Introduction To Probability And Statistics From A Bayesian Viewpoint Part 2 Inference 1st Edition D. V. Lindley - Solutions

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Under the two-parameter model (i.e., mdl groupinv(2, -2, 1, -1, 2)
The followings are for the additional exercises.(a) For each model:(i) Select a set of parameter values, and obtain the calculation table.(ii) Predict what the shape of the item characteristic curve will look like. It can be helpful to make a sketch of the item characteristic curve before the
This exercise illustrates the joint effect of the parameter values under the threeparameter model.(a) Set the values of the item parameters to a D 1:0, b D 2:0, and c D 0:10.Type in the following command line:> icccal(b=-2.0, a=1.0, c=.10)(b) The computer will display the table of computations.
This exercise uses the two-parameter model to illustrate the joint effect of item difficulty and item discrimination upon the shape of item characteristic curve.(a) Set the values of the item difficulty parameter to b D 2:0 and the item discrimination parameter to 1:0. Type in the following command
This exercise uses the Rasch model to illustrate how the item difficulty parameter locates an item along the ability scale.(a) Set the value of the item difficulty parameter to b D 2:0. Type in the following command line:> icccal(b=-2.0)(b) The computer will display the table of computations. Study
Experiment with various combinations of item difficulty and item discrimination of your own choice until you are confident that you can predict the shape of the item characteristic curve corresponding to the levels chosen. You may find it useful to make a rough sketch of what you think the curve
In this exercise, all the items will have the same level of item discrimination but different levels of item difficulty. The intent is to relate the location of the item on the ability scale to its level of item difficulty.(a) Use the function iccplot to plot an item characteristic curve of an item
In this exercise, all the items will have the same level of item difficulty but different levels of item discrimination. The intent is to relate the steepness of the curve to the level of item discrimination.(a) Use the function iccplot to plot an item characteristic curve of an item with medium
An item with medium item difficulty and low item discrimination is to be plotted.(a) Use the function iccplot to plot an item characteristic curve of an item with medium item difficulty and low item discrimination.(b) From the graph it can be seen that the probability of correct response will be
An item with hard item difficulty and low item discrimination is to be plotted.(a) Use the function iccplot to plot an item characteristic curve of an item with hard item difficulty and low item discrimination.(b) From the graph it can be seen that the probability of correct response will have a
An item with easy item difficulty and high item discrimination is to be plotted.(a) Use the function iccplot to plot an item characteristic curve of an item with easy item difficulty and high item discrimination.(b) From the graph it can be seen that the probability of correct response will be
An estimate is required of the number of individuals in a large population of size N, who possess a certain attribute A. Two methods are proposed. Method I would select a random sample of size aN and inter- view each member of the sample to determine whether he possesses A or not. Method II would
A subject answers n multiple-choice questions, each having k possible answers, and obtains R correct answers. Suppose that the subject really knows the answers to of the questions, where v is an unknown parameter, and that his answers to the remaining n-v questions are pure guesses, each having a
Show that the probability of needing exactly k independent trials to obtain a given number n of successes, when each trial can result in success or failure and the chance of success is p for all trials, is where q (k-1) p*g*-*, 1-p. Writing kn+s, show that the mean and variance of s are na/p and
In an investigation into the toxicity of a certain drug an experiment using k groups of animals is performed. Each of the n, animals in the ith group is given a dose x,, and the number of resulting deaths r, is recorded (i = 1, 2, ..., k). The probability of death for an animal in the ith group is
Let Z1, Z2, ..., Z be independent observations which are identically distributed with distribution function 1-exp(-x), where is an unknown parameter. Obtain an estimate of useful as mco and hence the approximate posterior distribution of . If the calculation of your esti- mator involves iteration,
In an experiment to measure the resistance of a crystal, independent pairs of observations (x, y) (i = 1, 2, ..., n) of current x and voltage y are obtained. These are subject to errors (X, Y), so that x = +X, y = n+ Yi, where (5i, n.) are the true values of current and voltage on the ith occasion,
There are two ways in which an item of equipment may fail. If failure has not occurred at time t, there is a chance 8t+o(8) of failure of type I and a chance A,8t+o(81) of failure of type II in (t, t+8t). A number n of items is placed on test and it is observed that r, of them fail from cause I at
Bacteria in suspension form into clumps, the probability that a random clump will contain n bacteria is 0-1(1-0) (n = 1, 2, ...). When subject to harmful radiation there is a probability At that a bacteria will be killed in any interval of length 8t irrespective of the age of the bacteria in the
Show that the posterior distribution of 0 has approximate variance g(1+g) (2+g) N(6+5g+4g2)' where g = (1-0). An alternative and arithmetically simpler estimation procedure would be to score 0 for each AA individual, 1 for each AB, and 2 for each BB, and then to equate the total score of the whole
A man selected at random from a population has probabilities (1-0), 20(1-0), 02 of belonging to the categories AA, AB, BB respectively. The laws of inheritance are such that the probabilities of the six possible pairs of brothers, instead of being the terms of a trinomial expansion, are: No. of
(Camb. Dip.)
