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Business Statistics
The condition of a machine, which is observed at the beginning of each day, can be represented by one of the following four states:The next state of the machine depends only on the present state and
Consider a machine that breaks down on a given day with probability p. A repair can be completed in 1, 2, 3, or 4 days with respective probabilities q1, q2, q3, and q4. The condition of the machine
Many products are classified as either acceptable or defective. Such products are often shipped in lots, which may contain a large number of individual items. A purchaser wants assurance that the
A credit counselor accepts customers who arrive at her office without appointments. Her office consists of a private room in which she counsels one customer, plus a small waiting room, which can hold
Consider a small clinic staffed by one nurse who injects patients with an antifl u vaccine. The clinic has space for only four chairs. One chair is reserved for the patient being vaccinated. The
Consider a small convenience store that is open 24 h a day, 7 days a week. The store is staffed by one checkout clerk. The convenience store has space for only three customers including the one who
Consider the following inventory system controlled by a (1, 3) ordering policy. An office supply store sells laptops. The state of the system is the inventory on hand at the beginning of the day. If
The following mortality data has been collected for a group of S0 integrated circuit chips (ICs) over a 4-year period:Observe that at the end of the 4-year period, none of the S0 ICs has survived,
A dam is used for generating electricity, controlling floods, and irrigating land. The dam has a capacity of 4 units of water. Assume that the volume of water stored in the dam is always an integer.
A certain college of management has five academic ranks for its faculty: instructor, assistant professor, associate professor, research professor, and applications professor. The first three ranks
A small optical shop is staffed by an optometrist and an optician. Customers may arrive at the shop every 30 min, immediately after the hour and immediately after the half-hour. Two stages of service
Three basketball players compete in a basketball foul shooting contest. The eligible players are allowed one foul shot per round. After each round, all players who miss their foul shots are
Three hockey players compete in a defensive contest. In each round, each eligible player is allowed one shot at the goal of the remaining player who has the highest career scoring average. After each
A woman needs \($5\),000 for a down payment on a condominium. She will try to raise the money for the down payment by gambling. She will place a sequence of bets until she either accumulates
Suppose that the woman seeking a \($5\),000 down payment in Problem 3.3 has the option of betting either \($1\),000 or \($2\),000. She chooses the following aggressive strategy. If she has \($1\),000
A consumer electronics dealer sells new flat panel televisions for $1,000. The dealer offers customers a 4-year warranty. The warranty provides free replacement of a television that fails within the
A production process contains three stages in series. An entering item starts in the first manufacturing stage. The output from each stage is inspected. An item of acceptable quality is passed on to
An unmanned expendable rocket is launched to place a communications satellite in either high earth orbit (HEO) or low earth orbit (LEO). HEO is the preferred destination. As the rocket is tracked, a
An investor buys a stock for \($10\). The monthly share price, rounded to the nearest \($5\), has been varying among \($0\), \($5\), \($10\), \($15\), and \($20\). She has been advised that the price
A married couple owns a consulting firm. The firm’s weekly income is a random variable, which may equal \($10\),000, \($15\),000, or \($20\),000. The income earned next week depends only on the
A dam is used for generating electric power and for irrigation. The dam has a capacity of 4 units of water. Assume that the volume of water stored in the dam is always an integer. At the beginning of
At 9:00 AM every day an independent financial advisor will see a client who is concerned about retirement. A client has one of the following three types of concerns: creating a new retirement plan,
Suppose that in a certain state, every licensed professional engineer (P.E.) is required to pass an annual profi ciency test in the practice of engineering in order to renew her license. A P.E. who
A woman wishes to sell a car within the next 4 weeks. She expects to receive one bid or offer each week from a prospective buyer. The weekly offer is a random variable, which has the following
A consumer electronics retailer can place orders for flat panel TVs at the beginning of each day. All orders are delivered immediately. Every time an order is placed for one or more TVs, the retailer
At the start of each day, the condition of a machine is classified as either excellent, acceptable, or poor. The daily behavior of the machine is modeled as a three-state absorbing unichain with the
Suppose that the condition of a machine can be described by one of the following three states: The daily behavior of the machine when it is left alone for 1 day is modeled as an absorbing unichain
Consider the following stochastic shortest route problem over a three-period planning horizon for the network shown below: The network has six nodes numbered 1 through 6. One time period is needed
At the beginning of every month, a woman executes diagnostic software on her personal computer to determine whether it is virus-free or is infected with a malignant or benign virus. She models the
Suppose that in a certain state, every registered professional engineer (P.E.) is required to complete 15 h of continuing professional development (CPD) every year. CPD hours may be earned by
Consider a regular Markov chain that has the following transition probability matrix: State 0 0 1 2 3 0 0.23 0.34 0.26 0.17 P = 1 0.08 0.42 0.18 0.32 2 0.31 0.17 0.23 0.29 3 0.24 0.36 0.34 0.06
Consider an absorbing multichain which has the following transition probability matrix expressed in canonical form: State 0 4 LO 5 1 2 3 0 1 0 0 0 0 0 4 0 1 0 0 0 0 P = 5 0 0 1 0 0 0 = where 1 0.15
T ∼ exp4.4. What are \(\mu_{T}, \sigma_{T}\), and \(P(T \leq 4)\) ?
