Introduction To Bayesian Statistics 1st Edition William M. Bolstad - Solutions

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(a) Find the population variance and standard deviation. (b) Find the mean absolute deviation. (e) Find the range. (d) Find the interquartile range.
Refer to the percentage assessments data of Exercise
Refer to the data of Exercise 1 on weekly complaints received by a store's customer ser- vice department over a 10-week period. (a) Find the population variance and standard deviation. (b) Find the mean absolute deviation. (c) Find the range. (d) Find the interquartile range.
Develop a realistic business example in which the most appropriate measure of central tendency is (a) The mean. (b) The median, (c) The mode.
A sample of twelve senior executives found the following results for percentage of total compensation derived from bonus payments: 15.8 7.3 28.4 18.2 15.0 24.7 13.1 10.2 29.3 34.7 16.9 25.3 (a) Find the sample median. (b) Find the sample mean.
A department store chain has ten stores in a state. After a review of sales records, it was found that compared with the same period last year, the following percentage increases in dollar sales had been achieved over the Christmas period this year: 10.2 3.1 5.9 7.0 3.7 2.9 6.8 7.3 8.2 4.3 For this
A corporation recruiting business graduates was particularly interested in hiring numerate graduates. To check on the numeracy of applicants, a test of fifty questions was developed. In a pilot study, this test was administered to a sample of ten recent business graduates, re- sulting in the
A sample of eight students, living in campus dormitories, were asked to rate on a scale from one (poor) to seven (excellent) the quality of dormitory food service. The ratings were 2 4 2 3 5 4 3 2 (a) Find the sample mean rating. (b) Find the sample median.
The ten business economists in a sample were asked to predict the percentage growth in the consumer price index over the next year. The forecasts were 3.6 3.1 3.9 3.7 3.5 3.7 3.4 3.0 3.6 3.4 (a) Find the sample mean prediction. (b) Find the sample median.
A company owns twelve commercially zoned parcels of land. The assessment rates (in) percentages) assigned to these in 1994 were 21 22 27 36 22 29 22 23 22 28 36 33 For this population: (a) Find the mean percentage assessment rate. (b) Find the median of these percentage rates. (c) Find the mode of
A department store manager is interested in the number of complaints received by the cus- tomer service department about the quality of electrical products sold by the store. Records over a 10-week period yield the data shown in the table. WEEK NUMBER OF COMPLAINTS WEEK NUMBER OF COMPLAINTS 3 4 8
Does minimum wage legislation affect the level of unemployment?
Are utilities more profitable in areas where they have local monopoly power than where they are subject to competition?
Are companies whose dividends are a high percentage of total cash flow viewed as high or low risk?
If General Motors increases the price of mid-size cars by 5%, what will be the effect on the sales of these cars?
Does the rate of growth of the money supply influence the inflation rate?
You are going take a sample of 500 gm blocks of cheese. You know they are approximately normal (µ, σ2) where σ2 = 22. You have a precise normal(502, 12) prior for µ because this is what the the process is set for. However, you have a nagging doubt that maybe the machine needs adjustment, so you
You are going take a sample of measurements of specific gravity of a chemical product being produced. You know the specific gravity measurements are approximately normal (µ, σ2) where σ2 = .0052. You have precise normal(1.10, .0012) prior for µ because the manufacturing process is quite
You are going to conduct a survey of the students in your university to find out whether they read the student newspaper regularly. Based on your friends opinions, you think that a strong majority of the students do read the paper regularly. However, you are not sure your friends are representative
You are going to conduct a survey of the voters in the city you live in. They are being asked whether or not the city should build a new convention facility.You believe that most of the voters will disapprove the proposal because it may lead to increased property taxes for residents. As a resident
In the previous two problems we found the posterior distribution of the slope of y on x, the rate of weevil infestation for endophyte infected and noninfected ryegrass. Let β1 be the slope for noninfected ryegrass, and let β2 be the slope for infected ryegrass(a) Find the posterior distribution
For ryegrass that was not infected with endophyte the following data were observed:(a) Plot a scatterplot of number of ryegrass plants versus the infestation rate.(b) The relationship between infestation rate and number of ryegrass plants is clearly nonlinear. Look at the transformed variable y =
Endophyte is a fungus Neotyphodium lolli that lives inside ryegrass plants. It does not spread between plants, but plants grown from endophyte-infected seed will be infected. One of its effects is that it produces a range of compounds that are toxic to Argentine stem weevil Listronotus bonariensis,
In Chapter 3, Exercise 7, we were looking at the relationship between log(mass)and log(length) for a sample of 100 New Zealand slugs of the species Limax maximus from a study conducted by Barker and McGhie (1984.) These data are in the Minitab worksheet slug.mtw. We identified observation 90 that
is known. Use a normal (0, 102)prior for β. What is the posterior distribution of β.(f) Find a 95% credible interval for β.(g) Perform a Bayesian test of H0 : β ≤ 0 versus H1 : β > 0 at the 95 % level of significance.
