Perfectly random sampling of truncated multinormal distributions

概率论与数理统计中的英文单词和短语概率论与数理统计Probability Theory and Mathematical Statistics第一章概率论的基本观点Chapter 1Introduction of Probability Theory不确立性indeterminacy必定现象certain phenomenon随机现象random phenomenon试验experiment结果outcome频次数frequency number样本空间sample space出现次数frequency of occurrencen 维样本空间n-dimensional sample space样本空间的点point in sample space随机事件random event / random occurrence基本领件elementary event必定事件certain event不行能事件impossible event等可能事件equally likely event事件运算律operational rules of events事件的包括implication of events并事件union events交事件intersection events互不相容事件、mutually exclusive exvents/互斥事件/incompatible events互逆的mutually inverse加法定理addition theorem古典概率classical probability古典概率模型classical probabilistic model几何概率geometric probability 乘法定理product theorem概率乘法multiplication of probabilities条件概率conditional probability全概率公式、全formula of total probability概率定理贝叶斯公式、逆Bayes formula概率公式后验概率posterior probability先验概率prior probability独立事件independent event独立随机事件independent random event独立实验independent experiment两两独立pairwise independent两两独立事件pairwise independent events第二章随机变量及其散布Chapter2Random Variables and Distributions随机变量random variables失散随机变量discrete random variables概率散布律law of probability distribution一维概率散布one-dimension probability distribution 概率散布probability distribution两点散布two-point distribution伯努利散布Bernoulli distribution二项散布 / 伯努Binomial distribution利散布超几何散布hypergeometric distribution三项散布trinomial distribution多项散布polynomial distribution泊松散布Poisson distribution泊松参数Poisson theorem散布函数distribution function概率散布函数probability density function连续随机变量continuous random variable概率密度probability density概率密度函数probability density function概率曲线probability curve平均散布uniform distribution指数散布exponential distribution指数散布密度函exponential distribution density 数function正态散布、高斯normal distribution散布标准正态散布standard normal distribution正态概率密度函normal probability density function数正态概率曲线normal probability curve标准正态曲线standard normal curve柯西散布Cauchy distribution散布密度density of distribution第三章多维随机变量及其散布Chapter 3 Multivariate Random Variables and Distributions二维随机变量two-dimensional random variable结合散布函数joint distribution function二维失散型随机two-dimensional discrete random 变量variable二维连续型随机two-dimensional continuous random 变量variable结合概率密度joint probability variablen 维随机变量n-dimensional random variablen 维散布函数n-dimensional distribution functionn 维概率散布n-dimensional probability distribution边沿散布marginal distribution边沿散布函数marginal distribution function边沿散布律law of marginal distribution边沿概率密度marginal probability density二维正态散布two-dimensional normal distribution二维正态概率密two-dimensional normal probability 度density 二维正态概率曲two-dimensional normal probability 线curve条件散布conditional distribution条件散布律law of conditional distribution条件概率散布conditional probability distribution条件概率密度conditional probability density边沿密度marginal density独立随机变量independent random variables第四章随机变量的数字特点Chapter 4 Numerical Characteristics fo Random Variables数学希望、均值mathematical expectation希望值expectation value方差variance标准差standard deviation随机变量的方差variance of random variables均方差mean square deviation有关关系dependence relation有关系数correlation coefficient协方差covariance协方差矩阵covariance matrix切比雪夫不等式Chebyshev inequality第五章大数定律及中心极限制理Chapter 5 Law of Large Numbers and Central Limit Theorem大数定律law of great numbers切比雪夫定理的special form of Chebyshev theorem特别形式依概率收敛convergence in probability伯努利大数定律Bernoulli law of large numbers同散布same distribution列维 - 林德伯格independent Levy-Lindberg theorem定理、独立同分布中心极限制理辛钦大数定律Khinchine law of large numbers利亚普诺夫定理Liapunov theorem棣莫弗 - 拉普拉De Moivre-Laplace theorem斯定理第六章样本及抽样分布Chapter 6 Samples and Sampling Distributions统计量statistic整体population个体individual样本sample容量capacity统计剖析statistical analysis统计散布statistical distribution统计整体statistical ensemble随机抽样stochastic sampling / random sampling 随机样本random sample简单随机抽样simple random sampling简单随机样本simple random sample经验散布函数empirical distribution function样本均值sample average / sample mean样本方差sample variance样本标准差sample standard deviation标准偏差standard error样本 k 阶矩sample moment of order k样本中心矩sample central moment样本值sample value样本大小、样本sample size容量样本统计量sampling statistics 随机抽样散布random sampling distribution抽样散布、样本sampling distribution散布自由度degree of freedomZ 散布Z-distributionU 散布U-distribution第七章参数预计Chapter 7Parameter Estimations统计推测statistical inference参数预计parameter estimation散布参数parameter of distribution参数统计推测parametric statistical inference点预计point estimate / point estimation整体中心距population central moment整体有关系数population correlation coefficient整体散布population covariance整体协方差population covariance点预计量point estimator预计量estimator无偏预计unbiased estimate/ unbiasedestimation预计量的有效性efficiency of estimator矩法预计moment estimation整体均值population mean整体矩population moment整体 k 阶矩population moment of order k整体参数population parameter极大似然预计maximum likelihood estimation极大似然预计量maximum likelihood estimator极大似然法maximum likelihood method /maximum-likelihood method似然方程likelihood equation似然函数likelihood function区间预计interval estimation置信区间confidence interval置信水平confidence level置信系数confidence coefficient单侧置信区间one-sided confidence interval置信上限置信下限U 预计正态整体整体方差的预计confidence upper limit confidence lower limitU-estimatornormal populationestimation of population variance置信度方差比degree of confidence variance ratio第八章假定查验Chapter 8Hypothesis Testings参数假定假定查验两类错误统计假定统计假定查验查验统计量明显性查验统计明显性parametric hypothesis hypothesis testingtwo types of errors statistical hypothesis statistical hypothesis testing test statisticstest of significance statistical significance单边查验、单侧one-sided test查验单侧假定、单边one-sided hypothesis 假定两侧假定两侧查验明显水平拒绝域 / 否认区two-sided hypothesis two-sided testing significant level rejection region域接受地区acceptance regionU 查验F 查验方差齐性的查验拟合优度查验U-testF-testhomogeneity test for variances test of goodness of fit。