T ∼ exp4.4. What are \(\mu_{T}, \sigma_{T}\), and \(P(T \leq 3.2)\) ?
T ∼ exp3.1. What are \(\mu_{T}, \sigma_{T}\), and \(P(T>1.2)\) ?
T ∼ exp4.4. What are \(\mu_{T}, \sigma_{T}\), and \(P(T \in\langle 0.15,0.28])\) ?
T ∼ expλ and \(E[T]=2.9\). What are \(\lambda, \mu_{T}\), and \(\sigma_{T}\) ?
T ∼ expλ. and \(\operatorname{Var}(T)=2.25\). What are \(\lambda, \mu_{T}\), and \(\sigma_{T}\) ?
T ∼ erl(2,5). What are \(\mu_{T}, \sigma_{T}\), and \(P(T \leq 1)\) ?
T ∼ erl(3,2.3). What are \(\mu_{T}, \sigma_{T}\), and \(P(T \leq 1.7)\) ?
T ∼ erl(5,3.1). What are \(\mu_{T}, \sigma_{T}\), and \(P(T>2.3)\) ?
T ∼ erl(2,4.4,λ). What are \(\mu_{T}, \sigma_{T}\), and \(P(T \in\langle 0.2,2])\) ?
T ∼ erl(2,λ). and \(E[T]=2.9\). What are \(\lambda, \mu_{T}\), and \(\sigma_{T}\) ?
T ∼ erl(3,λ). and \(\operatorname{Var}(T)=2.25\). What are \(\lambda, \mu_{T}\), and \(\sigma_{T}\) ?
X ∼ t(0,1,8). Find an \(x\) such that \(P(X \leq x)=0.005\).
X ∼ t(0,1,6). Find an \(x\) such that \(P(X \leq x)=0.9\).
X ∼ t(7,3.11). Find an \(x\) such that \(P(X \leq x)=0.95\).
X ∼ t(-3.14,7.2,23). Find an \(x\) such that \(P(X \leq x)=0.999\).
X ∼ t(4.4,3.1,7). \(Z=X-Y\) and \(Y \sim t_{(-0.1,2.2,18)}\). Find the probability distribution of \(Z=X-Y\).
X ∼ β(1,1). Find \(\mu_{X}, \sigma_{X}, P(X \leq 0.4)\), and graph the probability distribution.
X ∼ β(2,2). Find \(\mu_{X}, \sigma_{X}, P(X>0.6)\), and graph the probability distribution.
X ∼ β(2,3). Find \(\mu_{X}, \sigma_{X}, P(X \in\langle 0.3,0.6])\), and graph the probability distribution.
X ∼ β(3,2). Find \(\mu_{X}, \sigma_{X}, P(X \in\langle 0.4,0.65])\), and graph the probability distribution.
X ∼ β(53,22). Find \(P(X \leq 0.7)\), and \(p\) such that \(P(X \leq p)=80 \%\).
X ∼ β(108,72). Find \(P(X \leq 0.5)\), and \(p\) such that \(P(X \leq p)=95 \%\).
X ∼ β(17,42). Find \(P(X \leq 0.2)\) and \(p\) such that \(P(X \leq p)=10 \%\).
X ∼ β(43.19). Find \(P(X \geq 0.8)\), and \(p\) such that \(P(X \geq p)=90 \%\).
X ∼ β(128,81). Find \(P(X \geq 0.6)\), and \(p\) such that \(P(X \geq p)=99 \%\).
X ∼ β(491,396). Find \(\mu_{X}\) and \(\sigma_{X}\), and find \(P(X \in[0.45,0.50angle)\). (You may calculate this both exactly and with the Normal approximation.)
T ∼ weib(0.2,2). What are \(\mu_{T}, \sigma_{T}\), and \(P(T \geq 1)\) ?
T ∼ weib(2,5). What are \(\mu_{T}, \sigma_{T}\), and \(P(T \in(1,3))\) ?
T ∼ weib(0.5,3). What are \(\mu_{T}, \sigma_{T}\), and \(P(T \in(2,4))\) ?