A textile manufacturer is concerned about the strength of cotton yarn. In order to find out whether fiber length is an important factor in determining the strength of yarn, the quality control manager checked the fiber length (x) and strength (y) for a sample of 10 segments of yarn. The results
The Police Department is interested in determining the effect of alcohol consumption on driving performance. Twelve male drivers of similar weight, age, and driving experience were randomly assigned to three groups of four. The first group consumed two cans of beer within 30 minutes, the second
A researcher is investigating the relationship between fuel economy and driving speed. He makes six runs on a test track, each at a different speed, and measures the kilometers travelled on one liter of fuel. The speeds (in kilometers per hour)and distances (in kilometers) are recorded below:(a)
A researcher is investigating the relationship between yield of potatoes (y) and level of fertilizer (x.) She divides a field into eight plots of equal size and applied fertilizer at a different level to each plot. The level of fertilizer and yield for each plot is recorded below:(a) Plot a
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 be used in
One of the advantages of Bayesian statistics is that evidence from different sources can be combined. In Exercise 6 and Exercise 14, we found posterior distributions of µd using data sets from two different experiments. In the first experiment, the two treatments were given to two sets of cows,
The experiment described in Exercise 6 was repeated on another set of 7 cows, McLeay, Carruthers, and Neil (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 13C had been eliminated from
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 first child before the age of 25. Find the posterior distribution of π1, the proportion of New Zealand women in that
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 family
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 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 the
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 Zealand women who report having experienced sexual intercourse before age 18 for the two generations of New Zealand
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 25-29, 171
Have the educational qualifications of younger New Zealand women changed from those of previous generations of New Zealand women? To shed light on this question, we will compare the educational qualifications of two generations of New Zealand women 25 years apart. The women in the age group 25-29
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 is 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
Cattle being ruminants have multiple chambers in their stomaches. Stimulating specific receptors causes reflex contraction of the reticular groove and swallowed fluid then bypasses the reticulo-rumen and moves directly to the abomasum. Scientists wanted to develop a simple nonradioactive,
A thermal power station discharges its cooling water into river. An environmental scientist wants to determine if this has adversely affected the dissolved oxygen level. She takes samples of water one kilometer upstream from the power station, and one kilometer downstream from the power station,
Independent random samples of ceramic produced by two different processes were tested for hardness. The results were:(a) We will assume that the observations come from normal(µ1, σ2) and normal(µ2, σ2), where σ2 = .42. Use independent normal (m, s2) prior Find the posterior distributions of
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 the
A consumer testing organization obtained samples of size 12 from two brands of emergency flares, and measured the burning times. They are:(a) We will assume that the observations come from normal(µA, σ2) and normal(µB, σ2), where σ2 = 32. Use independent normal (m, s2) prior distributions for
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, and 10 of them are randomly assigned to be trained using method A, and the other 10 are assigned to be trained using method B. After
A medical researcher collected the systolic blood pressure reading for a random sample of n = 30 female students under the age of 21 who visited the Student’s Health Service. The blood pressures are:120 122 121 108 133 119 136 108 106 105 122 139 133 115 104 94 118 93 102 114 123 125 124 108 111
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 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 first bounce. The values are:Assume y, the height (in cm) a golf ball bounces when dropped from a one meter height is
A statistician buys a pack of 10 new golf balls, and drops each golf ball from a height of one meter, and measures the height in centimeters it returns on the first bounce. The ten values are:79.9 80.0 78.9 78.5 75.6 80.5 82.5 80.1 81.6 76.7 Assume y, the height (in cm) a golf ball bounces
We will use the Minitab macro NormGCP.mac to find the posterior distribution of the normal mean µ when we have a random samples of size n of normal(µ, σ2) observations with known σ2 = 22, and we have a general continuous prior for µ. Suppose the prior has the shape given by g(µ)
Repeat part (a) with a normal(30, 42) prior, storing the likelihood and posterior in c5 and c6.
