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

Despite a series of quality checks by a company that makes television sets, there is a probability of 0.0007 that when a purchaser unpacks a newly purchased television set it does not work properly. If the company sells 250,000 television sets a year, estimate the probability that there will be no
A multiple-choice test consists of a series of questions, each with four possible answers. (a) If there are 60 questions, estimate the probability that a student who guesses blindly at each question will get at least 30 questions right. (b) How many questions are needed in order to be 99% confident
Recall Problem 4.3.4 in which a day's sales in $1000 units at a gas station have a gamma distribution with k = 5 and λ = 0.9. If the sales on different days are distributed independently of each other, estimate the probability that in one year the gas station takes in more than $2 million.
Suppose that the random variable X has a lognormal distribution with parameter values μ = 1.2 and σ = 1.5. Find: (a) E(X) (b) Var(X) (c) The upper quartile of X (d) The lower quartile of X (e) The interquartile range (f) P(5 ≤ X ≤ 8)
Use your computer package to find the following critical points: (a) F0.04,7,37 (b) F0.87,17,43 (c) F0.035,3,8 If the random variable X has an F-distribution with degrees of freedom v1 = 5 and v2 = 33, use your computer package to find: (d) 5(X ≥ 2.35) (e) 5(0.21 ≤ X ≤ 2.92)
If the random variable X has a t-distribution with v degrees of freedom, explain why the random variable Y = X2 has an F-distribution with degrees of freedom 1 and v.
(a) There is a probability of 0.90 that a t random variable with 23 degrees of freedom lies between -x and x. Find the value of x. (b) There is a probability of 0.975 that a t random variable with 60 degrees of freedom is larger than y. Find the value of y. (c) What is the probability that a
Use the tables to put bounds on these probabilities, (a) P(F10,50 ≥ 2.5) (b) P(X217 ≤ 12) (c) P(t24 ≥ 3) (d) P(t14 ≥ −2)
Use the tables to put bounds on these probabilities. (a) P(t21 ≤ 2.3) (b) P(X26 ≥ 13.0) (c) P(t10 ≤ −1.9) (d) P(t7 ≥ −2.7)
Use the tables to put bounds on these probabilities (a) P(t16 ≤ 1.9) (b) P(X225 ≥ 42.1) (c) P(F9,14 ≤ 1.8) (d) P(−1.4 ≤ t29 ≤ 3.4)
Suppose that the random variable X has a lognormal distribution with parameter values μ = -0.3 and σ = 1.1. Find: (a) E(X) (b) Var(X) (c) The upper quartile of X (d) The lower quartile of X (e) The interquartile range (f) P(0.1 ≤ X ≤ 7.0)
A researcher grows cultures of bacteria. Suppose that after one day's growth, the size of the culture has a lognormal distribution with parameters μ = 2.3 and σ = 0.2. (a) What is the expected size of the culture after one day? (b) What is the median size of the culture after one day? (c) What is
Use your computer package to find the following critical points, and check that they match the values given in Table II. (a) X20.10,9 (b) X20.05,20 (c) X20.01,26 (d) X20.90,50 (e) X20.95,6
Use your computer package to find the following critical points: (a) X20.12,8 (b) X20.54,19 (c) X20.023,32 If the random variable X has a chi-square distribution with 12 degrees of freedom, use your computer package to find: (d) P(X ≤ 13.3) (e) P(9.6 ≤ X ≤ 15.3)
Use your computer package to find the following critical points, and check that they match the values given in Table III. (a) t0.10,7 (b) t0.05,19 (c) t0.01,12 (d) t0.025,30 (e) t0.005,4
Use your computer package to find the following critical points: (a) t0.27,14 (b) t0.09,22 (c) t0.016,7 If the random variable X has a t-distribution with 22 degrees of freedom, use your computer package to find: (d) P(X ≤ 1.78) (e) P(-0.65 ≤ X ≤ 2.98) (f) P(|X| ≥ 3.02)
Use your computer package to find the following critical points, and check that they match the values given in Table IV. (a) F0.10,9,10 (b) F0.05,6,20 (c) F0.01,15,30 (d) F0.05,4,8 (e) F0.01,20,13
The amount of sulfur dioxide escaping from the ground in a certain volcanic region in one day is normally distributed with a mean μ = 500 tons and a standard deviation σ = 50 tons under ordinary conditions. However, if a volcanic eruption is imminent, there are much larger sulfur dioxide
Recall Problem 3.4.8, where the number of misrecorded pieces of information in a scanning process has a Poisson distribution with parameter λ = 9.2. Estimate the probability that there are fewer than 1000 total pieces of misrecorded information when 100 different scans are performed.