审计学：⼀种整合⽅法阿伦斯英⽂版第12版课后答案Chapter15SolutionsManualChapter 15Audit Sampling for Tests of Controls andSubstantive Tests of TransactionsReview Questions15-1 A representative sample is one in which the characteristics of interest for the sample are approximately the same as for the population (that is, the sample accurately represents the total population). If the population contains significant misstatements, but the sample is practically free of misstatements, the sample is nonrepresentative, which is likely to result in an improper audit decision. The auditor can never know for sure whether he or she has a representative sample because the entire population is ordinarily not tested, but certain things, such as the use of random selection, can increase the likelihood of a representative sample.15-2Statistical sampling is the use of mathematical measurement techniques to calculate formal statistical results. The auditor therefore quantifies sampling risk when statistical sampling is used. In nonstatistical sampling, the auditor does not quantify sampling risk. Instead, conclusions are reached about populations on a more judgmental basis.For both statistical and nonstatistical methods, the three main parts are:1. Plan the sample2. Select the sample and perform the tests3. Evaluate the results15-3In replacement sampling, an element in the population can be included in the sample more than once if the random number corresponding to that element is selected more than once. In nonreplacement sampling, an element can be included only once. If the random number corresponding to an element is selected more than once, it is simply treated as a discard the second time. Although both selection approaches are consistent with sound statistical theory, auditors rarely use replacement sampling; it seems more intuitively satisfying to auditors to include an item only once.15-4 A simple random sample is one in which every possible combination of elements in the population has an equal chance of selection. Two methods of simple random selection are use of a random number table, and use of the computer to generate random numbers. Auditors most often use the computer to generate random numbers because it saves time, reduces the likelihood of error, and provides automatic documentation of the sample selected.15-5In systematic sampling, the auditor calculates an interval and then methodically selects the items for the sample based on the size of the interval. The interval is set by dividing the population size by the number of sample items desired.To select 35 numbers from a population of 1,750, the auditor divides 35 into 1,750 and gets an interval of 50. He or she then selects a random number between 0 and 49. Assume the auditor chooses 17. The first item is the number 17. The next is 67, then 117, 167, and so on.The advantage of systematic sampling is its ease of use. In most populations a systematic sample can be drawn quickly, the approach automatically puts the numbers in sequential order and documentation is easy.A major problem with the use of systematic sampling is the possibility of bias. Because of the way systematic samples are selected, once the first item in the sample is selected, other items are chosen automatically. This causes no problems if the characteristics of interest, such as control deviations, are distributed randomly throughout the population; however, in many cases they are not. If all items of a certain type are processed at certain times of the month or with the use of certain document numbers, a systematically drawn sample has a higher likelihood of failing to obtain a representative sample. This shortcoming is sufficiently serious that some CPA firms prohibit the use of systematic sampling. 15-6The purpose of using nonstatistical sampling for tests of controls and substantive tests of transactions is to estimate the proportion of items in a population containing a characteristic or attribute of interest. The auditor is ordinarily interested in determining internal control deviations or monetary misstatements for tests of controls and substantive tests of transactions.15-7 A block sample is the selection of several items in sequence. Once the first item in the block is selected, the remainder of the block is chosen automatically. Thus, to select 5 blocks of 20 sales invoices, one would select one invoice and the block would be that invoice plus the next 19 entries. This procedure would be repeated 4 other times.15-8 The terms below are defined as follows:15-8 (continued)15-9The sampling unit is the population item from which the auditor selects sample items. The major consideration in defining the sampling unit is making it consistent with the objectives of the audit tests. Thus, the definition of the population and the planned audit procedures usually dictate the appropriate sampling unit.The sampling unit for verifying the occurrence of recorded sales would be the entries in the sales journal since this is the document the auditor wishes to validate. The sampling unit for testing the possibility of omitted sales is the shipping document from which sales are recorded because the failure to bill a shipment is the exception condition of interest to the auditor.15-10 The tolerable exception rate (TER) represents the exception rate that the auditor will permit in the population and still be willing to use the assessed control risk and/or the amount of monetary misstatements in the transactions established during planning. TER is determined by choice of the auditor on the basis of his or her professional judgment.The computed upper exception rate (CUER) is the highest estimated exception rate in the population, at a given ARACR. For nonstatistical sampling, CUER is determined by adding an estimate of sampling error to the SER (sample exception rate). For statistical sampling, CUER is determined by using a statistical sampling table after the auditor has completed the audit testing and therefore knows the number of exceptions in the sample.15-11 Sampling error is an inherent part of sampling that results from testing less than the entire population. Sampling error simply means that the sample is not perfectly representative of the entire population.Nonsampling error occurs when audit tests do not uncover errors that exist in the sample. Nonsampling error can result from:1. The auditor's failure to recognize exceptions, or2. Inappropriate or ineffective audit procedures.There are two ways to reduce sampling risk:1. Increase sample size.2. Use an appropriate method of selecting sample items from thepopulation.Careful design of audit procedures and proper supervision and review are ways to reduce nonsampling risk.15-12 An attribute is the definition of the characteristic being tested and the exception conditions whenever audit sampling is used. The attributes of interest are determined directly from the audit program.15-13 An attribute is the characteristic being tested for in a population. An exception occurs when the attribute being tested for is absent. The exception