A radioactive sample emits particles randomly at a rate which decays with time, the rate beinge- after time t. The first n particles emitted are observed at successive times t, t,..., t. Set up equations for the maximum- likelihood estimates and , and show that satisfies the equation where ein-1
In a routine eyesight examination of 8-year-old Glasgow school- children in 1955 the children were divided into two categories, those who wore spectacles (A), and those who did not (B). As a result of the test, visual acuity was classed as good, fair or bad. The children wearing spectacles were
A sample of 175 school-children was classified in two ways, the result being given in the table. P denotes that the child was particularly able and not-P that it was not so. Is the apparent association between ability and family prosperity large enough to be regarded as statistically significant?
A group of individuals is classified in two different ways as shown in the following table: X Not-X Y a b a+b Not-Y C d c+d a+c b+d n Establish the formula x = n(ad-bc)/(a+c) (b+d) (a+b) (c+d) for testing the independence of the two classifications. An entomologist collects 1000 specimens of
k samples S, (i = 1, 2, ..., k) of an insect population are taken at k different sites in a certain habitat. S, consists of n, insects of whicha, are black and b,(=n-a) are brown. When the samples are combined the total number of specimens is n of which a are black and b(=n-a) are brown. It is
One hundred plants were classified with respect to the two contrasts large L versus small 1, white W versus coloured w, the numbers in the four resulting classes being as shown in the table: L 1 Totals W 40 15 55 W 20 25 45 Totals 60 40 100 Investigate the following questions: (a) Is there reason
Test whether there is any evidence for supposing an association between the dominance of hand and of eye in the following contingency table (i.e. of the 60 people, 14 were left-handed and left-eyed): Left-eyed Right-eyed Left-handed Right-handed Totals 14 6 Totals 20 19 21 40 33 27 60 How would you
Hence show that the probability distribution ofa, given r and m depends only on 011022/012021. Suggest a method of testing that the classifications into A and B are independent: that is p(AB) = p(A)p(B). (Wales Maths.)