When making a connection at an airport. Jasmine arrives on a plane that is due to arrive at 2:15 P.M. However, the amount by which her plane arrives late has a normal distribution with a mean μ = 32 minutes and a standard deviation σ = 11 minutes. Jasmine wants to transfer to a plane that is due
A clinic has four different physicians. A, B, C, and D, one of whom is selected by each new patient. If the new patients are equally likely to choose each of the four physicians independently of each other, estimate the probability that physician A will get at least 25 out of the next 80 new
An aircraft can seat 220 passengers, and each of the passengers booked on the Might has a probability of 0.9 of actually arriving at the gate to board the plane, independent of the other passengers. (a) Suppose the airline books 235 passengers on the flight. What is the probability that there will
(a) What is the probability that a random variable with a standard normal distribution takes a value between 0.6 and 2.2? (b) What is the probability that a random variable with a normal distribution with μ = 4.1 and σ = 0.25 takes a value between 3.5 and 4.5? (c) What is the probability that a
Components have lifetimes in minutes that are independent of each other with a lognormal distribution with parameters μ = 3.1 and σ = 0.1. Suppose that a random sample of 200 components is taken. What is the probability that 30 or more of the components will have a lifetime of at least 25 minutes?
Are the following statements true or false? (a) A t-distribution with 60 degrees of freedom has a larger variance than a standard normal distribution. (b) The probability that a normal random variable with mean 10 and standard deviation 2 is less than 14 is equal to the probability that a normal
When an order is placed with a company, there is a probability of 0.2 that it is an express order. Estimate the probability that 90 or more of the next 400 orders will be express orders.
In genetic profiling, the expression of a gene is measured for a set of different samples. Suppose that the expressions are modeled as being independently normally distributed with a mean 0.768 and a standard deviation 0.083. (a) If six samples are measured, what is the probability that at least
Suppose that electrical components have lifetimes that are independent and that come from a normal distribution with a mean of 8200 minutes and a standard deviation of 350 minutes. (a) If three components are selected, what is the probability that 1 lasts for less than 8000 minutes, 1 lasts between
The breaking strengths of nylon fibers are normally distributed with a mean of 12,500 and a variance of 200,000. (a) What is the probability that a fiber strength is more than 13,000? (b) What is the probability that a fiber strength is less than 11,400? (c) What is the probability that a fiber
The time taken by operator A to finish a task has a normal distribution with a mean 220 minutes and a standard deviation 11 minutes. The time taken by operator B to finish a task has a normal distribution with a mean 185 minutes and a standard deviation 9 minutes, independent of operator A.
When users connect to a server, the lengths of time in minutes that they are connected are independently distributed with a Weibull distribution with λ = 0.03 and a = 0.8. (a) Suppose that 5 users connect to the server. What is the probability that 2 of the users are connected for a time less than
Tiles have weights that are independently normally distributed with a mean of 45.3 and a standard deviation of 0.02. What is the probability that the total weight of three tiles is no more than 135.975?
Components of type A have lengths that are independently normally distributed with a mean of 67.2 and a standard deviation of 1.9. Components of type B have lengths that are independently normally distributed with a mean of 33.2 and a standard deviation of 1.1. What is the probability that two
Suppose that the failure time of a component is modeled with an exponential distribution with a mean of 32 days. A company acquires a batch of 240 components. If the failure times of these components are taken to be independent of each other, estimate the probability that at least half of the
Machine A produces components with holes whose diameter is normally distributed with a mean 56,000 and a standard deviation 10. Machine B produces components with holes whose diameter is normally distributed with a mean 56,005 and a standard deviation 8. Machine C produces pins whose diameter is
(a) What is the probability that a t random variable with 40 degrees of freedom lies between -1.303 and 2.021? (b) Use Table III to put bounds on the probability that a t random variable with 17 degrees of freedom is greater than 2.7.
