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Introduction To Bayesian Statistics 3rd Edition William M. Bolstad, James M. Curran - Solutions

In this exercise we will test the hypothesis that the mean vector for bottle number 5 is not contained in a credible interval centered around the mean of the remaining observations. We can think of this as a crude equivalent of a hypothesis test for a single mean from a normal distribution. It is
The data in this exercise and those that follow can be downloaded from http://www.stat.berkeley.edu/ statlabs/data/babies.data. The variables in the data set are:[Minitab:] The data on the webpage needs to be saved as as a text le (*.txt) import the data into Minitab. It is important to click on
It is important to make sure that we are working with only the complete data; otherwise we have to have a model for the missing values.[Minitab:] Select Copy > Columns to Columns. . . from the Data menu. Enter c1-c7 in the Copy from columns: text box. Click on the Subset the data... button.
It is always useful to plot the data before we contemplate models. This sometimes can reveal features which we may not have noticed in the data, and can warn us of potential issues. A scatterplot matrix is a good first choice for multiple regression.[Minitab:] Select Matrix Plot. . . from the Graph
Use the Minitab macro BayesMultReg or the R function bayes.lm with a multivariate normal prior to t a multiple linear regression model to this data set. An initial choice of prior might be b0 = 0 and V0 = 106I7, where I7 is a 7 x 7 identity matrix. This is a very diffuse prior centered on zero.
Use the posterior mean and covariance of the regression coeffcients to test the hypothesis Ho: Bage Bweight
The following 8 observations come from a simple linear regression model where the variance \({σ}^{2}=2^{2}\) is known.(a) Use BayesLinReg in Minitab, or bayes.lin.reg in R, to nd the posterior distribution of the slope β when we use a \(\operatorname{normal}\left(\begin{array}{lll}0 &
What does r2 determine about the statistical significance of a relationship between a dependent and an independent variable?
Every December 15, a certain company offers its employees a choice of one of the following three health care plans for the new year beginning on January 1: An employee may keep her current plan or switch to a different plan. During the past several years the Human Resources Department (HRD) has
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 is independent of its past history. Hence, the condition of the machine can be modeled as a
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 can be represented by one of the following five states:Since the next state of the machine depends
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 proportion of defective items in a lot is not excessive. Instead of inspecting each item in a lot, a
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 up to two additional customers. Any customer who arrives when the waiting room is full does not
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 other three chairs are reserved for patients who wait. The arrival process is deterministic. One new
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 is being served. Customers may arrive at the store every 30 min, immediately before the hour and
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 number of laptops on hand at the start of the day is less than 1 (in other words, equal to 0),
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, that is, all have failed. Every IC that fails is replaced with a new one at the end of the year in
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. Let Xn be the volume of water in the dam at the end of week n. During week n, a volume of water
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 are tenure-track. Only research professors and applications professors have tenure. A tenure-track
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 are provided in the following sequence. A customer is given an eye examination by the optometrist
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 eliminated, and the remaining players participate in the next round. The contest ends when a single player
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 round, all players who allow their opponent to score a goal are eliminated, and the remaining
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 \($5\),000 or loses all her money. She starts with \($2\),000, and will wager \($1\),000 on each bet. Each
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 or \($4\),000, she will bet \($1\),000, and will win \($1\),000 with probability 0.4 or lose
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 4-year warranty period, but does not cover the cost of a replaced TV. The dealer discloses that 5%
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 the next stage, a defective item is scrapped, and an item of marginal quality is reworked at the
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 sequence of course correction signals is sent to it. The system has fi ve states, which are labeled
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 will never exceed \($20\). She has been informed that the price of the stock can be modeled as a
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 income earned this week. In any week, the married couple may sell the firm to either their adult son or
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 every week, a volume of water denoted by Wn flows into the dam. The stationary probability
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, managing investments under an existing retirement plan, or generating income during retirement. The
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 passes the test has her license renewed for 1 year. A P.E. who fails the test has her license
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 stationary probability distribution: Once the woman accepts an offer, the bidding stops. Her objective is
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 pays a fixed cost of \($60\). Every TV ordered costs the retailer \($100\). The daily holding cost
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 following transition probability matrix:A machine in excellent condition earns revenue of \($600\)
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 with the following transition probability matrix:The machine is observed at the start of each day.
