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statistical sampling to auditing

Questions and Answers of Statistical Sampling To Auditing

28. Exponential families . The exponential family (12) with T(x) = x and Q(8) = 8 is STPoo' with 0 the natural parameter space and !l' = (- 00, 00). [That the determinant leUjxJI, i, j = 1, . . . ,
27. Totally positive families. A family of distributions with probability densities Pe(x), 8 and x real-valued and varying over n and !!£ respectively, is said to be totally positive of order r (TPr
26. For a random variable X with binomial distribution b(p, n), determine the constants Cj , 't, (i = 1,2) in the UMP test (24) for testing H : P s .2 or s .7 when a = .1 and n = 15. Find the power
25. Let FI , ••• , F", + I be real-valued functions defined over a space U. A sufficient condition for Uo to maximize Fm +1 subject to F;(u) C; (i = 1, ... , m) is that it satisfies these side
24. The following example shows that Corollary 4 does not extend to a countably infinite family of distributions. Let Pn be the uniform probability density on [0, 1 + lin], and Po the uniform density
23. Optimum selection procedures. On each member of a population n measurements (XI" . . , X,,) = X are taken, for example the scores of n aptitude tests which are administered to judge the
22. If fJ ((J) denotes the power function of the UMP test of Corollary 2, and if the function Q of (12) is differentiable, then fJ'((J) > 0 for all (J for which Q'«(J) > O. [To show that {J'«(Jo) >
21. Confidence bounds withminimumrisk. Let L«(J,~) be nonnegative and nonincreasing in its second argument for < (J , and equal to 0 for (J . If and ~* are two lower confidence bounds for (J such
20. (i) For n = 5,10 and 1 - a = .95, graph the upper confidence limits p and p* of Example 7 as functions of t = x + u. (ii) For the same values of n and a l = a2 = .05, graph the lower and upper
19. In the experiment discussed in Example 5, n binomial trials with probability of success P = 1 - -~ are performed for the purpose of testing A = Ao against A = AI ' Experiments corresponding to
18. For the 2 X 2 table described in Example 4, and under the assumption P s 'TT s ! made there, a sample from iJ is more informative than one from A. On the other hand, samples from B and iJ are not
17. Conditions for comparability. (i) Let X and X' be two random variables taking on the values 1 and 0, and suppose that P(X-1) Po, P(X-1) P or that P(X = 1} = P P(X 1) p. Without loss of generality
16. If the experiment (f, g) is more informative than (f', g'), then (g,f) is more informative than (g', f').
15. If FQ , FI are two cumulative distribution functions on the real line, then FI(x) s FQ(x) for all x if and only if Eo1/J(X) s EI1/J(X) for any nondecreasing function 1/J.
14. Extension of Lemma 2. Let Po and P, be two distributions with densities Po, P such that p(x)/Po(x) is a nondecreasing function of a real-valued statistic T(x). (i) If 7 has probability density p;
13. Let X be a single observation from the Cauchy density given at the end of Section 3. (i) Show that no UMP test exists for testing = 0 against > 0. (ii) Determine the totality of different shapes
12. Let X (X., X) be a sample from the uniform distribution U(0,0 + 1). (i) For testing H:00, against K: 0> 0 at level a there exists a UMP test which rejects when min(X,,..., X) > + C(a) or max(XX)
11. When a Poisson process with rate is observed for a time interval of length 'T, the number X of events occurring has the Poisson distribution P(~'T) Under an alternative scheme, the process is
10. Let Xl' . .. ' Xn be independently distributed with density (2fJ)-le- x / 2I1 , x 0, and let YI ;5; •• • ;5; y" be the ordered X's. Assume that YI becomes available first, then Y2 , and so
9. Let the probability density PII of X have monotone likelihood ratio in T(x), and consider the problem of testing H : 0 ;5; 00 against 0 > fJo. If the distribution of T is continuous, the p-value
8. (i) A necessary and sufficient condition for densities PII(x) to have monotone likelihood ratio in x, if the mixed second derivative a2IogplI(x)/ao ax exists, is that this derivative is 0 for all
