Primer Of Biostatistics 7th Edition Stanton Glantz - Solutions

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14.1 Derive and plot the symmetrical confidence limit curves shown in Fig. 14.6, for the binomial distribution for the case n = 20. (If you use a computer, you might as well have it plot the whole family of curves.)
13.4 For the toy model in Sect. 13.2, we had a naive solution for the estimate of the number of events in (13.7) and a solution in which the variance was defined as x; rather than the theoretically predicted x in (13.10). Show that for this toy model, if is defined by x2 = x(1+e), then the two
13.3 A set of n thin wire proportional chambers uniformly spaced at positions along the x axis has made a series of measurements of track position, i.e., x, y values for a straight track. We wish to estimate tan in the two-dimensional projection. The spacing between individual chambers is A.x. Find
13.2 Imagine that we measure n pairs of x, y values. Suppose y is a normally dis- tributed variable with constant variance o but with a mean that depends linearly on x. m = a +(x-x). where X is the sample mean. x does not have to be a random variable. We might fix x at x and sample y. Then we might
13.1 Suppose we have a visible detector such as a bubble chamber with a fast beam of short-lived particles of known energy passing through it. We might imagine, for instance, that we have a beam of 5.0 GeV particles. Some of the particles decay in the chamber and some pass through the chamber
11.1 Show that the Lindeberg criterion is satisfied for the binomial distribution and, therefore, the binomial distribution does approach the normal distribution for large n. (Let each xi be a single trial.)
10.2 Suppose we have a collection of diatomic molecules that we excite with laser beam pulses. The molecules can ionize giving up an electron or dissociate giving up an ion. At each pulse, we measure the number of ions (x) and the number of electrons (y) in appropriate detectors for each. We wish
10.1 Consider a two-dimensional density function proportional to exp −(2x 2 + 2x + 1/2 + 2x y + 8y2 ) . (a) Find m1, m2, σ1, σ2, ρ, λi j . (b) Write the correctly normalized density function. (c) Make a linear non-singular transformation such that the new variables are uncorrelated.
9.5 Derive Eq. 9.17 for the mean and variance for the number of failures preceding the mth success.
9.4 For Sect. 9.1, Item 5, repair people servicing machines, calculate for m = 3, n = 20, R = 0.1 a table for k = 1 − 10, showing the values for the number of machines being serviced, waiting to be serviced, repair people idle, and Pk .
9.3 Derive Eq. 9.14. Hint: Consider the equilibrium situation with k machines needing service for three cases, k = 0, 1 ≤ k ≤ m, and k > m. Show these equilibrium relations are satisfied by (k + 1)λO Pk+1 = (n − k)λI Pk , k ≤ m; (9.28) mλO Pk+1 = (n − k)λI Pk , k > m. Then show these
9.2 Show that the equilibrium equations, Eqs. 9.9 and 9.10 satisfy the equilibrium versions of Eq. 9.8.
9.1 Show that for the infinite buffer problem at the beginning of this chapter, E{k} = R/(1 − R) and var{k} = R/(1 − R)2.
8.5 Let us consider an example of the importance sampling variant of the acceptance rejection method. Consider again the problem of generating normally distributed pseudorandom numbers for x > 0. Start with a function g(x) = a1e−a2 x . This is not to be a normalized density function, but a
8.4 In this exercise, we will examine the efficiency of detection of events in an experiment. You will generate 300 events of D0 decays and see if each event is detectable. To be detectable, the D0 has to live sufficiently long that the decay point can be seen, it has to decay by the right mode,
8.3 From this point on, a number of the exercises will ask you to draw graphs and make fits to data. Both kinds of these problems ask you to calculate a set of data, say x vs y and then operate on them. I suggest that it may be simplest to just calculate the set of data in your favorite language
8.2 Devise a fast way to generate a Poisson distribution using Monte Carlo methods assuming λ is less than 20, using random numbers uniformly distributed between 0 and 1. (Hint: Consider using a table.)
