Mathematics And Statistics For Science 1st Edition James Sneyd, Rachel M. Fewster, Duncan McGillivray - Solutions

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32. In each case below, determine whether the set function, P, is a probability set function.a. PðAÞ ¼ 1 91 Xx2A x2 for A S; S ¼ f g 1; 2; 3; 4; 5; 6b. PðAÞ¼ Z x2A:25e:25xdx; where A is any Borel subset of S¼ ½ Þ 0;1c. PðAÞ ¼ X x2A:3x:71x for A S; S ¼ f g 0; 1d. PðAÞ ¼ Z x2A
31. Let Ai; i ¼ 1; :::; 4, represent four events in a sample space, S . For each of the situations below, determine which assignment of probabilities are actually possible(i.e., do not contradict Kolmogorov’s axioms), and which are not. Justify your answers.a. P Að Þ¼ 1 :3; P Að Þ¼ 2 :3; P
30. The Port Authority of a large East Coast City is investigating the traffic flow in and out of a large train station in the middle of the city. There are six entry gates and six exit gates that travelers can use to enter or leave the train station. An experiment is to be conducted to observe the
29. BuyOnLine is a large internet-based online retailer that maintains four different teams of sales representatives. The ages of unpaid invoices from each of the four sales teams is summarized in the table below.a. If an invoice is selected randomly from the pooled set of invoices, what is the
28. Consider the experiment of tossing a fair coin (meaning heads and tails are equally likely on each toss) three times and observing the sequence of heads and tails that results. Let H denote heads and T denote tails.a. Define the sample space, S, for this experiment.b. Let A be the event that at
27. SUPERCOMP, a retail computer store, sells personal computers and printers. The number of computers and printers sold on any given day varies, with the probabilities of the various possible sales outcomes being given by the following table:
26. Let S ¼ [0, 5] be a sample space containing all possible values of the daily quantity demanded of electric power for a large midwestern city in the summer months.The units of measurement are millions of megawatts, and the capacity of the power grid is five million megawatts.Answer the
25. The management of the AJAX Microchip Company(mentioned in Problem 24) is interested in increasing quality control at the plant, and is considering the purchase of a testing device that can determine when a memory chip is faulty. In particular, the specifications on the device are as
24. The AJAX Microchip Company produces memory chips for personal computers. The company’s entire production is generated from two assembly lines, labeled I and II. Assembly line I uses more rapid assembly techniques and produces 80 percent of the company’s output, while assembly line II
23. A box contains four different computer disks, labeled 1, 2, 3, and 4. Two disks are selected at random from the box “with replacement,” meaning that after the first selection is made, the selected disk is returned to the box before the second selection is made. “At random”means that all
22. The Smith Floor Wax Company manufactures and sells industrial-strength floor wax in the wholesale market for home care products. The factory produces 10,000 gal of floor wax daily and currently has an inventory of 5,000 gal of floor wax in its warehouse. If sales of floor wax exceed production,
21. For each case below, determine whether or not the real-valued set function P(A) is in fact a probability set function:a. Sample space S ¼ {1, 2, 3, 4, 5, 6, 7, 8}, Event space U ¼ fA : A Sg , Set Function P(A) ¼Sx2Aðx=36Þ for A 2 U.b. Sample space S ¼ [0, 4), Event Space U ¼ fA : A is an
20. A large sack contains 1,000 flower seeds consisting of 300 carnations and 700 impatiens. Of the 300 carnation seeds, 200 will produce red flowers and 100 will produce white flowers. Of the 700 impatiens seeds, 400 will produce red flowers and 300 will produce white flowers.a. If you randomly
19. The diagram below indicates how probabilities have been assigned to various subsets of the sample space S:
18. The ACME Computer Co. operates three plants that manufacture notebook computers. The plants are located in Seattle, Singapore, and New York. The plants produce 20, 30, and 50 percent of the company’s output, respectively. ACME attaches the labels “Seattle,” “SING,” or“NY” to the
