families have more males than females. Assuming that this percentage is still valid, what is the probability that among 1000 randomly selected families, the male members exceed the female in between 570 and 650 (both included) of the families?
When α is a positive integer n, the gamma distribution is also known as the Erlang distribution. Setting α = n in the gamma distribution on page 215, the Erlang distribution is It can be shown that if the times between successive events are independent, each having an exponential distribution
The exponential distribution is frequently applied to the waiting times between successes in a Poisson process. If the number of calls received per hour by a telephone answering service is a Poisson random variable with parameter λ = 4, we know that the time, in hours, between successive calls has
A non-governmental agency working on traffic awareness conducted a survey in the city last year. It found that 49% of the population had a basic knowledge of the traffic rules. What is the probability that among any 1000 randomly selected individuals of the city, between 482 and 510 (both included)
Show that the failure-rate function is given by if and only if the time to failure distribution is the Weibull distribution
Consider the information in Exercise 6.58.1. What is the probability that more than 1 minute elapses between arrivals?2. What is the mean number of minutes that elapse between arrivals?
The number of automobiles that arrive at a certain intersection per minute has a Poisson distribution with a mean of 5. Interest centers around the time that elapses before 10 automobiles appear at the intersection.1. What is the probability that more than 10 automobiles appear at the intersection
For Exercise 6.56, what is the mean power usage (average dB per hour)? What is the variance?
Rate data often follow a lognormal distribution. Average power usage (dB per hour) for a particular company is studied and is known to have a lognormal distribution with parametersμ = 4 and σ = 2. What is the probability that the company uses more than 270 dB during any particular hour?
According to a telephone operator, the average time for each call is 3.2 minutes. This time follows an exponential distribution.1. What is the probability that the call time exceeds 5 minutes?2. What is the probability that call completes in 2 minutes?
The lifetime, in weeks, of a certain type of transistor is known to follow a gamma distribution with mean 10 weeks and standard deviation weeks.1. What is the probability that a transistor of this type will last at most 50 weeks?2. What is the probability that a transistor of this type will not
In a biomedical research study, it was determined that the survival time, in weeks, of an animal subjected to a certain exposure of gamma radiation has a gamma distribution with α= 5 and β = 10.1. What is the mean survival time of a randomly selected animal of the type used in the experiment?2.
The lives of a certain automobile seal have the Weibull distribution with failure rate . Find the probability that such a seal is still intact after 4 years.
If the proportion of a brand of television requiring service during the first year of operation is a random variable having a beta distribution with α = 3 and β = 2, what is the probability that at least 80% of the new models of this brand sold this year will require service during their first
Suppose the random variable X follows a beta distribution with α = 1 and β = 3.1. Determine the mean and median of X.2. Determine the variance of X.3. Find the probability that X > 1/3.
Derive the mean and variance of the beta distribution.
Suppose that the service life, in years, of a hearing aid battery is a random variable having a Weibull distribution withα = 1/2 and β = 2.1. How long can such a battery be expected to last?2. What is the probability that such a battery will be operating after 2 years?
The life of a street bulb follows an exponential distribution, with an average life β = 3 years. The bulbs are replaced whenever they fail. Out of the 1000 street bulbs installed in a city, what is the probability that at most 250 of them will need to be replaced during the first year?
At a train reservation counter, one man completes his reservation with a mean time of 3 minutes. Service completion time is assumed to follow exponential distribution. Out of the 5 customers in queue, what is the probability that at least 4 will complete their reservation within 3 minutes?
The water supply board of a metropolitan city reveals that the each family consumes an average of 20 liters of drinking water per day, with a standard deviation of liters. Let X denote the drinking water consumption per family and follow the gamma distribution.1. Find α and β.2. Find the
1. Find the mean and variance of the daily water consumption in Exercise 6.40.2. According to Chebyshev’s theorem, there is a probability of at least 3/4 that the water consumption on any given day will fall within that interval?
Suppose that the time, in hours, required to service a motorbike is a random variable X having a gamma distribution, 1/4 with α = 2 and . What is the probability that on the next service call, 1. at most 2 hours of service will be required?2. at least 1 hour of service will be required?
If a random variable X has a gamma distribution, with α =1, β = 1, find P(1.6 < X < 2.8).
In a certain city, the daily consumption of water (in millions of liters) follows approximately a gamma distribution with α = 2 and β = 3. If the daily capacity of that city is 9 million liters of water, what is the probability that on any given day the water supply is inadequate?
Use the gamma function with to show that .
A telemarketing company has a special letter-opening machine that opens and removes the contents of an envelope. If the envelope is fed improperly into the machine, the contents of the envelope may not be removed or may be damaged. In this case, the machine is said to have “failed.”1. If the
The systolic blood pressure X of 30-year-old men is approximately normally distributed with a mean of 123 mmHg and standard deviation of 6 mmHg.1. Find the probability that the blood pressure of a randomly selected 30-year-old man exceeds 127 mmHg.2. Out of 250 randomly selected men of 30 years
A common practice of airline companies is to sell more tickets for a particular flight than there are seats on the plane, because customers who buy tickets do not always show up for the flight. Suppose that the percentage of no-shows at flight time is 2%. For a particular flight with 197 seats, a
A firm assembling electronics has a record of 98% perfect assemblies. They export the assembled items in a lot of 50 units.1. What is the probability that a lot will contain more than 2 defectives?2. What is the probability that a lot will contain at most one defective?
A pair of dice is rolled 360 times. What is the probability of obtaining a sum of 12 1. at least 5 times?2. between 12 and 18 times, both included?3. exactly 6 times?
Statistics released by the National Highway Traffic Safety Administration and the National Safety Council show that on an average weekend night, 1 out of every 10 drivers on the road is drunk. If 400 drivers are randomly checked next Saturday night, what is the probability that the number of drunk
A pin manufacturing company knows that the chance of producing a defective item is 4%. The company markets the items with a promise that in a pack of 250, no more than 10 pins will be defective. What are the chances of the company keeping up its claim? Use the normal approximation to the binomial
One-sixth of the male freshmen entering a large state school are out-of-state students. If the students are assigned at random to dormitories, 180 to a building, what is the probability that in a given dormitory at least one-fifth of the students are from out of state?
A village administrator claims that, on average, 75% of the villagers are literate. The authorities decided to verify the claim by testing 100 villagers selected at random. They decided to accept the claim if 70% or more of the villages were found to be literate.1. What is the probability that the
If 20% of the residents in a U.S. city prefer a white cell phone over any other color available, what is the probability that among the next 1000 cell phone purchased in that city 1. between 170 and 185 inclusive will be white?2. at least 210 but not more than 225 will be white?
As part of the research on “the role of English as a gateway to knowledge”, a survey is conducted among 1000 college students, in which 72% of the students agree with the statement. If 100 students are picked at random, what is the probability that 1. at least 80 of them agree with the
In a city, 4% of the adolescents are alcoholic. Out of the 100 adolescents randomly selected, what is the probability that 1. between 8 and 18 of them are alcoholics?2. fewer than 5 are alcoholics?
A certain batch contains 5% defectives defective. If 100 apples are randomly examined, what is the probability that the number of defective apples 1. exceeds 15?2. is less than 10?
In a textile manufacturing company, 1% of the items are known to be defective. The quality control team decides to select 100 items produced by the company for examination. If none of the units are found to be defective, the process continues. Use the normal approximation to the binomial to find 1.
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