8.1 A distribution of events as a function of energy is expected to have a Breit–Wigner resonance (M,) and a background density function proportional to 1/(a + bE)3. It is expected that the number of events in the resonance is one-third of the total number of events. Show how to generate a Monte
7.7 Find the mean and standard deviation of theχ2 distribution using the characteristic functions.
7.6 Prove Theorem 7.8 for generating functions (Eq. 7.25). Q(x) ≡ ∞ k=0 qk xk = P (x) P (0) , where qk = (k + 1)pk+1/ < k >.
7.5 Prove Eq. 7.22. h(x) = ∞ j=0 π(j)xj = ∞ j=0 xj ∞ k1=0 ··· ∞ km=0 δ(j, i ki) m i=1 p(i) ki = m i=1 Pi (x); where Pi (x) = ∞ k=0 p(i) k xk . The probability π(j) was π(j) = ∞ k1=0 ··· ∞ km=0 δ j, i ki m i=1 p(i) ki
7.4 Prove Eq. 7.20.
7.3 Prove the generating functions Eqs. 7.14, 7.18.
7.2 Suppose, for X , a discrete random variable, fn = Pr(X = n) has generating function F(s). Let gn = Pr(X > n) and have generating function G(s). Show that G(s) = (1 − F(s))/(1 − s).
7.1 If xi is a set of independent measurements, let xAV = xi/n. Prove that for the Breit–Wigner distribution the distribution of xAV is the same as that of xi. (Recall that for a normal distribution, the distribution of xAV has standard deviation decreasing with increasing n.) What is the σ of
6.5 Suppose we have N books numbered 1 to N on a shelf and randomly select two of them located at j1 and j2. The probability for the sum j1 + j2 is called the triangle distribution. The distribution is given by: p(x) = 1 − |x − μ| for μ − < x < μ + and 0 otherwise. (6.46) The mean
6.4 We have seen that the binomial distribution approaches the normal distribution for large n. Compare the exact calculation of the probability of having 20 heads out of 40 tosses of a fair coin with the approximation using the normal distribution.
6.3 Suppose we make five measurements of a quantity we know to be normally distributed with mean 0.05. We obtain 0.041, 0.064, 0.055, 0.046, 0.060. Estimate the variance of the distribution. Suppose we know that the variance is, in fact, 1.0 × 10−4. What is the value of χ2 for this set of
6.2 You are measuring a quantity and are uncertain of the error. You make three measurements obtaining 1.0, 2.0, 1.0. Estimate the variance. The next day you come back to your apparatus and measure 7.0. You suspect that something has changed, but worry that it might be only that the variance is
6.1 Suppose the probability that an electronic component fails to function in the time interval (t, t + dt) is φ(t) dt provided the component has functioned properly up to time t. Given ∞ 0 φ(τ ) dτ = ∞, show that p(t) = φ(t)e− t 0 φ(τ ) dτ (6.45) is the probability density
5.5 This problem has arisen in evaluating random backgrounds in an experiment involving many counting channels. Suppose one has three channels and each of the three channels is excited randomly and independently. Suppose at any instant the probability of channel i being on is pi. Imagine “on”
5.4 Suppose we are measuring second harmonic generation when pulsing a laser on a solid. However, the detector we have available is only a threshold detector and the time resolution is such that it can only tell us whether or not there were any second harmonic photons on that pulse. Assume the
5.3 We consider collisions of a high-energy proton with a proton at rest. Imagine the resultant products consist of the two protons and some number of pions. Suppose we imagine the number of pions is even and equal to 2N, where N is the number of pion pairs. As a crude model, we imagine the
5.2 A nuclear physics experiment is running. It requires 50 independent counters sensitive to ionizing radiation. The counters are checked between data runs and are observed to have a 1% chance of failure between checks (i.e., p = 0.01 for failure). What is the probability that all the counters
5.1 The negative binomial distribution gives the probability of having the mth success occur on the rth Bernoulli trial, where p is the probability of the occurrence of the success of a single trial. Show that the negative binomial distribution probability function is given by P(r, m) = (r − 1)!