for the audit procedure, the duplicate sales invoice has been initialed indicating the performance of internal verification, is the lack of initials on duplicate sales invoices.15-14 Tolerable exception rate is the result of an auditor's judgment. The suitable TER is a question of materiality and is therefore affected by both the definition and the importance of the attribute in the audit plan.The sample size for a TER of 6% would be smaller than that for a TER of 3%, all other factors being equal.15-15 The appropriate ARACR is a decision the auditor must make using professional judgment. The degree to which the auditor wishes to reduce assessed control risk below the maximum is the major factor determining the auditor's ARACR.The auditor will choose a smaller sample size for an ARACR of 10% than would be used if the risk were 5%, all other factors being equal.15-16 The relationship between sample size and the four factors determining sample size are as follows:a. As the ARACR increases, the required sample size decreases.b. As the population size increases, the required sample size isnormally unchanged, or may increase slightly.c. As the TER increases, the sample size decreases.d. As the EPER increases, the required sample size increases.15-17 In this situation, the SER is 3%, the sample size is 100 and the ARACR is 5%. From the 5% ARACR table (Table 15-9) then, the CUER is 7.6%. This means that the auditor can state with a 5% risk of being wrong that the true population exception rate does not exceed 7.6%.15-18 Analysis of exceptions is the investigation of individual exceptions to determine the cause of the breakdown in internal control. Such analysis is important because by discovering the nature and causes of individual exceptions, the auditor can more effectively evaluate the effectiveness of internal control. The analysis attempts to tell the "why" and "how" of the exceptions after the auditor already knows how many and what types of exceptions have occurred.15-19 When the CUER exceeds the TER, the auditor may do one or more of the following:1. Revise the TER or the ARACR. This alternative should be followed onlywhen the auditor has concluded that the original specifications weretoo conservative, and when he or she is willing to accept the riskassociated with the higher specifications.2. Expand the sample size. This alternative should be followed whenthe auditor expects the additional benefits to exceed the additionalcosts, that is, the auditor believes that the sample tested was notrepresentative of the population.3. Revise assessed control risk upward. This is likely to increasesubstantive procedures. Revising assessed control risk may bedone if 1 or 2 is not practical and additional substantive proceduresare possible.4. Write a letter to management. This action should be done inconjunction with each of the three alternatives above. Managementshould always be informed when its internal controls are notoperating effectively. If a deficiency in internal control is consideredto be a significant deficiency in the design or operation of internalcontrol, professional standards require the auditor to communicatethe significant deficiency to the audit committee or its equivalent inwriting. If the client is a publicly traded company, the auditor mustevaluate the deficiency to determine the impact on the auditor’sreport on internal control over financial reporting. If the deficiency isdeemed to be a material weakness, the auditor’s report on internalcontrol would contain an adverse opinion.15-20 Random (probabilistic) selection is a part of statistical sampling, but it is not, by itself, statistical measurement. To have statistical measurement, it is necessary to mathematically generalize from the sample to the population.Probabilistic selection must be used if the sample is to be evaluated statistically, although it is also acceptable to use probabilistic selection with a nonstatistical evaluation. If nonprobabilistic selection is used, nonstatistical evaluation must be used.15-21 The decisions the auditor must make in using attributes sampling are: What are the objectives of the audit test? Does audit sampling apply?What attributes are to be tested and what exception conditions are identified?What is the population?What is the sampling unit?What should the TER be?What should the ARACR be?What is the EPER?What generalizations can be made from the sample to thepopulation?What are the causes of the individual exceptions?Is the population acceptable?15-21 (continued)In making the above decisions, the following should be considered: The individual situation.Time and budget constraints.The availability of additional substantive procedures.The professional judgment of the auditor.Multiple Choice Questions From CPA Examinations15-22 a. (1) b. (3) c. (2) d. (4)15-23 a. (1) b. (3) c. (4) d. (4)15-24 a. (4) b. (3) c. (1) d. (2)Discussion Questions and Problems15-25a.An example random sampling plan prepared in Excel (P1525.xls) is available on the Companion Website and on the Instructor’s Resource CD-ROM, which is available upon request. The command for selecting the random number can be entered directly onto the spreadsheet, or can be selected from the function menu (math & trig) functions. It may be necessary to add the analysis tool pack to access the RANDBETWEEN function. Once the formula is entered, it can be copied down to select additional random numbers. When a pair of random numbers is required, the formula for the first random number can be entered in the first column, and the formula for the second random number can be entered in the second column.a. First five numbers using systematic selection:Using systematic selection, the definition of the sampling unit for determining the selection interval for population 3 is the total number of lines in the population. The length of the interval is rounded down to ensure that all line numbers selected are within the defined population.15-26a. To test whether shipments have been billed, a sample of warehouse removal slips should be selected and examined to see ifthey have the proper sales invoice attached. The sampling unit willtherefore be the warehouse removal slip.b. Attributes sampling method: Assuming the auditor is willing to accept a TER of 3% at a 10% ARACR, expecting no exceptions in the sample, the appropriate sample size would be 76, determined from