7. Let X be the number of successes in n independent trials with probability P of success, and let l/l(x) be the UMP test (9) for testing P s Po against P > Po at level of significancea. (i) For n =
6. Fully informative statistics. A statistic T is fully informative if for every decision problem the decision procedures based only on T form an essentially complete class. If 9 is dominated and T
5. If the sample space !l is Euclidean and Po, PI have densities with respect to Lebesgue measure, there exists a nonrandomized most powerful test for testing Po against PI at every significance
4. The following example shows that the power of a test can sometimes be increased by selecting a random rather than a fixed sample size even when the randomization does not depend on the
3. UMP test for exponential densities. Let XI' ... ' Xn be a sample from the exponential distribution E(a, b) of Chapter 1, Problem 18, and let ,\(1) = min( XI, .. . , Xn ) . (i) Determine the UMP
2. UMP test for U(0, 8). Let X = (Xl" .. , X;,) be a sample from the uniform distribution on (0, 8). (i) For testing H : 8 80 against K : 8 > 80 any test is UMP at level a for which Esoep(X) =a,
1. Let Xl' . . . • X" be a sample from the normal distribution N(t (1 2 ). (i) If 11 = 110 (known), there exists a UMP test for testing H: ~o against > ~o, which rejects when 1:(X; - ~o) is too
16. Let ° be the natural parameter space of the exponential family (35), and for any fixed tr + 1, ... , tk (r < k) let 0e,.....9, be the natural parameter space of the family of conditional
15. For any 8 which is an interior point of the natural parameter space, the expectations and covariances of the statistics in the exponential family (35) are given by E[~(X)] alog C(8) a~ (j=I, ...
14. Life testing. Let XI" ' " XII be independently distributed with exponential density (20) -le- x / 28 for x 0, and let the ordered X's be denoted by Y1 :::; Y2 :::; • • • :::; It is assumed
13. Let Xi (i = 1, .. . , s) be independently distributed with Poisson distribution P(A;), and let To = LXj, T; = Xi' A = LAj . Then To has the Poisson distribution P( A), and the conditional
12. For a decision problem with a finite number of decisions, the class of procedures depending on a sufficient statistic T only is essentially complete. [For Euclidean sample spaces this follows
11. If a statistic T is sufficient for 9, then for every function I which is (d, Pu)-integrable for all 8 e n there exists a determination of the conditional expectation function Eu[f(X)lt] that is
10. Pairwise sufficiency. A statistic T is pairwise sufficientfor 9 if it is sufficient for every pair of distributions in 9 . (i) If 9 is countable and T is pairwise sufficient for 9, then T is
9. Sufficiency of likelihood ratios. Let Po, PI be two distributions with densities Po'Pi - Then T(x) = PI(x )Ipo(x) is sufficient for 9 = {Po' PI}' [This follows from the factorization criterion by
8. Symmetric distributions. (i) Let 9 be any family of distributions of X = (XI " .. , X,,) which are symmetric in the sense that p{ (.\';" ... ,.\';J E A} = P{ (XI' " '' X,,) E A} for all Borel sets
7. Let!l' = '!!Ix.r, and suppose that Po, PI are two probability distributions given by dPo(y , t) = f(y)g(t) dp.(y) dv(t), dPI(y , r) = h(y , r) dp.(y) dv(t) , where h(y, t)lf(y)g(t) < 00 . Then
6. Prove Theorem 4 for the case of an n-dimensional sample space. [The condition that the cumulative distribution function is nondecreasing is replaced by P{x,continuous on the right can be stated as
5. (i) Let be any family of distributions X = (X...., X) such that P{(X, XXX,, X,) A} = P{(x,..., X) = A} for all Borel sets A and all i=1,..., n. For any sample point (x1,...,x,,) define (y)(x,
4. Let (~ Jaf) be a measurable space, and Jafo a a-field contained in Jaf. Suppose that for any function T, the a-field gj is taken as the totality of sets B such that T - 1(B) E Jaf. Then it is not
3. If f(x) > 0 for all xES and J1. is a-finite, then fsfdJ1. = 0 implies J1.(S) = o. [Let SII be the subset of S on which f(x) l in. Then J1.(S) .s LJ1.(S,,) and J1.(SII) nfsJdJ1. s nfsfdJ1. = 0.]