17. The ZAP Electric Co. manufactures electric circuit breakers. The circuit breakers are produced on two different assembly lines in the company’s Spokane plant.Assembly line I is highly automated and produces 85 percent of the plant’s output. Assembly line II uses older technology that is
16. Answer the following questions regarding the validity of probability assignments. If you answer false, explain why the statement is false.a. If P(A) ¼ .2, P(B) ¼ .3, and A\B ¼ ; , then P(A[B)¼ .06. True or False?b. If A\B ¼ ; and P(B) ¼ .2, then P(A|B) ¼ 0. True or False?c. If P(B) ¼
15. A company manages three different mutual funds.Let Ai be the event that the ith mutual fund increases in value on a given day. Probabilities of various events relating to the mutual funds are given as follows:
14. A clerk is maintaining three different files containing job applications submitted for three different positions currently open in the firm at which the clerk is employed. One file contains two completed applications, one file contains one complete and one incomplete application, and the third
13. A large-scale firm specializing in providing temporary secretarial services to corporate clients has completed a study of the main reason why secretaries become dissatisfied with their work assignments, and how likely it is that a dissatisfied secretary will quit her job. It was found that 20
12. A large midwestern bank has devised a math aptitude test that it claims provides valuable input into the hiring decision for bank tellers. The bank’s research indicates that 60 percent of all tellers hired by midwestern banks are classified as performing satisfactorily in the position at
11. Prove that the set function defined by PðAjBÞ ¼ PðA \ BÞPðBÞ for PðBÞ 6¼ 0 is a valid probability set function in the probability space{B,UB,P(|B)}, where UB is the event space for the sample space B.
10. The probability that a stereo shop sells at least one amplifier on a given day is .75; the probability of selling at least one CD player is .6; and the probability of selling at least one amplifier and at least one CD player is .5.a. What is the probability that the stereo shop will sell at
9. A computer manufacturing firm accepts a shipment of CPU chips from its suppliers only if an inspection of 5 percent of the chips, randomly chosen from the shipment, does not contain any defective chips. If a shipment contains five defective chips and there are 1,000 chips in the shipment, what
8. A large electronics firm is attempting to hire six new electrical engineers. It has been the firm’s experience that 35 percent of the college graduates who are offered positions with the firm have turned down the offer of employment. After interviewing candidates for the positions, the firm
7. A computer manufacturing firm produces three product lines: (1) desktop computer systems, (2) notebook computers, and (3) subnotebook computers. The sales department has convened its monthly meeting in which the four staff members of the department provide the department manager with their
6. Schneider’s Plumbing and Heating, located in Fargo, North Dakota, has 300 accounts receivable distributed as follows:Current 1–30 days past due 31–60 days past due 61–90 days past due Sent for collection 140 80 40 25 15 An auditor is coming to inspect Schneider’s financial records.
5. The manager of the cost accounting department of a large computer manufacturing firm always tells three jokes during her monthly report to the board of directors in an attempt to inject a bit of levity into an otherwise sobering presentation. She has an inventory of a dozen different jokes from
4. Assign probabilities to the events a to f in the preceding question, but include the condition “given that the individual chosen is 30 years old.”
3. The sales team of a large car dealership in Seattle consists of the following individuals:Name Sales experience Age Education Married Tom 4 years 34 High school Yes Karen 12 years 31 < High school No Frank 21 years 56 College grad Yes A customer visiting the dealership randomly chooses one of
2. For each of the sample spaces you have defined above, indicate whether the sample space is finite, countably infinite, or uncountably infinite. Justify your answers.