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 to traverse an arc between two adjacent nodes. The origin, node 1, is entered at epoch 0. Nodes 2 and
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 condition of her computer as a Markov chain with the following three states:If her computer is
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 successfully completing continuing education courses offered by a professional or trade organization, or
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 0.05 0.20 0.2 0.3 0.1 2 0.09 0.03 0.18 0.1 0.4 0.2 3 0.12 0.02 0.06 0.5 0.2 0.1 [1 0 0 1[0.15 0.05
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 \(\mu, \tau\), and \(X_{+}\). Further, calculate (if relevant) the probabilities \(P(\mu
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 calculate \(P\left(T_{+k} \leq l\right)\) and \(P\left(N_{+1} \leq m\right)\).(a) Posterior
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 catch was 111 grouse. Use a neutral prior, and find the posterior probability distribution for
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 complex with 70 flats, and have looked into the documentation for the last semester (26 weeks), and find
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 in \(10 \mathrm{~km}\). Use neutral prior, and calculate the posterior probability distribution of
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 fields. The first field is \(t_{1}=1.9 \mathrm{~m}^{2}\) in area, and has \(n_{1}=0\) four-leaf
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 colonies. Use a neutral prior, and calculate the posterior probability distribution of \(\lambda\).
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 complex with 70 flats, and have looked into the documentation for the last semester (26 weeks), and find
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 colonies. Use neutral prior, and test the alternative hypothesis \(H_{1}: \lambda
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 \(\lambda_{\text {Lowlands }} \sim \gamma_{(111,23)}\). You have now spoken to 8 hunters from the Highlands, and their catch on
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 \(y\), the height (in \(\mathrm{cm}\) ) a golf ball bounces when dropped from a one-meter height, is
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 values are:Assume that \(y\), the height (in \(\mathrm{cm}\) ) a golf ball bounces when dropped from a
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 residential electricity accounts and looked at the total cost of electricity used over the three months of
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 to be trained using method \(\mathrm{A}\), and the other 10 are assigned to be trained using method
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 \(\left(A^{2}\right)\) and normal( \(\left(B^{2}\right)\), where \({ }^{2}=3^{2}\). Use independent 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 each of the factories' output and measures the butterfat. The results are:(a) We will assume that
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 \(\operatorname{normal}\left({ }_{1}{ }^{2}\right)\) and normal \(\left(2^{2}\right)\), where \({ }^{2}=4^{2}\). Use independent
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 one kilometer upstream from the power station, and one kilometer downstream from the power station,
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 and moves directly to the abomasum. Scientists wanted to develop a simple nonradioactive, noninvasive
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 glass from the crime scene. To make this comparison rigorous, we need to know the variability the index
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 control of these aspects of their lives. During those years, phrases such as women's liberation
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 generations of New Zealand women 25 years apart.(a) Out of 314 women in the age group 2529,171 were
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 women who report having experienced sexual intercourse before age 18 for the two generations of New
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 been married before age 22 for the two generations of New Zealand women.(a) Out of the 314 women in
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 instead of legally marrying). When we take this into account, are younger New Zealand women forming
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 age of 25. Find the posterior distribution of 1 , the proportion of New Zealand women in that age
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 circumcision status and self reported STI history using data from 25-year longitudinal study of a
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 the first treatment, at a later time when the first dose of \({ }^{13} \mathrm{C}\) had been
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 predict oxygen uptake. If so, then the estimated oxygen uptake based on the measured heart rate can

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