Table 15-8.Nonstatistical sampling method: There is no one right answer to this question because the sample size is determined using professional judgment. Due to the relatively small TER (3%), the sample size should not be small. It will most likely be similar in size to the sample chosen by the statistical method.c. Systematic sample selection:22839 = Population size of warehouse removal slips(37521-14682).76 = Sample size using statistical sampling (students’answers will vary if nonstatistical sampling wasused in part b.300 = Interval (22839/76) if statistical sampling is used (students’ answers will vary if nonstatisticalsampling was used in part b).14825 = Random starting point.Select warehouse removal slip 14825 and every 300th warehouse removal slip after (15125, 15425, etc.)Computer generation of random numbers using Excel (P1526.xls): =RANDBETWEEN(14682,37521)The command for selecting the random number can be entered directly onto the spreadsheet, or can be selected from the function menu (math & trig) functions. It may be necessary to add the analysis tool pack to access the RANDBETWEEN function. Once the formula is entered, it can be copied down to select additional random numbers.d. Other audit procedures that could be performed are:1. Test extensions on attached sales invoices for clerical accuracy. (Accuracy)2. Test time delay between warehouse removal slip date and billing date for timeliness of billing. (Timing)3. Trace entries into perpetual inventory records to determinethat inventory is properly relieved for shipments. (Postingand summarization)15-26 (continued)e. The test performed in part c cannot be used to test for occurrenceof sales because the auditor already knows that inventory wasshipped for these sales. To test for occurrence of sales, the salesinvoice entry in the sales journal is the sampling unit. Since thesales invoice numbers are not identical to the warehouse removalslips it would be improper to use the same sample.15-27a. It would be appropriate to use attributes sampling for all audit procedures except audit procedure 1. Procedure 1 is an analyticalprocedure for which the auditor is doing a 100% review of the entirecash receipts journal.b. The appropriate sampling unit for audit procedures 2-5 is a line item,or the date the prelisting of cash receipts is prepared. The primaryemphasis in the test is the completeness objective and auditprocedure 2 indicates there is a prelisting of cash receipts. All otherprocedures can be performed efficiently and effectively by using theprelisting.c. The attributes for testing are as follows:d. The sample sizes for each attribute are as follows:15-28a. Because the sample sizes under nonstatistical sampling are determined using auditor judgment, students’ answers to thisquestion will vary. They will most likely be similar to the samplesizes chosen using attributes sampling in part b. The importantpoint to remember is that the sample sizes chosen should reflectthe changes in the four factors (ARACR, TER, EPER, andpopulation size). The sample sizes should have fairly predictablerelationships, given the changes in the four factors. The followingreflects some of the relationships that should exist in student’ssample size decisions:SAMPLE SIZE EXPLANATION1. 90 Given2. > Column 1 Decrease in ARACR3. > Column 2 Decrease in TER4. > Column 1 Decrease in ARACR (column 4 is thesame as column 2, with a smallerpopulation size)5. < Column 1 Increase in TER-EPER6. < Column 5 Decrease in EPER7. > Columns 3 & 4 Decrease in TER-EPERb. Using the attributes sampling table in Table 15-8, the sample sizesfor columns 1-7 are:1. 882. 1273. 1814. 1275. 256. 187. 149c.d. The difference in the sample size for columns 3 and 6 result from the larger ARACR and larger TER in column 6. The extremely large TER is the major factor causing the difference.e. The greatest effect on the sample size is the difference between TER and EPER. For columns 3 and 7, the differences between the TER and EPER were 3% and 2% respectively. Those two also had the highest sample size. Where the difference between TER and EPER was great, such as columns 5 and 6, the required sample size was extremely small.Population size had a relatively small effect on sample size.The difference in population size in columns 2 and 4 was 99,000 items, but the increase in sample size for the larger population was marginal (actually the sample sizes were the same using the attributes sampling table).f. The sample size is referred to as the initial sample size because it is based on an estimate of the SER. The actual sample must be evaluated before it is possible to know whether the sample is sufficiently large to achieve the objectives of the test.15-29 a.* Students’ answers as to whether the allowance for sampling error risk is sufficient will vary, depending on their judgment. However, they should recognize the effect that lower sample sizes have on the allowance for sampling risk in situations 3, 5 and 8.b. Using the attributes sampling table in Table 15-9, the CUERs forcolumns 1-8 are:1. 4.0%2. 4.6%3. 9.2%4. 4.6%5. 6.2%6. 16.4%7. 3.0%8. 11.3%c.d. The factor that appears to have the greatest effect is the number ofexceptions found in the sample compared to sample size. For example, in columns 5 and 6, the increase from 2% to 10% SER dramatically increased the CUER. Population size appears to have the least effect. For example, in columns 2 and 4, the CUER was the same using the attributes sampling table even though the population in column 4 was 10 times larger.e. The CUER represents the results of the actual sample whereas theTER represents what the auditor will allow. They must be compared to determine whether or not the population is acceptable.15-30a. and b. The sample sizes and CUERs are shown in the following table:a. The auditor selected a sample size smaller than that determinedfrom the tables in populations 1 and 3. The effect of selecting asmaller sample size than the initial sample size required from thetable is the increased likelihood of having the CUER exceed theTER. If a larger sample size is selected, the result may be a samplesize larger than needed to satisfy TER. That results in excess auditcost. Ultimately, however, the comparison of CUER to TERdetermines whether the sample size was too large or too small.b. The SER and CUER are shown in columns 4 and 5 in thepreceding table.c. The population