Each of n individuals is classified as A or not-A (A) and also as B or B. The probabilities are: P(AB) = P(AB) = 0, p(AB)=021 p(AB) = 0229 with 0,1; and the individuals are independent. Show that the joint 1,3 distribution of m, the number classified as A; r, the number classified as B; anda, the
The following data (from H. E. Wheeler, Amer. J. Bot. 46, 361, 1959) give the numbers of fertile and infertile perithecia from cultures of Glomerella grown in different media: Medium Fertile Infertile Total Oatmeal 179 56 235 Cornmeal 184 19 203 Potato dextrose 161 39 200 Synthetic 176 26 202
Each engineering apprentice entering a certain large firm is given a written test on his basic technical knowledge, his work being graded A, B, C, or D (in descending order of merit). At the end of his third year with the firm his section head gives him a rating based on his current performance;
Birds of the same species, kept in separate cages, were observed at specified times and it was found that each bird had a probability of of being on its perch when observed. Nine of these birds were then put together into an aviary with a long perch. At each of the next 100 times of observation the
Antirrhinum flowers are either white, or various shades of red. Five seeds, taken from a seed-pod of one plant, were sown, and the number of seeds which produced plants with white flowers was noted. The process was repeated with five seeds from each of 100 seed-pods, and the following table shows
If trees are distributed at random in a wood the distance r from a tree to its nearest neighbour is such that the probability that it exceeds a is exp[-Ta/4], where is the mean distance. The following data (from E. C. Pielou, J. Ecol. 48, 575, 1960) give the frequencies of the nearest neighbour
Antirrhinum flowers are either white, or various shades of red. Five seeds, taken from a seed-pod of one plant, were sown, and the number of seeds which produced plants with white flowers was noted. The process was repeated with five seeds from each of 100 seed-pods, and the following table shows
If trees are distributed at random in a wood the distance from a tree to its nearest neighbour is such that the probability that it exceeds a is exp[-na/4], where is the mean distance. The following data (from E. C. Pielou, J. Ecol. 48, 575, 1960) give the frequencies of the nearest neighbour
In connexion with a certain experiment an instrument was devised which, in each consecutive time period of th of a second, either gave an impulse or did not give an impulse. It was hoped that the probability, p, of the impulse occurring would be the same for every time period and independent of
The table given below, containing part of data collected by Parkes from herd-book records of Duroc-Jersey Pigs, shows the distribution of sex in litters of 4, 5, 6 and 7 pigs. Examine whether these data are consistent with the hypothesis that the number of males within a litter of given size is a
A sequence is available of n+1 geological strata of four different types in all, denoted bya, b, c andd. The null hypothesis assumes that the occur- rence of the four types occur randomly (with possibly different probabilities Pa, Pb, Pe and pa), except that two consecutive strata of the same type
Two fuses included in the same apparatus are assumed to have independent probabilities p and q of being blown during each repetition of a certain experiment. Show that the probabilities of neither, one only, or both being blown during any repetition are given respectively by 1-p-q+pa, p+q-2pq, pq.
The following table gives the results of 7776 sequences of throws of an ordinary die. For s = 1, 2, 3, n, is the number of sequences in which a 6 was first thrown at the sth throw, and n is the number of sequences in which a 6 was not thrown in any of the first 3 throws. Investigate if there is
A suspension of particles in water is thoroughly shaken. 1 c.c. of the suspension is removed and the number of particles in it is counted. The remainder of the suspension is diluted to half concentration by adding an equal volume of water, and after being thoroughly shaken another c.c. is removed
An observer records the arrival of vehicles at a road bridge during a period of just over an hour. In this period 54 vehicles reach the bridge, the intervals of time between their consecutive instants of arrival (in seconds) being given in numerical order in the table below. It is postulated that
The independent random variables X,..., X, follow Poisson distri- butions with means ,, . The hypotheses H, H, H, are defined as 1, M. follows: H,, are arbitrary; H = a+i (i = 1,..., n); H = Y, wherea, , y are unknown. Develop tests for: (i) the null hypothesis H, with H as alternative; (ii) the
In an experiment to investigate whether certain micro-organisms tended to move in groups or not, 20 micro-organisms were placed in the field of view A of a microscope and were free to move in a plane of which A was a part. After a lapse of time the number of micro-organisms remaining within A was
In some classical experiments on pea-breeding Mendel obtained the following frequencies for different kinds of seed in crosses with round yellow seeds and wrinkled green seeds: Observed Theoretical Round and yellow 315 312-75 Wrinkled and yellow 101 104-25 Round and green 108 104-25 Wrinkled and
In an investigation of the incidence of death due to a particular cause it is required to find whether death is more likely at some times of the day than at others. In 96 cases the times of death are distributed as follows in 3-hourly intervals beginning at the stated times. Mid- night 3 a.m. 6a.m.
Bacterial populations in a fluid suspension are being measured by a technique in which a given quantity of the fluid is placed on a microscope slide and the number of colonies on the slide is counted. If the laboratory technique is correct, replicate counts from the same culture should follow a
The following table shows the result of recording the telephone calls handled between 1 p.m. and 2 p.m. on each of 100 days, e.g. on 36 days no calls were made. Show that this distribution is consistent with calls arriving independently and at random. Obtain (on this assumption) a 99% confidence
Past observations of a bacterial culture have shown that on the average 2% of the cells are undergoing division at any instant. After treatment of the culture 200 cells are counted and only 1 is found to be dividing. Is this evidence that the treatment has decreased the ability of the cells to
Cakes of a standard size are made from dough with currants randomly distributed throughout. A quarter is cut from a cake and found to contain 25 currants. Find a 95% confidence interval for the mean number of currants per cake. (Camb. N.S.)