1. Define an appropriate sample space for each of the experiments described below:a. At the close of business each day, the Acme Department Store’s accountant counts the number of customer transactions that were made in cash. On a particular day, there were 100 customer transactions at the
The data at the beginning of Chapter 36 showed IQ scores from 38 subjects who were selected at random from a population of university students. The mean IQ score in the data is 111.3, and the sample standard deviation is 22.6. Specify an appropriate test for whether the mean IQ in the population of
The Avogadro constant is a fundamental quantity in chemistry specifying the number of molecules in one mole of a substance.It can only be determined by experiment, for example using an electrochemical cell. The accepted value is 6.022 × 1023 .A chemist conducts five repeats of an experiment to
A popular experiment in physics classes is to estimate the acceleration due to gravity, g, for example using pendulums or ticker-tape machines. The accepted value of g is 9.81 ms−2.A student runs an experiment to estimate g ten times and gets the estimates 9.62 9.75 9.86 9.58 9.85 9.82 9.83 10.03
Following from Exercise 41.6, suppose Y ∼ Gamma(k =24, λ = 1).a. What is E(Y)?b. Again we wish to test H0 : µ = 25. Compared with the simulation in Exercise 41.6, what do you expect to happen to the P-value for this test?c. Simulate n = 10 observations fromY and test H0 using the t.test
Suppose X ∼ Gamma(k = 20, λ = 1).a. What is E(X)?b. Give R commands to simulate a sample of 10 observations from the distribution of X.c. Run the commands in R to generate your data of 10 observations. Using direct calculation, test whether the sample gives evidence against H0 : µ = 25, and
If there is an outbreak of measles in a school, suppose that children who have not been immunised have probability 0.9 of catching it, whereas children who have been immunised have probability 0.05 of catching it. In the population under study, the immunisation rate of children against measles is
Standardised IQ tests are constructed such that human IQ scores follow a normal distribution with mean 100 and standard deviation 15. The organisation Mensa is for people with an IQ in the top 2% of the population. Using the R command qnorm, specify the command you would use to find the qualifying
Let X be a continuous random variable with PDF fX(x) =(2x for 0 < x < 1 , 0 otherwise.a. Find E(X) and Var(X).b. Suppose X1, X2, . . . , X40 are independent, each with the same distribution as X, and let Xsum = X1 + X2 + . . . + X40 .What is the approximate distribution of Xsum?c. What is the
By looking at Fig. 41.11, say what answers you expect for each of the following computations. Confirm your answers by running the commands in R.a. qt(0.025, 19)b. qt(0.025, 49)c. qt(0.975, 49)d. pt(2.09, 19)e. pt(2.09, 19) - pt(-2.09, 19)
Sketch the PDF of the Normal(µ = −3, σ2 = 16) distribution.Mark the following features on your sketch.a. Approximate start and end points for the distribution, within which nearly all observations will fall.b. The mean, µ.c. An area corresponding to a probability of 0.5.
As we saw in Exercise 40.19, the half-life of a radioactive substance is the time taken for half of its atoms to decay. If atoms decay according to an exponential distribution with rateλ per unit time, then the half-life can be calculated from t1/2 =− ln(0.5)λ.In this question we’ll use our
The dive depths of sperm whales range from a few hundred metres to as much as 2 km. Suppose dives follow a Gamma(k =2, λ) model. We want to estimate λ.a. Let X ∼ Gamma(2, λ). Sketch on a single plot the PDF of X for two values of λ, labelling which curve corresponds to the larger value of λ
In this question we’ll use simulations to explore maximum likelihood estimation for the waiting time, λ, in a Poisson process.Suppose we have k See Section 40.4.1 for more details. independent observations from the Exponential(λ) distribution. We can add these up and treat them as a single
In this question we’ll see how the memoryless property of the exponential distribution works.We’ll use X ∼ Exponential(λ =4) for illustration.a. Find an expression for P(X ≥ x) for any x > 0.b. Now suppose that we condition on the event {X ≥ 2}.Explain in words what P(X ≥ 2 + x | X ≥
The atomic nucleus of plutonium-239 (239Pu) is unstable, and it spontaneously decays to form uranium-235 (235U):This is the first step in the radioactive decay chain of nuclear reactors. 239Pu −→ 235U.The time for an atom to decay can be modelled by the Exponential(λ) distribution, where the
As well as following a Poisson process in time, we can also use a Poisson point process to model the location of volcanic eruptions in space. The Auckland volcanic field has a total area of 600 km2. We know the next eruption will happen eventually, but we don’t know where. Assume it’s equally
Assume that volcanic eruptions in the Auckland volcanic field follow a Poisson process with rate λ =1 1000 eruptions year−1.Eruptions can happen anywhere in the volcanic field. The last eruption, about 600 years ago, created a new island called Rangitoto.a. Let Y be the time from Rangitoto to
Suppose we observe a Poisson process for a length of time t.The number of events is X ∼ Poisson(λt). We can show that the maximum likelihood estimator of λ isλb=X t.Say whether each of the following statements is true or false.Show your working or reasoning.a. E(X) = λt and Var(X) = λt.b.