4.2 Assume that we start with r red and b black balls in an urn. We randomly draw balls out one at a time without looking into the urn. (a) We draw them out and place them to one side of the urn (no replacement). Find the probability of drawing n1 black balls in N drawings. Assume that n1 < b; N
4.1 Consider the probability of players having complete suits, A,...,K in bridge hands in a given round. (a) Find the probability that one particular player has a complete suit. (b) Find the probability that two specific players of the four bridge players have complete suits (c) Find the
3.6 Two measurements for κ, the compressibility of liquid He, gave 3.87 ± 0.04 × 10−12 and 3.95 ± 0.05 × 10−12 cm2/dyn. From these two experiments, find the best estimate and its error for the value of κ. Are these two experiments consistent with each other? The standard value for κ is
3.5 Atomic physicists have measured parity violation effects in a series of delicate experiments. In one kind of experiment, they prepare two samples identically except for reversed external fields and look at the intensity of fluorescence from each when illuminated by a laser pulse. Suppose they
3.4 Suppose that we are measuring counting rates with a scintillator and that we are measuring a signal and a background. First we measure a background plus a signal, then a background only, obtaining R1 = m1/T1 and R2 = m2/T2, respectively, as results. m1, m2 are the number of counts obtained and
3.3 Suppose one has measured a counting rate in a scintillation counter twice, obtaining R1 = m1/T1 and R2 = m2/T2, respectively, as results. m1, m2 are the number of counts obtained and T1, T2 are the times used for the measurements. Suppose the errors in T1, T2 are negligible and that σ2 m1 ≈
3.2 Imagine that a charged particle is passing through matter undergoing multiple scattering. Neglect energy loss, i.e., let θ 2 y = Kx. We wish to estimate the direction of motion of the particle in the x-y plane by measuring its y position independently at two points, x = 0 and x = L. We
3.1 Suppose we obtain n independent results xi from a distribution F(x). Let xAV be the mean of these n values, the sample mean. Define σ2 s ≡ 1 n − 1 n i = 1 (xi − xAV) 2 . (3.41) Show that the expectation value of σ2 s is σ2. Remark It can also be shown by similar (but lengthy
2.4 A coin is tossed until for the first time the same result appears twice in succession. To every possible pattern of n tosses, attribute probability 1/2". Describe the sample space. Find the probability of the following events: (a) the experiment ends before the sixth toss, (b) an even number of
2.3 Suppose one has two independent radioactive sources. Disintegrations from each are counted by separate scintillators with expected input rates N and N2. Ignore the dead times of the array. Suppose one is measuring the coincidence rate, that is, asking how often the two scintillators are excited
2.2 Let ,, = x" be the nth moment of a distribution, , be the nth central moment of the distribution, and m = be the mean of the distribution. Show that 3=3-3m2+2m.
2.1 A 1-cm-long matchstick is dropped randomly onto a piece of paper with parallel lines spaced 2 cm apart marked on it. You observe the fraction of the time that a matchstick intersects a line. Show that from this experiment you can determine the value of . (This problem is known as the Buffon's
11-4 The BALANCE investigators who conducted the study of drug treatments for bipolar people in Problem 5-9 also collected data on when experimental subjects had emergent mood episodes for 36 months. Table 11-9 gives the observations. What is the median time to an event for the two treatment
11-3 What is the sample size for each experimental group to obtain .80 power using a log rank test to detect a significant difference (with a = .05) in steady state survival rates between .40 and .20?
11-2 Taking care of old people on an outpatient basis is less costly than caring for them in nursing homes or hospitals, but health professionals have expressed concern about how well it is possible to predict clinical outcomes among people cared for on an outpatient basis. As part of an
11-1 Surgery is an accepted therapeutic approach for treating cancer patients with metastases in their lungs.Philippe Girard and colleagues†collected data on 35 people who had metastases removed from their lungs (see Table 11-7). Estimate the survival curve and associated 95% confidence interval.
10-8 In his continuing effort to become famous, the author of an introductory biostatistics text invented a new way to test if some treatment changes an individual’s response.Each experimental subject is observed before and after treatment, and the change in the variable of interest is computed.
10-7 Rework Problem 9-1 using the Wilcoxon signed-rank test.