2. Radon-Nikodym derivatives. (i) If X and J1. are a-finite measures over (~, Jaf) and J1. is absolutely continuous with respect to X, then ffdJ1. = ff:~ dX for any J1.-integrable function f . (ii)
1. Monotone class. A class !F of subsets of a space is a field if it contains the whole space and is closed under complementation and under finite unions; a class vi( is monotone if the union and
19. A statistic T satisfying (17}-(19) is sufficient if and only if it satisfies (20).
18. (i) Let Xl' . . . ' Xn be a sample from the uniform distribution V(O, 8), o< 8 < 00, and let T = max(Xl' ... , Xn ) . Show that T is sufficient, once by using the definition of sufficiency and
17. In n independent trials with constant probability p of success, let X; = 1 or 0 as the i th trial is a success or not. Then E7-1 X; is minimal sufficient. [Let T = EX; and suppose that V = f(T)
16. (i) Let X take on the values 8 - 1 and 8 + 1 with probability t each. The problem of estimating 8 with loss function L(8,d) = min(18 - dl,l) remains invariant under the transformation gX = X +c,
15. Admissibility of invariant procedures. If a decision problem remains invariant under a finite group, and if there exists a procedure 80 that uniformly minimizes the risk among all invariant
14. Admissibility of unbiased procedures. (i) Under the assumptions of Problem 10, if among the unbiased procedures there exists one with uniformly minimum risk, it is admissible. (ii) That in
13. (i) Let XI" ' " Xn be a sample from N( t ( 2 ), and consider the problem of deciding between Wo: < 0 and WI : O. If x = Ex;/n and C = (a l /ao) 2/n, the likelihood-ratio procedure takes decision
12. (i) Let X have probability density plJ(x) with 8 one of the values 81" " , 8n , and consider the problem of determining the correct value of 8, so that the choice lies between the n decisions dl
11. Invariance and minimax. Let a problem remain invariant relative to the groups G, G, and G* over the spaces !!C, 0 , and D respectively. Then a randomized procedure y. is defined to be invariant
10. Unbiasedness and minimax. Let °= 00 U °1 where 00' °1 are mutually exclusive, and consider a two-decision problem with loss function L(8, d;) = a; for 8 E OJ (j '" i) and L(8, d;) = 0 for 8 E
9. (i) As an example in which randomization reduces the maximum risk, suppose that a coin is known to be either standard (HT) or to have heads on both sides (HH). The nature of the coin is to be
8. Structure of Bayes solutions. (i) Let be an unobservable random quantity with probability density p(0), and let the probability density of X be p(x) when = 0. Then 8 is a Bayes solution of a given
7. Unbiasedness in interval estimation . Confidence intervals 1= (1:, I) are unbiased for estimating ° with loss function L(O, /) = (0 -1:)2 + (I - 0)2 provided E[t(1: + I)] = ° for all 0, that
6. Relation of unbiasedness and inuariance. (i) If 60 is the unique (up to sets of measure 0) unbiased procedure with uniformly minimum risk, it is almost invariant. (ii) If G is transitive and G*
5. Let CC be any class of procedures that is closed under the transformations of a group G in the sense that 6 E 'i implies g*6g- 1 E 'i for all g E G. If there exists a unique procedure 60 that
4. Nonexistence of unbiased procedures. Let Xl " '" Xn be independently distributed with density (l/a)j(x - ~)/a), and let 0 =a, a) . Then no estimtor of exists which is unbiased with respect to the
3. Median unbiasedness. (i) A real number m is a median for the random variable Y if P{ Y m} t, P{Y m} t. Then all real ai' a2 such that m s a\ s a2 or m a\ a2 satisfy ElY - ad ElY - a21· (ii) For
2. Unbiasedness in point estimation . Suppose that 'I is a continuous real-valued function defined over n which is not constant in any open subset of n, and that the expectation h(O) = Eg8(X) is a
1. The following distributions arise on the basis of assumptions similar to those leading to (1)-(3). (i) Independent trials with constant probability p of success are carried out until a preassigned
The history of statistical hypothesis testing really began with a tea-tasting experiment(Fisher, 1935), so it seems fitting for this book to end with one. The owner of a small tearoom does not think
For the data in Exercise 18.5, we could say that 3 out of 10 residents used fewer hypotheses the second time and 7 used more. We could test this with x2. How would this differ from Friedman’s test
It would be possible to apply Friedman’s test to the data in Exercise 18.5. What would we lose if we did?