results are unacceptable for populations 1, 4, and 6.In each of those cases, the CUER exceeds TER.The auditor's options are to change TER or ARACR, increase the sample size, or perform other substantive tests to determine whether there are actually material misstatements in thepopulation. An increase in sample size may be worthwhile inpopulation 1 because the CUER exceeds TER by only a smallamount. Increasing sample size would not likely result in improvedresults for either population 4 or 6 because the CUER exceedsTER by a large amount.d. Analysis of exceptions is necessary even when the population isacceptable because the auditor wants to determine the nature andcause of all exceptions. If, for example, the auditor determines thata misstatement was intentional, additional action would be requiredeven if the CUER were less than TER.15-30 (Continued)e.15-31 a. The actual allowance for sampling risk is shown in the following table:b. The CUER is higher for attribute 1 than attribute 2 because the sample sizeis smaller for attribute 1, resulting in a larger allowance for sampling risk.c. The CUER is higher for attribute 3 than attribute 1 because the auditorselected a lower ARACR. This resulted in a larger allowance for sampling risk to achieve the lower ARACR.d. If the auditor increases the sample size for attribute 4 by 50 items and findsno additional exceptions, the CUER is 5.1% (sample size of 150 and three exceptions). If the auditor finds one exception in the additional items, the CUER is 6.0% (sample size of 150, four exceptions). With a TER of 6%, the sample results will be acceptable if one or no exceptions are found in the additional 50 items. This would require a lower SER in the additional sample than the SER in the original sample of 3.0 percent. Whether a lower rate of exception is likely in the additional sample depends on the rate of exception the auditor expected in designing the sample, and whether the auditor believe the original sample to be representative.15-32a. The following shows which are exceptions and why:b. It is inappropriate to set a single acceptable tolerable exception rate and estimated population exception rate for the combined exceptions because each attribute has a different significance tothe auditor and should be considered separately in analyzing the results of the test.c. The CUER assuming a 5% ARACR for each attribute and a sample size of 150 is as follows:15-32 (continued)d.*Students’ answers will most likely vary for this attribute.e. For each exception, the auditor should check with the controller todetermine an explanation for the cause. In addition, the appropriateanalysis for each type of exception is as follows:15-33a. Attributes sampling approach: The test of control attribute had a 6% SER and a CUER of 12.9%. The substantive test of transactionsattribute has SER of 0% and a CUER of 4.6%.Nonstatistical sampling approach: As in the attributes samplingapproach, the SERs for the test of control and the substantive testof transactions are 6% and 0%, respectively. Students’ estimates ofthe CUERs for the two tests will vary, but will probably be similar tothe CUERs calculated under the attributes sampling approach.b. Attributes sampling approach: TER is 5%. CUERs are 12.9% and4.6%. Therefore, only the substantive test of transactions resultsare satisfactory.Nonstatistical sampling approach: Because the SER for the test ofcontrol is greater than the TER of 5%, the results are clearly notacceptable. Students’ estimates for CUER for the test of controlshould be greater than the SER of 6%. For the substantive test oftransactions, the SER is 0%. It is unlikely that students will estimateCUER for this test greater than 5%, so the results are acceptablefor the substantive test of transactions.c. If the CUER exceeds the TER, the auditor may:1. Revise the TER if he or she thinks the original specificationswere too conservative.2. Expand the sample size if cost permits.3. Alter the substantive procedures if possible.4. Write a letter to management in conjunction with each of theabove to inform management of a deficiency in their internalcontrols. If the client is a publicly traded company, theauditor must evaluate the deficiency to determine the impacton the auditor’s report on internal control over financialreporting. If the deficiency is deemed to be a materialweakness, the auditor’s report on internal control wouldcontain an adverse opinion.In this case, the auditor has evidence that the test of control procedures are not effective, but no exceptions in the sample resulted because of the breakdown. An expansion of the attributestest does not seem advisable and therefore, the auditor shouldprobably expand confirmation of accounts receivable tests. Inaddition, he or she should write a letter to management to informthem of the control breakdown.d. Although misstatements are more likely when controls are noteffective, control deviations do not necessarily result in actualmisstatements. These control deviations involved a lack ofindication of internal verification of pricing, extensions and footingsof invoices. The deviations will not result in actual errors if pricing,extensions and footings were initially correctly calculated, or if theindividual responsible for internal verification performed theprocedure but did not document that it was performed.e. In this case, we want to find out why some invoices are notinternally verified. Possible reasons are incompetence,carelessness, regular clerk on vacation, etc. It is desirable to isolatethe exceptions to certain clerks, time periods or types of invoices.Case15-34a. Audit sampling could be conveniently used for procedures 3 and 4 since each is to be performed on a sample of the population.b. The most appropriate sampling unit for conducting most of the auditsampling tests is the shipping document because most of the testsare related to procedure 4. Following the instructions of the auditprogram, however, the auditor would use sales journal entries asthe sampling unit for step 3 and shipping document numbers forstep 4. Using shipping document numbers, rather than thedocuments themselves, allows the auditor to test the numericalcontrol over shipping documents, as well as to test for unrecordedsales. The selection of numbers will lead to a sample of actualshipping documents upon which tests will be performed.。