10-5 To determine whether or not offspring of parents with type II diabetes have abnormal glucose levels compared to offspring without a parental history of type II diabetes. Gerald Berenson and colleagues* collected data on whether these offspring had different cholesterol levels.The data for 30
10-4 Rework Problem 9-6 using methods based on ranks.
10-3 Rework Problem 9-5 using methods based on ranks.
10-2 The inappropriate and overuse of antibiotics is a well-recognized problem in medicine. To test whether it was possible to encourage more appropriate use of antibiotics in elderly hospitalized patients, Monika Lutters and her colleagues* monitored the number of patients receiving antibiotics in
10-1 Despite progresses in technique, adhesions (the abnormal connection between tissues inside the body formed during healing following surgery) continue to be a problem in abdominal surgery, such as when operating on the uterus. To see if it would be possible to reduce adhesions following uterine
9-10 Review all original articles published in the New England Journal of Medicine during the last 12 months. How many of these articles present the results of experiments that should be analyzed with a repeated measures analysis of variance? What percentage of these articles actually did such an
9-9 The data in Problem 9-8 could also be presented in the following form as shown in Table 9-13. How would these data be analyzed? If this result differs from the analysis in Problem 9-9, explain why and decide which approach is correct.
9-8 In the fetus, there is a connection between the aorta and the artery going to the lungs called the ductus arteriosus that permits the heart to bypass the nonfunctioning lungs and circulate blood to the placenta to obtain oxygen and nourishment and dispose of wastes. After the infant is born and
9-7 What is the power of the test in Problem 9-7 to find a 100 mL change in food intake with 95% confidence?
9-6 In general, levels of the hormone testosterone decrease during periods of stress. Because physical and psychological stressors are inevitable in the life of soldiers, the military is very interested in assessing stress response in soldiers. Many studies addressing this issue suffer from taking
9-5 In addition to measuring FEV1 (the experiment described in conjunction with Fig. 9-6), Michel and colleagues took measurements of immune response in their subjects, including measuring the amount of C-reactive protein (CRP), a protein that is elevated when tissue is inflamed. Their results are
9-4 Rework Problem 9-2 as a repeated measures analysis of variance. What is the arithmetic relationship between F and t ?
9-3 What are the chances of detecting a halving of the heart rate variability in Problem 9-2 with 95% confidence?Note that the power chart in Figure 6-9 also applies to the paired t test.
9-2 Secondhand tobacco smoke increases the risk of a heart attack. In order to investigate the mechanisms for this effect, C. Arden Pope III and his colleagues†studied whether breathing secondhand smoke affected autonomic (reflex) nervous system control of the heart. At rest the heart beats
9-1 Several epidemiological studies have shown that people who have a diet high in flavenols (which are in tea, wine, cocoa products and various fruits) have lower rates of dying from coronary artery disease. To investigate whether this effect of flavinols is mediated, at least in part, by
8-11 Clinical and epidemiologic studies have demonstrated an association between high blood pressure, diabetes, and high levels of lipids measured in blood. In addition, several studies demonstrated that people with high blood pressure have lower insulin sensitivity than people with normal blood
8-10 What sample size is necessary to have an 80% power for detecting a correlation between journal circulation and selectivity with 95% confidence if the actual correlation in the population is .6?
8-9 What is the power of the study of journal circulation and selectivity described in Figure 8-12 to detect a correlation of .6 with 95% confidence? (There are 113 journals in the sample.)