26.3 28.03 28.5 26.3 31.9 28.5 27.2 20.9 27.5 28.0 18.6 22.3 25.0 31.5 Use NormNP.mac to find the posterior distribution g(µ|y1,...,y15, where we choose a normal (m = 20, s2 = 52) prior for µ. The details for invoking NormNP.mac are in Appendix 3. Store the likelihood and posterior in
We will use the Minitab macro NormNP.mac to find the posterior distribution of the normal mean µ when we have a random sample of size n from a normal (µ, σ2) distribution with known σ2, and we use a normal (m, s2)prior for µ. The normal family of priors is the conjugate family for normal
Instead of thinking of a random sample of size n = 16, let’s think of the sample mean as a single observation from its distribution.(a) What is the distribution of y¯? Calculate the observed value of y¯?(b) Use NormDP.mac to find the posterior distribution g(µ|y¯).(c) What does this show us?
Instead, combine all the observations together to give a random sample of size n = 16, and use NormDP.mac to find the posterior distribution where we go back the original prior that had all the possible values equally likely. What do the results of the last two problems show us?
Suppose another 6 random observations come later. They are: 6.22 3.99 3.67 6.35 7.89 6.13 Use NormDP.mac to find the posterior distribution, where we will use the posterior after the first ten observations y1,...,y10, as the prior for the next six observations y11,...,y16.
Use the Minitab macro NormDP.mac to find the posterior distribution of the mean µ when we have a random sample of observations from a normal (µ, σ2), where σ2 is known, and we have a discrete prior for µ.Suppose we have a random sample of n = 10 observations from a normal(µ, σ2) distribution
The Houhora site in Northland (top of North Island) New Zealand is one of the sites of early human inhabitation that New Zealand archeologists have investigated, in trying to determine when the Polynesian migration to New Zealand occurred and documenting local adaptations to New Zealand conditions.
New Zealand was the last major land mass to be settled by human beings.The Shag River Mouth in Otago (lower South Island), New Zealand, is one of the sites of early human inhabitation that New Zealand archeologists have investigated, in trying to determine when the Polynesian migration to New
The Houhora site in Northland (top of North Island) New Zealand is one of the sites of early human inhabitation that New Zealand archeologists have investigated, in trying to determine when the Polynesian migration to New Zealand occurred and documenting local adaptations to New Zealand conditions.
New Zealand was the last major land mass to be settled by human beings.The Shag River Mouth in Otago (lower South Island), New Zealand, is one of the sites of early human inhabitation that New Zealand archeologists have investigated, in trying to determine when the Polynesian migration to New
An engineer takes a sample of 5 steel I beams from a batch, and measures the amount they sag under a standard load. The amounts in mm are: 5.19 4.72 4.81 4.87 4.88 It is known that the sag is normal (µ, σ2) where the standard deviation σ = .25 is known.(a) Use a normal (5, .52) prior for µ.