The table shows the numbers of births (rounded to nearest 100) and the numbers of sets of triplets born in Norway between 1911 and 1940. Period Total births Set of triplets 1911-15 308,100 52 1916-20 319,800 52 1921-25 303,400 40 1926-30 253,000 30 1931-35 222,900 24 1936-40 227,700 20 Apply x2 to
The table shows the number of motor vehicles passing a specified point between 2.00 p.m. and 2.15 p.m. on 6 days in 2 successive weeks. Week 1 Week 2 Mon. Tues. Wed. Thurs. Fri. Sat. 50 65 52 63 84 102 56 49 60 45 112 90 The flow of vehicles on any day is believed to conform to a Poisson
Initially, at time = 0, a Poisson process is set to produce on the average one incident per unit time. It is observed for a time T and incidents occur at t, 1, ..., . It is then learnt from other information that at time T the Poisson process is working at a rate of two incidents per unit time. If
You are going to play a pachinko machine at a fair and can either win or lose a penny on each play. If is the probability of winning, you know that since otherwise the fair owner would not continue to use the machine (though there is always the slight chance that it has gone wrong, for him, and he
A random sample from B(20, 0) gave the value r = 7. Determine 50, 95 and 99% confidence limits for by the following methods and compare their values: (i) the exact method of theorem 7.2.1 and its first corollary: (ii) the approximation of the second corollary to that theorem; (iii) the inverse-sine
A random sample of size n from P(0) gave values 1, 2, ...,a, and the prior knowledge of 0 was small. Determine the probability that a further random sample of size m from P(20) will have means.
A random sequence of n trials with initially unknown constant probability of success gives successes. What is the probability that the (n+1)st trial will be successful? Consider in particular the cases r = 0 and r = n.
The random variables Y, Z have the form Y = X+U, Z = X+V, where X, U, V, are independently normally distributed with zero means and variances 0, 1, 1. Given " independent pairs (1, 1), (yn, Zn), obtain the maximum likelihood estimate 6. What is the posterior distri- bution of 0? (Lond. M.Sc.)
A new type of electric light bulb has a length of life, whose probability density function is of the form x f(x) = (x > 0). Forty bulbs were given a life test and failures occurred after the following times (in hours): 196, 327, 405, 537, 541, 660, 671, 710, 786, 940, 954, 1004, 1004, 1006, 1202,
A survey is carried out to investigate the incidence of a disease. The number of cases occurring in a fixed period of time in households of size k is recorded for those households having at least one case. Suppose that the disease is not infectious and that one attack confers immunity. Let p = 1-q
A household consists of two persons, either of whom is susceptible to a certain infectious disease. After one person has contracted the infection a period of time U elapses until symptoms appear, when he is immediately isolated, and during this period it is possible for him to transmit the
A new grocery product is introduced at time t = 0. Observations are made for a period T on a random sample of n households. It is observed that m of these buy the product for the first time at times [1, ..., fm and that in the remaining n-m households the product has not been bought by the end of
Serum from each of a random sample of n individuals is mixed with a certain chemical compound and observed for a time T, in order to record the time at which a certain colour change occurs. It is observed that individuals respond at times t, fa, ..., tr, and that the remaining (n-r) have shown no
In an experiment on the time taken by mice to respond to a drug, the log dose x, received by mouse i is controlled exactly, and the log response timet, is measured individually for all mice responding before log time T. The results of the experiment consist of (n-r) pairs (x, t) with 1, 8(t) = +xi,
Independent observations are made on a Poisson variable of unknown mean 0. It is known only that of n observations, no have the value 0, n have the value 1, and the remaining n-no-n observations have values greater than one. Obtain the equation satisfied by the maximum likelihood estimate and
Calculate the maximum likelihood estimate, 0, of 0, the parameter of the Cauchy distribution 1 1 dF = ! dx (-00 < x < 0), 771+(x-0)2 given the sample of 7 observations, 3-2, 2-0, 2-3, 10-4, 1-9, 0-4, 2-6. Describe how you could have used this calculation to determine approxi- mate confidence limits
A continuous random variable x, defined in the range 0 < x 0. Find the density function of x. Given a random sample of n observations from this distribution, derive the maximum likelihood equation for &, the estimate ofa. Indicate very briefly how this equation can be solved numerically. Also prove
A random sample is drawn from a population with density function f(x|0) = -e-e-x (0 x 1). Show that the mean of the sample is a sufficient statistic for the parameter and verify that the maximum likelihood estimate is a function of the sample mean. Let this estimate be denoted by 01. Suppose that
An individual taken from a certain very large biological population is of type A with probability (1+F) and of type B with probability (1-F). Give the probability that a random sample of n such individuals will consist of a of type A and b of type B. Find , the maximum likelihood estimate of F.