大气科学系微机应用基础Primer of microcomputer applicationFORTRAN77程序设计FORTRAN77 Program Design大气科学概论An Introduction to Atmospheric Science大气探测学基础Atmospheric Sounding流体力学Fluid Dynamics天气学Synoptic Meteorology天气分析预报实验Forecast and Synoptic analysis生产实习Daily weather forecasting现代气候学基础An introduction to modern climatology卫星气象学Satellite meteorologyC语言程序设计 C Programming大气探测实验Experiment on Atmospheric Detective Technique云雾物理学Physics of Clouds and fogs动力气象学Dynamic Meteorology计算方法Calculation Method诊断分析Diagnostic Analysis中尺度气象学Meso-Microscale Synoptic Meteorology边界层气象学Boundary Layer Meteorology雷达气象学Radar Meteorology数值天气预报Numerical Weather Prediction气象统计预报Meteorological Statical Prediction大气科学中的数学方法Mathematical Methods in Atmospheric Sciences专题讲座Seminar专业英语English for Meteorological Field of Study计算机图形基础Basic of computer graphics气象业务自动化Automatic Weather Service空气污染预测与防治Prediction and Control for Air Pollution现代大气探测Advanced Atmospheric Sounding数字电子技术基础Basic of Digital Electronic Techniqul大气遥感Remote Sensing of Atmosphere模拟电子技术基础Analog Electron Technical Base大气化学Atmospheric Chemistry航空气象学Areameteorology计算机程序设计Computer Program Design数值预报模式与数值模拟Numerical Model and Numerical Simulation接口技术在大气科学中的应用Technology of Interface in Atmosphere Sciences Application海洋气象学Oceanic Meteorology现代实时天气预报技术（MICAPS系统）Advanced Short-range Weather Forecasting Technique(MICAPS system)1) atmospheric precipitation大气降水2) atmosphere science大气科学3) atmosphere大气1.The monitoring and study of atmosphere characteristics in near space as an environment forspace weapon equipments and system have been regarded more important for battle support.随着临近空间飞行器的不断发展和运用,作为武器装备和系统环境的临近空间大气特性成为作战保障的重要条件。

15CHAPTER 3TEACHING NOTESFor undergraduates, I do not work through most of the derivations in this chapter, at least not in detail. Rather, I focus on interpreting the assumptions, which mostly concern the population. Other than random sampling, the only assumption that involves more than populationconsiderations is the assumption about no perfect collinearity, where the possibility of perfect collinearity in the sample (even if it does not occur in the population should be touched on. The more important issue is perfect collinearity in the population, but this is fairly easy to dispense with via examples. These come from my experiences with the kinds of model specification issues that beginners have trouble with.The comparison of simple and multiple regression estimates – based on the particular sample at hand, as opposed to their statistical properties – usually makes a strong impression. Sometimes I do not bother with the “partialling out” interpretation of multiple regression.As far as statistical properties, notice how I treat the problem of including an irrelevant variable: no separate derivation is needed, as the result follows form Theorem 3.1.I do like to derive the omitted variable bias in the simple case. This is not much more difficult than showing unbiasedness of OLS in the simple regression case under the first four Gauss-Markov assumptions. It is important to get the students thinking aboutthis problem early on, and before too many additional (unnecessary assumptions have been introduced.I have intentionally kept the discussion of multicollinearity to a minimum. This partly indicates my bias, but it also reflects reality. It is, of course, very important for students to understand the potential consequences of having highly correlated independent variables. But this is often beyond our control, except that we can ask less of our multiple regression analysis. If two or more explanatory variables are highly correlated in the sample, we should not expect to precisely estimate their ceteris paribus effects in the population.I find extensive treatments of multicollinearity, where one “tests” or somehow “solves” the multicollinearity problem, to be misleading, at best. Even the organization of some texts gives the impression that imperfect multicollinearity is somehow a violation of the Gauss-Markovassumptions: they include multicollinearity in a chapter or part of the book devoted to “violation of the basic assumptions,” or something like that. I have noticed that master’s students who have had some undergraduate econometrics are often confused on the multicollinearity issue. It is very important that students not confuse multicollinearity among the included explanatory variables in a regression model with the bias caused by omitting an important variable.I do not prove the Gauss-Markov theorem. Instead, I emphasize its implications. Sometimes, and certainly for advanced beginners, I put a special case of Problem 3.12 on a midterm exam, where I make a particular choice for the function g (x . Rather than have the students directly 课后答案网ww w.kh d aw .c om16compare the variances, they should appeal to the Gauss-Markov theorem for the superiority of OLS over any other linear, unbiased estimator.SOLUTIONS TO PROBLEMS3.1 (i Yes. Because of budget constraints, it makes sense that, the more siblings there are in a family, the less education any one child in the family has. To find the increase in the number of siblings that reduces predicted education by one year, we solve 1 = .094(Δsibs , so Δsibs = 1/.094 ≈ 10.6.(ii Holding sibs and feduc fixed, one more year of mother’s education implies .131 years more of predicted education. So if a mother has four more years of education, her son is predicted to have about a half a year (.524 more years of education. (iii Since the number of siblings is the same, but meduc and feduc are both different, the coefficientson meduc and feduc both need to be accounted for. The predicted difference in education between B and A is .131(4 + .210(4 = 1.364.3.2 (i hsperc is defined so that the smaller it is, the lower the student’s standing in high school. Everything else equal, the worse the student’s standing in high school, the lower is his/her expected college GPA. (ii Just plug these values into the equation:n colgpa= 1.392 − .0135(20 + .00148(1050 = 2.676.(iii The difference between A and B is simply 140 times the coefficient on sat , because hsperc is the same for both students. So A is predicted to have ascore .00148(140 ≈ .207 higher.