Use the tables to put bounds on these probabilities, (a) P(F16,20 ≤ 2) (b) P(X228 ≥ 47) (c) P(t29 ≥ 1.5) (d) P(t7 ≤ −1.3) (e) P(t10 ≥ −2)
Use the tables to put bounds on these probabilities. (a) P(X240 > 65.0) (b) P(t20 < −1.2) (c) P(t26 < 3.0) (d) P(F8,14 > 4.8)
A patient has a doctor's appointment that is scheduled for 9:40 A.M. However, the amount of time after the scheduled time that the doctor's consultation actually starts has a normal distribution with a mean of 22 minutes and a standard deviation of 4 minutes. The doctor's consultation lasts for a
Adult salmon have lengths that are normally distributed with a mean of μ = 70 cm and a standard deviation of σ = 5.4 cm. (a) What is the probability that an adult salmon is longer than 80 cm? (b) What is the probability that an adult salmon is shorter than 55 cm? (c) What is the probability that
Consider again Problem 5.6.3 where the lengths of adult salmon have N(70, 5.42) distributions. (a) If you go fishing with a friend, what is the probability that the first adult salmon you catch is longer than the first adult salmon your friend catches? (b) What is the probability that the first
Suppose that the lengths of plastic rods produced by a machine are normally distributed with a mean of 2.30 m and a standard deviation of 2 cm. If two rods are placed side by side, what is the probability that the difference in their lengths is less than 3 cm?
A new 1.5-volt battery has an actual voltage that is uniformly distributed between 1.43 and 1.60 volts. Estimate the probability that the sum of the voltages from 120 new batteries lies between 180 and 182 volts.
The germination time in days of a newly planted seed is exponentially distributed with parameter λ = 0.31. If the germination times of different seeds are independent of one another, estimate the probability that the average germination time of 2000 seeds is between 3.10 and 3.25 days.
A publisher sends out advertisements in the mail asking people to subscribe to a magazine. Suppose that there is a probability of 0.06 that a recipient of the advertisement does subscribe to the magazine. If 350,000 advertisements are mailed out, estimate the probability that the magazine gains at
Suppose that if I invest $1000 today in a high-risk new-technology company, my return after 10 years has a lognormal distribution with parameters μ = 5.5 and σ = 2.0. (a) What are the median, upper, and lower quartiles of my 10-year return? (b) What is the probability that my 10-year return is at
One Friday morning at a television manufacturing company the quality inspector recorded the grades assigned to the pictures on the television sets that were ready to be shipped. The grades, presented in DS 6.1.2, are perfect," "good," "satisfactory," or "fail." (a) Define the population from which
DS 6.1.3 presents the eye colors of a group of students who are registered for a course on computer programming. (a) Define the population from which the sample is taken. Do you think that it is a representative sample? (b) Are there any other factors that should be taken into account in
One Saturday a researcher recorded the times taken to serve customers at a fast-food restaurant. DS 6.1.4 shows the service times in seconds for all the customers who were served between 2:00 and 3:00 in the afternoon. (a) Define the population from which the sample is taken. Do you think that it
Every day in the summer months a supermarket receives a shipment of peaches. The supermarket's quality inspector arranges to have one box randomly selected from each shipment for which the number of "spoiled" peaches (out of 48 peaches in the box), which cannot be put out on the supermarket
A researcher records the number of calls received by a switchboard during a one-minute period. These one-minute intervals are chosen at evenly spaced times during a working week. The data set obtained by the researcher is shown in DS 6.1.6. (a) Define the population from which the sample is taken.
A builder orders a large shipment of paving slabs from a particular company. The weights of a sample of randomly selected slabs are given in DS 6.1.7. (a) Define the population from which the sample is taken. Do you think that it is a representative sample? (b) Are there any other factors that
The data set of plastic panel bending capabilities given in DS 6.1.9. Use a statistical software package to obtain appropriate graphical presentations of each of the following data sets. Obtain more than one graphical presentation where appropriate. Indicate any data observations that might be
The data set of die rolls given in DS 6.1.1. Use a statistical software package to obtain appropriate graphical presentations of each of the following data sets. Obtain more than one graphical presentation where appropriate. Indicate any data observations that might be considered to be outliers.
Consider the data set given in DS 6.3.1. Calculate by hand the sample mean, sample median, sample trimmed mean, and sample standard deviation. Calculate the upper and lower sample quartiles, and draw a boxplot of the data set.
The data set of plastic panel bending capabilities given in DS 6.1.9. Use a statistical software package to obtain sample statistics and boxplots for the following data sets. What do the sample statistics and boxplots tell you about the data set?
Consider the data set 6 7 12 18 22 together with a sixth value x. What value of x minimizes the sample standard deviation of all six data points?