A complex organic compound contains a chemical which decays when the compound is removed from living tissue. If p is the proportion of the chemical present at the time of removal (t = 0), the proportion present at time t is pet where (>0) is known. To determine p a quantity of the compound is
Two independent Poisson processes are each observed for a unit time. m and n incidents are observed respectively and the unknown rates (the Poisson means) are 0, and 0, respectively. If 0, and 0, have independent uniform prior distributions over the positive real line show that the posterior
A random sample x1, x2, ..., xm from N(0, 0) has sufficient statistics m =(x,-) x=x/m and S = (x-x). 1=1 It is proposed to take a second independent random sample of size n from the same distribution and to calculate from the values y, y2, ..., y, the two statistics = y= yn and S-(-). 1-1 Show that
In the experiment discussed in detail in 6.5 a possible prior distribution might be one in which the four methods were thought unlikely to differ much among themselves but might differ substantially from the control (in either direction). On the lines of the example discussed in 6.6 put this prior
In an experiment to determine an unknown value for a substance, independent measurements are made on the substance at timesf, 12, ..., In A measurement made at time t has mean 0e-t, where > 0 is a known constant, and has known constant coefficient of variation v. Determine the posterior
An apparatus for determining g, the acceleration due to gravity, has standard deviation , when used at one place and , at another. It is proposed to make N determinations-some at one place, some at the other-to find the difference between g at the two places. Determine the optimum allocation of the
Four independent determinations are made of a physical constant using apparatuses of known standard deviations. The means and standard deviations of the posterior distributions associated with the four deter- minations are, in the form : 1-000-10, 1-21 0-08, 1-270-11, 1-110.06. Find the most
An experiment was carried out to investigate the efficacies of four compounds; two, P1, P2, based on one antibiotic and two, S1, S2, based on another antibiotic. A control C was also used. The compounds were each used five times in a suitable arrangement to ensure independence, with the following
Twelve tomato plants are planted in similar pots and are grown under similar conditions except that they are divided into three groups A, B, C of four pots each, the mixture of soil and fertilizer being the same within each group but the groups differing in the type of fertilizer. The yields (in
The table gives the time in seconds for each of six rats (A, B, ..., F) to run a maze successfully in each of four trials (1, 2, 3, 4). Perform an analysis of variance to determine whether the rats give evidence of differing in their ability. A 1234 15 2 10 15 3 13 4 14 38566 B D E F 18 11 18 19 14
The following table gives the values of the cephalic index found in two random samples of skulls, one consisting of fifteen and the other of thirteen individuals: Sample I: 74-1, 77-7, 74-4, 74-0, 73-8, 79-3, 75-8, 82-8, 72-2, 75-2, 78-2, 77-1, 78-4, 76-3, 76-8 Sample II: 70-8, 74-9, 74-2, 70-4,
Use the result of exercise 5 in chapter 2 to show that tables of the F-distribution can be used to obtain tables of the binomial distribution. Specifically, in an obvious notation show that if s is B(n, p) then p(s > r) = p[F(2(nr+1), 2r) > qr p(n-r+1)
A random sample x1, x2, ..., xm is available from N(0,) and a second independent random sample y, ya, ..., ya from N(0, 20). Obtain, under the usual assumptions, the posterior distribution of 01-02, and of o.