(iv With hsperc fixed, n colgpaΔ = .00148Δsat . Now, we want to find Δsat such that n colgpaΔ = .5, so .5 = .00148(Δsat or Δsat = .5/(.00148 ≈ 338. Perhaps not surprisingly, a large ceteris paribus difference in SAT score – almost two and one-half standard deviations – is needed to obtain a predicted difference in college GPA or a half a point.3.3 (i A larger rank for a law school means that the school has less prestige; this lowers starting salaries. For example, a rank of 100 means there are 99 schools thought to be better.课后答案网ww w.kh d aw .c om17(ii 1β > 0, 2β > 0. Both LSAT and GPA are measures of the quality of the entering class. No matter where better students attend law school, we expect them to earn more, on average. 3β, 4β > 0. The numbe r of volumes in the law library and the tuition cost are both measures of the school quality. (Cost is less obvious than library volumes, but should reflect quality of the faculty, physical plant, and so on. (iii This is just the coefficient on GPA , multiplied by 100: 24.8%. (iv This is an elasticity: a one percent increase in library volumes implies a .095% increase in predicted median starting salary, other things equal. (v It is definitely better to attend a law school with a lower rank. If law school A has a ranking 20 less than law school B, the predicted difference in starting salary is 100(.0033(20 = 6.6% higher for law school A.3.4 (i If adults trade off sleep for work, more work implies less sleep (other things equal, so 1β < 0. (ii The signs of 2β and 3β are not obvious, at least to me. One could argue that more educated people like to get more out of life, and so, other things equal,they sleep less (2β < 0. The relationship between sleeping and age is more complicated than this model suggests, and economists are not in the best position to judge such things.(iii Since totwrk is in minutes, we must convert five hours into minutes: Δtotwrk = 5(60 = 300. Then sleep is predicted to fall by .148(300 = 44.4 minutes. For a week, 45 minutes less sleep is not an overwhelming change. (iv More education implies less predicted time sleeping, but the effect is quite small. If we assume the difference between college and high school is four years, the college graduate sleeps about 45 minutes less per week, other things equal. (v Not surprisingly, the three explanatory variables explain only about 11.3% of the variation in sleep . One important factor in the error term is general health. Another is marital status, and whether the person has children. Health (however we measure that, marital status, and number and ages of children would generally be correlated with totwrk . (For example, less healthy people would tend to work less.3.5 Conditioning on the outcomes of the explanatory variables, we have 1E(θ =E(1ˆβ + 2ˆβ = E(1ˆβ+ E(2ˆβ = β1 + β2 = 1θ.3.6 (i No. By definition, study + sleep + work + leisure = 168. Therefore, if we change study , we must change at least one of the other categories so that the sum is still 168. 课后答案网ww w.kh d aw .c om18(ii From part (i, we can write, say, study as a perfect linear function of the otherindependent variables: study = 168 − sleep − work − leisure . This holds for every observation, so MLR.3 violated. (iii Simply drop one of the independent variables, say leisure :GPA = 0β + 1βstudy + 2βsleep + 3βwork + u .Now, for example, 1β is interpreted as the change in GPA when study increases by one hour, where sleep , work , and u are all held fixed. If we are holding sleep and work fixed but increasing study by one hour, then we must be reducing leisure by one hour. The other slope parameters have a similar interpretation.3.7 We can use Table 3.2. By definition, 2β > 0, and by assumption, Corr(x 1,x 2 < 0.Therefore, there is a negative bias in 1β: E(1β < 1β. This means that, on average across different random samples, the simple regression estimator underestimates the effect of thetraining program. It is even possible that E(1β is negative even though 1β > 0.3.8 Only (ii, omitting an important variable, can cause bias, and this is true only when the omitted variable is correlated with the included explanatory variables. The homoskedasticity assumption, MLR.5, played no role in showing that the OLS estimators are unbiased.(Homoskedasticity was used to o btain the usual variance formulas for the ˆjβ. Further, the degree of collinearity between the explanatory variables in the sample, even if it is reflected in a correlation as high as .95, does not affect the Gauss-Markov assumptions. Only if there is a perfect linear relationship among two or more explanatory variables is MLR.3 violated.3.9 (i Because 1x is highly correlated with 2x and 3x , and these latter variables have largepartial effects on y , the simple and multiple regression coefficients on 1x can differ by largeamounts. We have not done this case explicitly, but given equation (3.46 and the discussion with a single omitted variable, the intuition is pretty straightforward.(ii Here we would expect 1β and 1ˆβ to be similar (subject, of course, to what we mean by “almost uncorrelated”. The amount of correlation between 2x and 3x does not directly effect the multiple regression estimate on 1x if 1x is essentially uncorrelated with 2x and 3x .(iii In this case we are (unnecessarily introducing multicollinearity into the regression: 2x and 3x have small partial effects on y and yet 2x and 3x are highly correlated with 1x . Adding2x and 3x like increases the standard error of the coefficient on 1x substantially, so se(1ˆβis likely to be much larger than se(1β . 课后答案网ww w.kh d aw .c om19(iv In this case, adding 2x and 3x will decrease the residual variance without causingmuch collinearity (because 1x is almost uncorrelated with 2x and 3x , so we should see se(1ˆβ smaller than se(1β. The amount of correlation between 2x and 3x does not directly affect se(1ˆβ.3.10 From equation (3.22 we have111211ˆ,ˆni ii ni i r yr β===∑∑where the 1ˆi rare defined in the problem. As usual, we must plug in the true model for y i : 1011223311211ˆ(.ˆni i i i ii ni i r x x x u r βββββ==++++=∑∑The numerator of this expression simplifies because 11ˆni i r=∑ = 0, 121ˆni i i r x =∑ = 0, and 111ˆni i i r x =∑ = 211ˆni i r =∑. These all follow from the fact that the 1ˆi rare the residuals from the regression of 1i x on 2i x : the 1ˆi rhave zero sample average and are uncorrelated in sample with 2i x . So the numerator of 1βcan be expressed as2113131111ˆˆˆ.n n ni i i i i i i i rr x r u ββ===++∑∑∑Putting these back over the denominator gives 13111113221111ˆˆ.