A shuttle bus waits at the airport until it is loaded with passengers. Passengers arrive singly according to a Poisson process with rate λ = 20 passengers per hour. The bus is full when eight passengers have arrived.a. LetT be the time the driver has to wait for the first passenger to arrive, in
A car windscreen is a rectangle measuring 1.5 m by 0.8 m. One wiper clears an area of 0.6 m2, the other wiper clears 0.4 m2, and the area of overlap cleared by both wipers is 0.1 m2. If a splat lands anywhere on the windscreen with equal probability, what is the probability it will lie within reach
A roulette wheel has 37 blocks of equal sizes marked around the rim. Each block is marked with a unique number from 0 t 36. Eighteen of the blocks are red, eighteen are black, and one is green. When the wheel is spun, it’s equally likely to land at any angle between 0◦and 360◦. Players can
Following from Exercise 40.11, let Y be the number of meteors you fail to see before you’ve seen three meteors. What is the distribution of Y?
Meteors can be seen in the night sky as shooting stars according to a Poisson process with rate λ = 3 meteors per hour. Suppose you’re watching for meteors, and you see each one with probability p = 0.4, independently of all others. Say whether each of the following statements is true or false,
Following from Exercise 40.9, let X be the distribution of the time waited between customers, and let T be the time waited until three customers have arrived. Say whether each of the following statements is true or false, showing any working.a. T = X1 + X2 + X3 where X1, X2, X3 are independent with
Customers arrive at a bank according to a Poisson process with rate λ = 60 customers per hour. Say whether each of the following statements is true or false.a. The Poisson(60) distribution gives the number of customers to arrive in one hour.b. The Poisson(30) distribution gives the number of
The chi-squared distribution with parameter ν is a special case of the gamma distribution, with k = ν/2 and λ = 1/2. Using your knowledge of the gamma distribution, write down expressions for the mean and variance of the Chisquared(ν)distribution.
Let X ∼ Exponential(λ), with observation X = x. Find an expression for the maximum likelihood estimator of λ.
Let X ∼ Exponential(λ). Using the method of integration by parts from Chapter 23, show that E(X) =1λ; Var(X) =1λ2.
Sketch the PDF curves for the Exponential(2) and the Exponential(4) distributions on the same plot, marking any known values of the curve and clearly indicating which curve is which.Similarly, sketch the two CDF curves on the same plot, marking all important features.
Let W ∼ Uniform[−1, 4].a. Using the formulas given in Exercise 40.3, find the mean and variance of W. Quote your answers to 1 decimal place.b. Write down suitable R commands to check your answers for E(W) and Var(W) by simulation. Run the commands in R and verify that they give the results you
Let Y ∼ Uniform(a, b). Write down the PDF, fY(y). Hence use integration to show that E(Y) =b + a 2; Var(Y) =(b − a)2 12.You might wish to use the factorisation b3 − a 3 = (b − a)(b 2 + ab + a 2).
Let X ∼ Uniform(a, b). The PDF of X is a constant k for a < x < b:fX(x) = k, for a < x < b.a. Use integration to show that k =1 b−a.b. Using integration where necessary, show that the CDF of X is FX(x) = P(X ≤ x) =0, if x ≤ a, x−a b−a, if a < x < b, 1, if x
Let X ∼ Uniform(0, 1). The PDF of X is a constant, k, for 0 < x < 1:fX(x) = k, for 0 < x < 1.Use integration to show that the constant is k = 1.
Let X be the distribution of car speeds along a stretch of highway, and let Y be the corresponding stopping distances. The stopping distance is the distance it takes for a car to come to a stop in an emergency. X is measured in kilometres per hour, and Y is measured in metres. The two variables are
The Pareto distribution is often used to model the distribution of personal income in a society. It has a lower limit m, which could for example denote the minimum wage, and a long right tail reflecting very high incomes at the upper end. A Pareto distribution with a minimum income of $20 per hour
Fig. 39.12 shows the frequency histogram of elite male 100 m sprint times up to 2002. Based on the histogram, write a short description of the most important features of the distribution, and draw a rough sketch of the PDF and CDF curves for the underlying distribution of elite male sprint times.