Two methods, I and II, are available for determining the sulphur dioxide content of a soft drink. The variances associated with single determinations are and respectively, but it is usual to use several independent determinations with one of the two methods. Show that if I costs c times as much to
Two analysts each made six micro-analytical determinations of the carbon content of a chemical, giving the values: Analyst I: Analyst II: 59-09, 59-17, 59-27, 59-13, 59-10, 59-14 59-06, 59-40, 59-00, 59-12, 59-01, 59-25 Discuss whether these results provide evidence that the two analysts differ in
Two workers carried out an experiment to compare the repeatabilities with which they used a certain chemical technique. The first worker made eight determinations on the test substance and estimated the standard deviation of a single measurement as 0-74; the second worker made fifteen
It is expected that a certain procedure for estimating the percentage amount of a substance X in a sample of material will give a standard error approximately equal to 0-12. Two chemists take samples from a given bulk of material and each performs five independent determinations of the percentage
Two normal populations of known equal variance have means 01, 0. It is required to calculate, from random samples of size n taken from each population, a 100% confidence interval for A = 0-0. The following procedure is suggested: 100 % confidence intervals are calcu- lated separately for 0, and 0,
A new wages structure is introduced throughout a certain industry. The following values of output per man-hour are obtained from a number of factories just before and just after its introduction: Output per man-hour Old wages New wages structure structure 122-3 119-0 141-0 131-8 120-1 118-7 109-6
The following results relate to the yields of two varieties of tomato grown in pairs in nine different positions in a greenhouse: Posi- Variety tion, X, Variety Y, 1123 x 1.39 1.17 1-41 1-22 3 1-07 1-08 4 1-65 1-42 5 1-57 1.39 Posi- Variety Variety tion, X, Y, 1698 Yi 1-63 1-22 7 0-88 0-84 8 1-14
The table gives the yields of two varieties A and B of wheat grown in strictly comparable conditions at each of twelve Canadian testing stations. Obtain the posterior distribution of the difference in average yield of the two varieties: 25 28 28 B 17 15 21 Station... 1 2 3 4 5 6 7 8 9 10 11 12 A 20
Observations are made as follows on a Poisson process in which events occur at unknown rate per unit time. In the time period (0, u] it is observed that m events occur at instants x < x < ... < xm, whereas in the time interval (u, 2u] it is observed only that n events occur, the instants of
Certain radioactive substances emit particles at random at an average rate A. n intervals between such emissions are observed for each of two such substances, the mean intervals being and respectively. Explain how you could test whether the rates of emission are the same. (Leic. Gen.)
A Geiger counter has a probability 08u+ O(u) of clicking during the time interval u, u+8u, independent of u and all previous clicks. If the average interval between consecutive members of a sequence of 11 clicks is 2-2 sec, show that may be asserted to lie in the range 0-25 (sec) to 0-71 (sec)-
Bacterial organisms are known to be present in a large volume of water but it is not known whether there are many or few. Accordingly it is decided to examine samples each of 10 cm of water for the presence (or absence) of the organism, to go on taking samples until k samples have been found
In a large town, the size n of families, is distributed with probability density function p(n) (1-P)p" (0 0, i = 1,..., m) but whose sizes are unknown, whilst the rest contain no boys, obtain the posterior distribution of p. (Lond. Dip.)
The breakdowns of a machine occur in such a way that the probability of a breakdown in the interval (t, t+8t) is (t)8t+o(dt) independently of stoppages prior to 1. The machine is observed in the interval (0, 7) and the time of breakdowns recorded. If (t) = aet obtain a test of the hypo- thesis that
Prove that if x1, x2, ..., x, is a random sample of size n from N(, ), then x-x is independent of x and hence (x,-x) of x. Use this result to prove an extension of the result of $5.3 that (x;-)/02 is x with n degrees of freedom; namely that (x-x)/0 is x with (n-1) degrees of freedom.

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Introduction To Probability And Statistics From A Bayesian Viewpoint Part 2 Inference 1st Edition D. V. Lindley - Solutions

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