The standard process for making a polymer has mean yield 35%. A chemical engineer has developed a modified process. He runs the process on 10 batches and measures the yield (in percent) for each batch. They are:38.7 40.0 37.2 36.6 35.9 34.7 37. 35.1 37.5 35.6 Assume that yield is normal (µ, σ2)
The city health inspector wishes to determine the mean bacteria count per liter of water at a popular city beach. Assume the number of bacteria per liter of water is normal with mean µ and standard deviation known to be σ = 15. She collects 10 water samples and found the bacteria counts to be:
The city health inspector wishes to determine the mean bacteria count per liter of water at a popular city beach. Assume the number of bacteria per liter of water is normal with mean µ and standard deviation known to be σ = 15. She collects 10 water samples and found the bacteria counts to be:(a)
You are the statistician responsible for quality standards at a cheese factory.You want the probability that a randomly chosen block of cheese labelled"1 kg" is actually less than 1 kilogram (1000 grams) to be 1% or less. The weight (in grams) of blocks of cheese produced by the machine is
Comparing Bayesian and frequentist estimators for π. In Chapter 1 we learned that the frequentist procedure for evaluating a statistical procedure, namely looking at how it performs in the long run, for a (range of) fixed but unknown parameter values can also be used to evaluate a Bayesian
In the same study of water quality, n = 87 samples were taken from streams in municipal catchments. Out of these y = 8 had a high Campylobacter level.Let π be the true probability that a sample of water from this type of stream has a high Campylobacter level.(a) Find the frequentist estimator for
In the same study of water quality, n = 176 samples were taken from streams having a high environmental impact from sheep farming. Out of these y = 24 had a high Campylobacter level. Let π be the true probability that a sample of water from this type of stream has a high Campylobacter level.(a)
In the same study of water quality, n = 145 samples were taken from streams having a high environmental impact from dairying. Out of these y = 9 had a high Campylobacter level. Let π be the true probability that a sample of water from this type of stream has a high Campylobacter level.(a) Find the
In the study of water quality in New Zealand streams documented in McBride et al. (2002) a high level of Campylobacter was defined as a level greater than 100 per 100 ml of stream water. n = 116 samples were taken from streams having a high environmental impact from birds. Out of these y = 11 had a
The standard method of screening for a disease fails to detect the presence of the disease in 15% of the patients who actually do have the disease. A new method of screening for the presence of the disease has been developed.A random sample of n = 75 patients who are known to have the disease is
Let π be the proportion of students at a university who approve the governments policy on students allowances. The students newspaper is going to take a random sample of n = 30 students at a university and ask if they approve of the governments policy on student allowances.(a) What is the
Repeat the previous question with a uniform prior for π.
We will use the Minitab macro BinoGCP.mac or the equivalent R function to find the posterior distribution of the binomial probability π when the observation distribution of Y |π is binomial (n, π) and we have a general continuous prior for π. Suppose the prior has the shape given by g(π)
Repeat part (a) with a beta (2, 4) prior, storing the likelihood and posterior in c5 and c6.