8-8 As part of a study of the nature of cancers of the gum and lower jaw, Eiji Nakayama and his colleagues†were interested in relating the extent of cancer invasion, determined by direct histological investigation with the levels of invasion measured on a computed tomographic scan of people with
8-7 Erectile dysfunction is widely recognized to be associated with diabetes and cardiovascular disease. To investigate whether erectile dysfunction was associated with lower urinary tract infections, Wo-Sik Chung and colleagues* administered standard questionnaires to men between ages 40 and 70
8-6 Arteries adjust their size on a minute-to-minute basis to meet the needs of the body for blood to carry oxygen to the tissues and to remove waste products. A substantial part of this response is mediated by the one cell thick lining of the arteries known as the vascular endothelium responding
8-5 The ability to measure hormone levels based on a blood spot (like that used by diabetics for glucose monitoring) has several advantages over measurements based on a blood draw. First, blood spots allow for repeated measurements over the time course of minutes and hours.Second, they can be
8-4 Polychlorinated biphenyls (PCBs) are compounds that were once used as an insulating material in electrical transformers before being banned in the United States during the 1970s because of concerns about their toxicity.Despite the ban, PCBs can still be detected in most people because they are
8-3 The plots in Box 8-3 show data from four different experiments together with the associated observations.Compute the regression and correlation coefficients for each of these four sets of data. Discuss the similarities and differences among these sets of data. Include an examination of the
8-2 Plot the data and compute the linear regression of Y on X and correlation coefficient for each of the sets of observations shown in Table 8-10. In each case, draw the regression line on the same plot as the data. Discuss the results.
8-1 Plot the data and compute the linear regression of Y on X and correlation coefficient for each of the sets of observations shown in Table 8-9. In each case, draw the regression line on the same plot as the data. What stays the same and what changes? Why?
7-11 Rework Problem 5-14 using confidence intervals
7-10 Rework Problem 5-13 using confidence intervals.
7-9 Rework Problem 5-12 using confidence intervals.
7-8 Rework Problem 5-11 using confidence intervals.
7-7 Rework Problem 5-5 using confidence intervals.
7-6 Use the data in Problem 2-4 to find the 95% confidence interval for 90% and 95% of the population of PCB concentrations in Japanese adults. Plot these intervals together with the observations.
7-5 Find the 95% confidence intervals for the percentages of articles with favorable results in the two classes of studies in Problem 5-6.
7-4 Find the 95% confidence intervals for the mean difference in the six minute walk distances for the two test groups in Problem 4-3. Compare this result with the hypothesis test in Problem 4-3.
7-3 Find the 95% confidence intervals for the proportions of adverse outcomes as well as the 95% confidence interval for the difference in rates of adverse outcomes in Problem 5-1. Compare this result with the hypothesis test in Problem 5-1.
7-2 Find the 95% confidence interval for the difference in mean adenosine triphosphate (ATP) production per gram in the two groups of children in Problem 3-1. Based on this confidence interval is the difference significant with P < .05?
7-1 Find the 90% and 95% confidence intervals for the mean levels of polychlorinated biphenyl (PCB) levels in Problem 2-3.
6-10 How large would the sample size need to be in Problem 6-9 to reach 80% power?
6-9 What is the power of the experiment in Problem 5-4 to detect a situation in which nefazodone and psychotherapy each causes remission one-third of the time, and nefazodone and psychotherapy combined cause remission onehalf of the time? Assume that the same number of people take each treatment as
6-8 How large must each sample group be to have an 80%power to detect a change of 80 W with 95% confidence?
6-7 Use the data in Problem 3-3 to find the power of detecting an increase in a change in stair climbing power of 50 and 100 W with 95% confidence.
6-6 Use the data in Problem 3-2 to find the power of detecting a change in mean forced midexpiratory flow of 0.25 L/s with 95% confidence.
6-5 How large a sample would be necessary to be 90%confident that men have vertebral bone densities that differ by at least 30% of the values for women when you wish to be 95% confident in any conclusion that vertebral bone densities differ between men and women?
6-4 In Problem 3-5 (and again in Prob. 4-5), we decided that there was insufficient evidence to conclude that men and women who have had at least one vertebral fracture differ in vertebral bone density. What is the power of this test to detect average (with a = 0.05) bone density in men 20% lower
6-3 What is the minimum detectable effect one could obtain with 20 people in each group and 80% power?
6-2 How large a sample size would be necessary to increase the power of this study to detect a 25 mg/dL change in total cholesterol?
6-1 Both diabetes and high cholesterol interact to increase the risk of heart disease. Changing diet affects both blood sugar and cholesterol. To investigate how different diets used to control diabetes affects cardiovascular risk factors, Neal Barnard and colleagues†compared the effects of a

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Primer Of Biostatistics 7th Edition Stanton Glantz - Solutions

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