By taking readings from the curves shown in Figs 39.6 and 39.7, give approximate answers to the following questions for the 1960s and for the 2010s. For each question, say whether you’re using the PDF or the CDF, and explain how you’re using it to get the answer.a. For someone who died in that
Let bθ be a maximum likelihood estimator for a parameter θ.Say whether each of the following statements is true or false.a. θ is a random variable.b. bθ is a random variable.c. Var(bθ) is a random variable.d. sd(bθ) is a random variable.e. se(bθ) is a random variable.f. The upper and lower
The Loch Ness monster is a fearsome dragon-like beast, but fortunately it is rarely seen. It is, however, sometimes heard.The call of the Loch Ness monster is a terrifying screech, with loudness X, measured in Nessiebels. The PDF of X is fX(x) = k x2 for 0 < x < 2, where k is a normalising
Let X be a continuous random variable with PDF fX(x) =(x 29, for 0 < x < 3, 0, otherwise.Let FX(x) be the CDF of X. For each of the following statements, say whether it is correctly written. If not, say what is wrong with it.a. FX(x) =∫ s−∞fX(x) dx.b. FX(x) =∫ x−∞fX(s) ds.c. FX(x) =∫
Researchers conducted an investigation into the impact of safety adverts on driver behaviour. Either 0, 1, or 2 large safety adverts were installed on a 5 km stretch of motorway preceding a speed camera. In each case, the adverts were installed for one week, and the number of speeding drivers
Many patients drop out of IVF programmes without achieving a successful pregnancy. In this question we’ll see how to use the data from these patients when we calculate maximum likelihood estimates, along with the data from patients who did achieve a successful outcome.a. Let Z ∼ Geometric(p).
Reproduction can also be a tricky business for humans, leading to widespread adoption of assisted fertility practices such as in vitro fertilization (IVF). Each cycle of IVF might or might not lead to a pregnancy. Success rates are currently in the region of 30% per cycle for women in their late
A model for reproduction works as follows. A population starts with a single individual. At the end of one generation, this This type of model is relevant in many and varied scenarios. The population might consist of animals, or diseased cells in a tumour, or a computer virus that aims to spread
You observe a fish making 5 failed jumps before reaching the top of a waterfall. How would you estimate the probability of success, p, for each jump?
The list (a) to (d) below describes some real situations. Each situation can be modelled by one of the following approaches:Option Q is technically known as a mixture model, because the observed outcomes are a mixture of observations from different classes.Option R is called a regression model.P.
Let X ∼ Poisson(λ), and suppose the maximum likelihood estimate of λ based on X is λb = 30 with 95% confidence interval (19, 41). Say whether each of the following statements is true or false. If it is false, say what is wrong with it.a. About 95% of the time, the true value of λ lies between
Let X ∼ Geometric(p), with observation X = x. Find an expression for the maximum likelihood estimator of p.
As we mentioned for the binomial distribution (see Chapter 34, Exercise 34.3), we usually work with the log-likelihood function rather than the likelihood function for finding the MLE. Write out an expression for the Poisson log-likelihood, log {L(λ ; x)}, where as usual log indicates the natural
Write out the R command to plot the likelihood function L(λ ; 95) for an observation X = 95 from X ∼ Poisson(λ).Plot the likelihood function for 50 ≤ λ ≤ 150, and verify by inspection that the curve appears to reach its maximum where you think it should.
Let X ∼ Poisson(λ), with observation X = x. Write down the likelihood function, L(λ ; x), and find expressions for both the maximum likelihood estimate and the maximum likelihood estimator of λ.