We will use the Minitab macro BinoBP.mac or the equivalent R function to find the posterior distribution of the binomial probability π when the observation distribution of Y |π is binomial(n, π) and we have a beta (a,b) prior for π. The beta family of priors is the conjugate family for binomial
The same study found that y = 6 out of n = 87 samples within municipal catchments contained Giardia cysts.(a) What is the distribution of y, the number of samples containing Giardia cysts?(b) Let π be the true probability that a one-liter water sample from a site within a municipal catchment
The same study found that y = 10 out of n = 174 samples identified as having a heavy environmental impact from pastoral (sheep) farms contained Giardia cysts.(a) What is the distribution of y, the number of samples containing Giardia cysts?(b) Let π be the true probability that a one-liter water
The same study found that y = 12 out of n = 145 samples identified as having a heavy environmental impact from dairy farms contained Giardia cysts.(a) What is the distribution of y, the number of samples containing Giardia cysts?(b) Let π be the true probability that a one-liter water sample from
In a research program on human health risk from recreational contact with water contaminated with pathogenic microbiological material, the National Institute of Water and Atmosphere (NIWA) instituted a study to determine the quality of New Zealand stream water at a variety of catchment types.This
You are going to take a random sample of voters in a city in order to estimate the proportion π who support stopping the fluoridation of the municipal water supply. Before you analyze the data, you need a prior distribution for π. You decide that your prior mean is .4, and your prior standard
Sophie, the editor of the student newspaper, is going to conduct a survey of students to determine the level of support for the current president of the students association. She needs to determine her prior distribution for π, the proportion of students who support the president. She decides her
A city is considering building a new museum. The local paper wishes to determine the level of support for this project, and is going to conduct a poll of city residents. Out of the sample of 120 people, 74 support the city building the museum.(a) What is the distribution of y, the number who
In order to determine how effective a magazine is at reaching its target audience, a market research company selects a random sample of people from the target audience and interviews them. Out of the 150 people in the sample, 29 had seen the latest issue.(a) What is the distribution of y, the
let Y be distributed according to the beta (15, 10) distribution.(a) Find E(Y ).(b) Find V ar(Y ).(c) Find P(Y
Let Y be distributed according to the beta (10, 12) distribution.(a) Find E(Y ).(b) Find V ar(Y ).(c) Find P(Y >.5) using the normal approximation.
Let Y be normally distributed with mean µ = 860 and variance σ2 = 576.(a) Find P(Y ≤ 900).(b) Find P(Y ≥ 825).(c) Find P(840 ≤ Y ≤ 890).
Let Y be normally distributed with mean µ = 120 and variance σ2 = 64.(a) Find P(Y ≤ 130).(b) Find P(Y ≥ 135).(c) Find P(114 ≤ Y ≤ 127).
Let Z have the standard normal distribution.(a) Find P(0 ≤ Z ≤ 1.52).(b) Find P(Z ≥ 2.11).(c) Find P(−1.45 ≤ Z ≤ 1.74).
Let Z have the standard normal distribution.(a) Find P(0 ≤ Z ≤ .65).(b) Find P(Z ≥ .54).(c) Find P(−.35 ≤ Z ≤ 1.34).
Let X be a random variable having probability density function f(x)=2x for 0 ≤ x ≤ 1 .(a) Find P(X ≥ .75).(b) Find P(.25 ≤ X ≤ .6).
Let X have the uniform distribution.(a) Find E(X).(b) Find V ar(X).(c) Find P(X ≤ .25).(d) Find P(.33
Let X have a beta (12, 4) distribution.(a) Find E(X).(b) Find V ar(X).
Let X have a beta (3, 5) distribution.(a) Find E(X).(b) Find V ar(X).
Suppose we take an additional 7 trials, and achieve 2 successes.(a) Let the posterior after the 8 trials and 3 successes in the previous problem be the prior and use BinoDP.mac or the equivalent R function to find the new posterior distribution for π.(b) In total, we have taken 15 trials and
Use the Minitab macro BinoDP.mac to find the posterior distribution of the binomial probability π when the observation distribution of Y |π is binomial(n, π) and we have a discrete prior for π.Suppose we have 8 independent trials and each has one of two possible either success or failure. The
After the hand, the cards are discarded, and the next hand continues with the remaining cards in the deck. The player has had an opportunity to see some of the cards in the previous hand, those that were dealt face-up. Suppose he saw a total of 4 cards, and none of them were aces, nor were any of
In the game of "blackjack" also known as "twenty-one," the player and the dealer are dealt one card face-down, and one card face-up. The object is to get as close as possible to the score 21, without exceeding that. Aces count either 1 or 11, face cards count 10, and all other cards count at their
Suppose we look at the two draws from the urn (without replacement) as a single experiment. The results were first draw red, second draw green. Find the posterior distribution of X by filling in the simplified table.

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Introduction To Bayesian Statistics 1st Edition William M. Bolstad - Solutions

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