You are given a prior or prior hyperparameters for a Gaussian process, and observational data, and values \(z\) and \(s\). Find (if relevant) the posterior (predictive) probability distribution of
You are given posterior distributions or hyperparameters for the rate \(\lambda\) of a Poisson process, and values \(k, l, m\). Calculate the predictive distribution for \(N_{+}\)and \(T_{+k}\), and
The daily catch for grouse hunters in an area is considered to be Poisson distributed with rate \(\lambda\). One day, you talked to 23 hunters from a certain part of the Lowlands, and their total
The number of plumbing gaskets that need changing every week in an apartment complex is assumed to follow a Poisson process with rate \(\lambda\). You are estimating this need for an apartment
The number of cracks in the tarmac per kilometer of road is assumed to follow a Poisson process with rate \(\lambda\).Your job is to find this rate for a lesser highway, and you have found 13 cracks
How many four-leaf clovers are there per square meter in a field of leaf clovers? Assume that the occurrence follows a Poisson process with rate \(\lambda\). You look at three independent leaf clover
The number of bacterial colonies per cubic centimeter in a certain polluted lake is assumed to be Poisson distributed with parameter \(\lambda\). You sample 1 deciliter, and find 157 bacterial
The number of plumbing gaskets that need changing every week in an apartment complex is assumed to follow a Poisson process with rate \(\lambda\). You are estimating this need for an apartment
The number of bacterial colonies per cubic centimeter in a certain polluted lake is assumed to be Poisson distributed with parameter \(\lambda\). You sample 1 deciliter, and find 157 bacterial
You have been told that 23 grouse hunters from a certain part of the Lowlands had a total catch of 111 grouse in a single day, and concluded that the catch rate there was \(\lambda_{\text {Lowlands
A statistician buys a pack of 10 new golf balls, drops each golf ball from a height of one meter, and measures the height in centimeters it returns on the rst bounce. The ten values are:Assume that
The statistician buys ten used balls that have been recovered from a water hazard. He drops each from a height of one meter and measures the height in centimeters it returns on the rst bounce. The
The local consumer watchdog group was concerned about the cost of electricity to residential customers over the New Zealand winter months (Southern Hemisphere). They took a random sample of 25
The Human Resources Department of a large company wishes to compare two methods of training industrial workers to perform a skilled task. Twenty workers are selected: 10 of them are randomly assigned
A consumer testing organization obtained samples of size 12 from two brands of emergency ares and measured the burning times. They are:(a) We will assume that the observations come from normal
The quality manager of a dairy company is concerned whether the levels of butterfat in a product are equal at two dairy factories which produce the product. He obtains random samples of size 10 from
Independent random samples of ceramic produced by two di erent processes were tested for hardness. The results were:(a) We will assume that the observations come from \(\operatorname{normal}\left({
A thermal power station discharges its cooling water into a river. An environmental scientist wants to determine if this has adversely a ected the dissolved oxygen level. She takes samples of water
Cattle, being ruminants, have multiple chambers in their stomachs. Stimulating speci c receptors causes re ex contraction of the reticular groove and swallowed uid then bypasses the reticulo-rumen
Glass fragments found on a suspect's shoes or clothes are often used to connect the suspect to a crime scene. The index of refraction of the fragments are compared to the refractive index of the
The last half of the twentieth century saw great change in the role of women in New Zealand society. These changes included education, employment, family formation, and fertility, where women took
Are younger New Zealand women more likely to be in paid employment than previous generations of New Zealand women? To shed light on this question, we will look at the current employment status of two
Are younger New Zealand women becoming sexually active at an earlier age than previous generations of New Zealand women? To shed light on this question, we look at the proportions of New Zealand
Are younger New Zealand women marrying at a later age than previous generations of New Zealand women? To shed light on this question, we look at the proportions of New Zealand women who report having
Family formation patterns in New Zealand have changed over the time frame covered by the survey. New Zealand society has become more accepting of couples co-habiting (living together before or
Are young New Zealand women having their children at a later age than previous generations?(a) Out of the 314 women in the age group 25 29, 136 report having given birth to their rst child before the
Previous research has suggested that the childhood circumcision of males may be a protective factor against the acquisition of sexually transmitted infections (STI). Fergusson et al. (2006) relate
The experiment described in Exercise 13.6 was repeated on another set of 7 cows (McLeay et al., 1997). However, in this case, the second treatment was given to the same set of 7 cows that were given
A researcher measured heart rate \((x)\) and oxygen uptake \((y)\) for one person under varying exercise conditions. He wishes to determine if heart rate, which is easier to measure, can be used to
A researcher is investigating the relationship between yield of potatoes \((y)\) and level of fertilizer \((x\).\() She divides a eld into eight plots of equal\) size and applied fertilizer at a di
A researcher is investigating the relationship between fuel economy and driving speed. He makes six runs on a test track, each at a di erent speed, and measures the kilometers traveled on one liter
The Police Department is interested in determining the e ect of alcohol consumption on driving performance. Twelve male drivers of similar weight, age, and driving experience were randomly assigned
A study on air pollution in a major city measured the concentration of sulfur dioxide on 25 summer days. The measurements were:(a) Form a stem-and-leaf diagram of the sulfur dioxide measurements.(b)
Dutch elm disease is spread by bark beetles that breed in the diseased wood. A sample of 100 infected elms was obtained, and the number of bark beetles on each tree was counted. The data are
A manufacturer wants to determine whether the distance between two holes stamped into a metal part is meeting speci cations. A sample of 50 parts was taken, and the distance was measured to nearest
The manager of a government department is concerned about the efciency in which his department serves the public. Speci cally, he is concerned about the delay experienced by members of the public
A random sample of 50 families reported the dollar amount they had available as a liquid cash reserve. The data have been put in the following frequency table:(a) Construct a histogram of the
In this exercise we see how the default settings in for producing boxplots in Minitab and in \(\mathrm{R}\) can be misleading because they do not take the sample size into account. We will generate
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