Consider the data set of 30 piston rod lengths given in DS 6.2.3. Calculate the sample mean, sample median, sample trimmed mean, and sample standard deviation. Calculate the upper and lower sample quartiles, and draw a boxplot of the data set.
Consider the data set of physical training course completion limes given in DS 6.2.4. Calculate the sample mean, sample median, sample trimmed mean, and sample standard deviation. Calculate the upper and lower sample quartiles, and draw a boxplot of the data set.
The data set of die rolls given in DS 6.1.1. Use a statistical software package to obtain sample statistics and boxplots for the following data sets. What do the sample statistics and boxplots tell you about the data set?
The data set of service times given in DS 6.1.4. Use a statistical software package to obtain sample statistics and boxplots for the following data sets. What do the sample statistics and boxplots tell you about the data set?
The data set of spoiled peaches given in DS 6.1.5. Use a statistical software package to obtain sample statistics and boxplots for the following data sets. What do the sample statistics and boxplots tell you about the data set?
The data set of calls received by a switchboard given in DS 6.1.6. Use a statistical software package to obtain sample statistics and boxplots for the following data sets. What do the sample statistics and boxplots tell you about the data set?
The data set of paving slab weights given in DS 6.1.7. Use a statistical software package to obtain sample statistics and boxplots for the following data sets. What do the sample statistics and boxplots tell you about the data set?
The data set of paint thicknesses given in DS 6.1.8. Use a statistical software package to obtain sample statistics and boxplots for the following data sets. What do the sample statistics and boxplots tell you about the data set?
Three species of bird inhabit an island and they are classified as having either brown, grey, or black markings. DS 6.6.1 shows the types of birds observed by an ornithologist during a stay on the island. The data set can be used to practice the generation and interpretation of summary statistics
DS 6.6.2 presents the number of accidents occurring on a collection of oil rigs for each month during a two-year span. The data set can be used to practice the generation and interpretation of summary statistics and graphical representations.
A software development company keeps track of the number of errors found in the programs written by the company employees. DS 6.6.3 shows the number of errors found in the 30 programs that were written during a particular month. The data set can be used to practice the generation and interpretation
DS 6.6.4 shows the heights in inches of 60 adult males with osteoporosis who visit a medical clinic during a particular week. The data set can be used to practice the generation and interpretation of summary statistics and graphical representations.
A researcher grows bamboo under controlled conditions in a greenhouse. DS 6.6.5 presents the heights of a set of bamboo shoots 40 days after planting. The data set can be used to practice the generation and interpretation of summary statistics and graphical representations.
The knowledge of soil behavior is an important issue in civil engineering. When soil is subjected to a load, there is a change in the volume of the soil due to drainage of water. A consolidation test can be performed to evaluate the compressibility of soil, so that the amount of settlement of
Suppose that E(X1) = Î¼, Var(X1) = 10, E(X2) = Î¼, and Var(X2) = 15, and consider the point estimates(a) Calculate the bias of each point estimate. Is any one of them unbiased? (b) Calculate the variance of each point estimate. Which one has the smallest variance? (c)
Suppose that E(X1) = Î¼, Var(X1) = 7, E(X2) = Î¼, Var(X2) = 13, E(X3) = Î¼, and Var(X3) = 20, and consider the point estimates(a) Calculate the bias of each point-estimate. Is any one of them unbiased? (b) Calculate the variance of each point estimate. Which
Suppose that E(X1) = Î¼, Var(X1) = 4, E(X2) = Î¼, and Var(X2) = 6.(a) What is the variance of(b) What value of p minimizes the variance of = pX1 + (1 - p)X2? (c) What is the relative efficiency of 1 to the point estimate with the smallest variance that you have found?
Repeat Problem 7.2.3 with Var(X1) = 1 and Var(X2) = 7.Problem 7.2.3Suppose that E(X1) = Î¼, Var(X1) = 4, E(X2) = Î¼, and Var(X2) = 6.(a) What is the variance of(b) What value of p minimizes the variance of = pX1 + (1 - p)X2? (c) What is the relative efficiency of 1 to the
If which point estimate would you prefer to estimate Î¸? Why?