The police set up a breath-testing checkpoint on a Saturday night. All motorists going through the checkpoint are stopped and breath-tested. Assume each motorist tests positive for excess alcohol with probability p, independently of other motorists, and an average of 300 motorists per hour pass
Volcanic eruptions in the Auckland region are modelled using a Poisson distribution with rate λ = 1 eruption every 1000 years. Let Y ∼ Poisson(1). Which of the following statements gives the correct interpretation of Y?a. Y is the number of eruptions that occur in a 1000-year period.b. Y is the
Using R, plot the probability function for the following Poisson distributions using a line plot.Use plot(..., type="o") to get a line plot with plotted points highlighted. For each Poisson plot, pick a large value of n and an appropriate value of p, and superimpose a binomial probability
Let X ∼ Binomial(5, p), and let Y ∼ Geometric(p). You run two independent sets of Bernoulli trials to make one observation from X and one from Y. Suppose you observe X = 1 and Y = 4.Use S to denote a success in any trial, and F to denote a failure.Say whether each of the following statements is
A box contains 10 balls, of which 5 are white and 5 are black.You take out 4 balls, one after the other, without putting them back. Let X be the number of white balls you take out. Does X have a binomial distribution? If so, give its parameters. If not, say why.
Specify the distribution and parameters of the random variable in each of the following scenarios.a. You practise shooting a ball through a net. You succeed with probability p on each attempt, independently of other attempts. Define X to be the number of misses before you score ten hits.b. You
In Exercise 34.9 we met an apple-grower who treated his apples to reduce the skin disorder called scald. His apples developed scald with probability 1/5 if treated, but with probability 3/5 if untreated. Treatment could not be observed directly, but we were able to observe the number X of apples
In Exercise 34.8 we saw a tactic that researchers can use to encourage honest answers to sensitive questions, for example when investigating drug use among teenagers. The answer of each respondent is known, but not the question to which they are responding. Defining X to be the number of "yes"
Explain why histograms become smooth and curve-like in large samples. You might find it helpful to consider the height of a bar on the interval [a,b) as a random variable, whose distribution depends on a parameter p that you should specify. As the sample size gets larger, how does the standard
In Exercise 30.7, we investigated whether snorers are more likely to have nightmares than non-snorers. We suspected there was no relationship between snoring and nightmares. Now that we’ve studied variance, we have the tools to investigate this pressing question in more depth.The sample data
Who can catch a liar? Not everyone you might think, according to a well-known study which investigated the ability of police, judges, secret service agents, and university students to distinguish between videos of liars and truth-tellers. By random guesswork each participant would be expected to
The Brexit poll featured at the beginning of this chapter surveyed 3766 voters, of whom 51% said they were voting to remain in the EU. Let p be the true proportion of voters supporting the remain option. Find the margin of error of the poll, and hence calculate an approximate 95% confidence
The Drake equation is a lighthearted way of estimating the number of extraterrestrial civilisations that currently exist in the Milky Way galaxy, using probabilistic arguments such as the fraction of stars with planets, the probability a planet is suitable for supporting life, the probability a
Let X and Y be random variables. Say whether each of the following statements is true or false, in general.a. Var(X + Y) = Var(X) + Var(Y) always.b. Var(X + Y) = Var(X) + Var(Y) if X and Y have the same distribution.c. Var(X +Y) = Var(X)+ Var(Y) if X and Y are independent.d. Var(5X + 3) = 25Var(X).
Let X andY be respectively the number of cups of tea and coffee a randomly-chosen classmate of yours drank at breakfast-time today. Assume that none of your classmates would ever drink both tea and coffee at a single breakfast event, but that some of them drink tea and some drink coffee. Say
Again let X ∼ Binomial(200, p), where p is to be estimated. In the following list, some of the quantities are known numbers, some are unknown but fixed numbers, and some are random variables that depend upon the observation X. Give the correct classification for each quantity.a. p.b. pb.c.
Again let X ∼ Binomial(200, p), and suppose we make the observation X = 81. Which of the following statements correctly describes the maximum likelihood estimate of p?a. The MLE is the probability that X = 81.b. The MLE is the value of p that gives X = 81 the lowest variance.c. The MLE is the

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Mathematics And Statistics For Science 1st Edition James Sneyd, Rachel M. Fewster, Duncan McGillivray - Solutions

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