What advantage does the study described in Exercise 18.18 have over the study described in Exercise 18.17?
The test referred to in Exercise 18.19 is available on in the Window’s program on the book’s Web site. Run that program on the data for Exercise 18.18 and report the results. (There is
I did not discuss randomization tests on the evaluation of data that are laid out like a one-way analysis of variance (as in Exercise 18.17), but you should be able to suggest an analysis that would
As an alternative method of evaluating a group home, suppose that we take 12 adolescents who have been declared delinquent. We take the number of days truant (1) during the month before they are
A psychologist operating a group home for delinquent adolescents needs to show that it is successful at reducing delinquency. He samples nine adolescents living in their parents’ home that the
Three rival professors teaching English I all claim the honor of having the best students. To settle the issue, eight students are randomly drawn from each class and are given the same exam, which is
Why is rejection of the null hypothesis using a t test a more specific statement than rejection of the null hypothesis using the appropriate nonparametric test?
One of the arguments put forth in favor of nonparametric tests is that they are more appropriate for ordinal-scale data. This issue was addressed earlier in the book in a different context. Give a
What is the difference between the null hypothesis tested by Wilcoxon’s matched-pairs signed-ranks test and the corresponding t test?
What is the difference between the null hypothesis tested by Wilcoxon’s rank-sum test and the corresponding t test?
The results in Exercise 18.8 are not quite as clear-cut as we might like. Plot the differences as a function of the first-born’s score. What does this figure suggest?
How would we run a standard resampling test for the data in Exercise 18.8?
Rerun the analysis in Exercise 18.8 using the normal approximation.
It has been argued that first-born children tend to be more independent than later-born children.Suppose we develop a 25-point scale of independence and rate each of 20 first-born children and their
How would you go about applying a resampling procedure to test the difference between Before and After scores in Exercise 18.6?
Refer to Exercise 18.5.a. Repeat the analysis using the normal approximation.b. How well do the two answers (18.5a and 18.6a) agree? Why do they not agree exactly?
Nurcombe and Fitzhenry-Coor (1979) have argued that training in diagnostic techniques should lead a clinician to generate (and test) more hypotheses in coming to a decision about a case. Suppose we
Repeat the analysis in Exercise 18.2 using the appropriate one-tailed test.
Repeat the analysis in Exercise 18.2 using the normal approximation.
Kapp, Frysinger, Gallagher, and Hazelton (1979) have demonstrated that lesions in the amygdala can reduce certain responses commonly associated with fear (e.g., decreases in heart rate). If fear is
McConaughy (1980) has argued that younger children organize stories in terms of simple descriptive (“and then . . .”) models, whereas older children incorporate causal statements and social
Why might it not make much sense to examine the results for heterogeneity of effects?
Compute confidence limits on the mean risk ratio and draw the appropriate conclusions.
For the Bloch et al. study what would you conclude about the risk of increasing tic behavior using methylphenidate?
Create a forest plot for the data in Bloch’s study.
Why would it not make sense to try to treat this meta-analysis as a random model?
Calculate the mean effect size estimate and its confidence limits for the data in this table.
Recompute the mean effect size and its confidence limits assuming that the underlying model is fixed.
Test the hypothesis that there are no differences between the two subgroups in this table.
Calculate confidence limits on the mean effect size computed in the previous exercise.
Assuming a random model, calculate a mean effect size and its standard error.
Create a forest plot of these data.
Make up or find an example with respect to Exercise 16.25 where the slope is not nearly 1.0.Analyze it using both the analysis of covariance and a t test on difference scores. Do either of these
I said that in any experiment where we have pretest and posttest scores we could either look at the difference scores (compared across groups) or use the pretest as a covariate. These two analyses

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