Suppose that X ~ N(Î¼, Ïƒ2) and consider the point estimatefor some fixed value Î¼0. Show that this point estimate has a smaller mean square error than X when |Î¼ - Î¼0| ‰¤ ˆš3Ïƒ Explain why it is not
Suppose that X ~ B(10, p) and consider the point estimate = X/11 (a) What is the bias of this point estimate? (b) What is the variance of this point estimate? (c) Show that this point estimate has a mean square error of 10p - 9p2 / 121 (d) Show that this mean square error is smaller than the mean
Suppose that X1 is an estimate of a parameter θ with a standard deviation 5.39, and that X2 is an estimate of θ with a standard deviation 9.43. If the estimates X1 and X2 are independent, what is the standard deviation of the estimate (X1 + X2)/2?
Suppose that X1 ~ B(n1, p) and X2 ~ B(n2, p). What is the relative efficiency of the point estimate X1/n1 to the point estimate X2/n2 for estimating the success probability p?
Consider the data set of die rolls given in DS 6.1.1. Construct a point estimate of the probability of scoring a 6. What is the standard error of your point estimate?
Consider the data set of television picture grades given in DS 6.1.2. Construct a point estimate of the probability that a television picture is satisfactory. What is the standard error of your point estimate?
Consider the data set of eye colors given in DS 6.1.3. Construct a point estimate of the probability that a student has blue eyes. What is the standard error of your point estimate?
Consider the data set of service times given in DS 6.1.4. Construct a point estimate of the average service time. What is the standard error of your point estimate?
Consider the data set of spoiled peaches given in DS 6.1.5. Construct a point estimate of the average number of spoiled peaches per box. What is the standard error of your point estimate?
Consider the data set of calls received by a switchboard given in DS 6.1.6. Construct a point estimate of the average number of calls per minute. What is the standard error of your point estimate?
Consider the data set of paving slab weights given in DS 6.1.7. Construct a point estimate of the average slab weight. What is the standard error of your point estimate?
Consider the data set of paint thicknesses given in DS 6.1.8. Construct a point estimate of the average paint thickness. What is the standard error of your point estimate?
Consider the data set of plastic panel bending capabilities given in DS 6.1.9. Construct a point estimate of the average deformity angle. What is the standard error of your point estimate?
Unknown to an experimenter, the probability of a prototype etching procedure producing a defective part is p = 0.24. The experimenter examines 100 randomly selected parts and finds out whether or not each one is defective. What is the probability that the experimenter's point estimate of p is
Consider a sample X1,..., Xn of normally distributed random variables with mean μ and variance σ2 = 1. (a) If n = 10, what is the probability that |μ - | ≤ 0.3? (b) What is this probability when n = 30?
The capacitances of certain electronic components have a normal distribution with a mean μ = 174 and a standard deviation σ = 2.8. If an engineer randomly selects a sample of n = 30 components and measures their capacitances, what is the probability that the engineer's point estimate of the mean
Unknown to an experimenter, when a coin is tossed there is a probability of p = 0.63 of obtaining a head. The experimenter tosses the coin 300 times in order to estimate the probability p. What is the probability that the experimenter's point estimate of p will be within the interval (0.62, 0.64)?
The weights of bricks are normally distributed with μ = 110.0 and σ = 0.4. If the weights of 22 randomly selected bricks are measured, what is the probability that the resulting point estimate of μ will be in the interval (109.9, 110.1)?
Suppose that components have weights that are normally distributed with μ = 341 and σ = 2. An experimenter measures the weights of a random sample of 20 components in order to estimate p. What is the probability that the experimenter's estimate of μ will be less than 341.5?
Unknown to an experimenter, the corrosion rate of a certain type of chilled cast iron has a standard deviation of 5.2. The experimenter measures the corrosion rates of 18 random samples of the chilled cast iron and estimates the mean corrosion rate. What is the probability that the experimenter's
Unknown to an experimenter, the failure time of a component has an exponential distribution with parameter λ = 0.02 per minute. The experimenter takes 110 components, and finds out how main of them last longer than one hour. This allows the experimenter to estimate the probability that a component
The pH levels of food items prepared in a certain way are normally distributed with a standard deviation of σ = 0.82. An experimenter estimates the mean pH level by averaging the pH levels of a random sample of n items. (a) If n = 5, what is the probability that the experimenter's estimate is
A company has installed 3288 flow meters throughout an extensive sewer system. Unknown to the company, 592 of these meters are operating outside acceptable tolerance limits, whereas the other 2696 meters are operating satisfactorily. The company decides to estimate the unknown proportion p of the
Consider a sample X1,..., Xn of normally distributed random variables with mean μ and variance σ2 = 7. (a) If n = 15, what is the probability that |μ - | ≤ 0.4? (b) What is this probability when n = 50?

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