ˆˆnni i ii i nni i i i r x rur r βββ=====++∑∑∑∑课后答案网ww w.kh d aw .c om20Conditional on all sample values on x 1, x 2, and x 3, only the last term is random due to its dependence on u i . But E(u i = 0, and so131113211ˆE(=+,ˆni i i ni i r xr βββ==∑∑which is what we wanted to show. Notice that the term multiplying 3β is the regressioncoefficient from the simple regression of x i 3 on 1ˆi r.3.11 (i 1β < 0 because more pollution can be expected to lower housing values; note that 1β isthe elasticity of price with respect to nox . 2β is probably positive because rooms roughlymeasures the size of a house. (However, it does not allow us to distinguish homes where each room is large from homes where each room is small. (ii If we assume that rooms increases with quality of the home, then log(nox and rooms are negatively correlated when poorer neighborhoods have more pollution, something that is often true. We can use Ta ble 3.2 to determine the direction of the bias. If 2β > 0 andCorr(x 1,x 2 < 0, the simple regression estimator 1βhas a downward bias. But because 1β < 0, this means that the simple regression, on average, overstates the importance of pollution. [E(1β is more negative than 1β.] (iii This is what we expect from the typical sample based on our analysis in part (ii. The simple regression estimate, −1.043, is more negative (larger in magnitude than the multiple regression estimate, −.718. As those estimates are only for one sample, we can never know which is closer to 1β. But if this is a “typical” sample, 1β is closer to −.718.3.12 (i For notational simplicity, define s zx = 1(;ni i i z z x =−∑ this is not quite the samplecovariance between z and x because we do not divide by n – 1, but we are only using it tosimplify notation. Then we can write 1β as11(.niii zxz z ys β=−=∑This is clearly a linear function of the y i : take the weights to be w i = (z i −z /s zx . To show unbiasedness, as usual we plug y i = 0β + 1βx i + u i into this equation, and simplify: 课后答案网w w w .k h d aw .c o m21 11 1 011 111(( (((n ii i i zxnni zx i ii i zxniii zxz z x u s z z s z z u s zz u s ββββββ====−++=−++−=−=+∑∑∑∑where we use the fact that 1(ni i z z =−∑ = 0 always. Now s zx is a function of the z i and x i and theexpected value of each u i is zero conditional on all z i and x i in the sample. Therefore, conditional on these values,1111(E(E(niii zxz z u s βββ=−=+=∑because E(u i = 0 for all i . (ii From the fourth equation in part (i we have (again conditional on the z i and x i in the sample,2111222212Var ((Var(Var((n ni i i i i i zx zxnii zxz z u z z u s s z z s βσ===⎡⎤−−⎢⎥⎣⎦==−=∑∑∑because of the homoskedasticit y assumption [Var(u i = σ2 for all i ]. Given the definition of s zx , this is what we wanted to show.课后答案网ww w.kh d aw .c om22(iii We know that Var(1ˆβ = σ2/21[(].ni i x x =−∑ Now we can rearrange the inequality in the hint, drop x from the sample covariance, and cancel n -1everywhere, to get 221[(]/ni zx i z z s =−∑ ≥211/[(].ni i x x =−∑ When we multiply through by σ2 we get Var(1β ≥ Var(1ˆβ, which is what we wanted to show.3.13 (i The shares, by definition, add to one. If we do not omit one of the shares then the equation would suffer from perfect multicollinearity. The parameters would not have a ceteris paribus interpretation, as it is impossible to change one share while holding all of the other shares fixed. (ii Because each share is a proportion (and can be at most one, when all other shares are zero, it makes little sense to increase share p by one unit. If share p increases by .01 – which is equivalent to a one percentage point increase in the share of property taxes in total revenue – holding share I , share S , and the other factorsfixed, then growth increases by 1β(.01. With the other shares fixed, the excluded share, share F , must fall by .01 when share p increases by .01.SOLUTIONS TO COMPUTER EXERCISESC3.1 (i Prob ably 2β > 0, as more income typically means better nutrition for the mother and better prenatal care. (ii On the one hand, an increase in income generally increases the consumption of a good, and cigs and faminc could be positively correlated. On the other, family incomes are also higher for families with more education, and more education and cigarette smoking tend to benegatively correlated. The sample correlation between cigs and faminc is about −.173, indicating a negative correlation.(iii The regressions without and with faminc aren 119.77.514bwghtcigs =−21,388,.023n R ==and n 116.97.463.093bwghtcigs faminc =−+21,388,.030.n R ==课后答案网ww w.kh d aw .c om23The effect of cigarette smoking is slightly smaller when faminc is added to the regression, but the difference is not great. This is due to the fact that cigs and faminc are not very correlated, and the coefficient on faminc is practically small. (The variable faminc is measured in thousands, so $10,000 more in 1988 income increases predicted birth weight by only .93 ounces.C3.2 (i The estimated equation isn 19.32.12815.20price sqrft bdrms =−++288,.632n R ==(ii Holding square footage constant, n price Δ = 15.20 ,bdrms Δ and so n price increases by 15.20, which means $15,200.(iii Now n price Δ = .128sqrft Δ + 15.20bdrms Δ = .128(140 + 15.20 = 33.12, or $33,120. Because the size of the house is increasing, this is a much larger effect than in (ii. (iv About 63.2%. (v The predicted price is –19.32 + .128(2,438 + 15.20(4 = 353.544, or $353,544. (vi From part (v, the estimated value of the home based only on square footage and number of bedrooms is $353,544. The actual selling price was $300,000, which suggests the buyer underpaid by some margin. But, of course, there are many other features of a house (some that we cannot even measure that affect price, and we have not controlled for these.C3.3 (i The constant elasticity equation isn log( 4.62.162log(.107log(salary sales mktval =++ 2177,.299.n R ==(ii We cannot include profits in logarithmic form because profits are negative for nine of the companies in the sample. When we add it in levels form we getn log( 4.69.161log(.098log(.000036salary sales mktval profits =+++2177,.299.n R ==The coefficient on profits is very small. Here, profits are measured in millions, so if profits increase by $1 billion, which means profits Δ = 1,000 – a huge change – predicted salaryincreases by about only 3.6%. However, remember that we are holding sales and market value fixed.课后答案网ww w.kh d aw .c om24Together, these variables (and we could drop profits without losing anything explain almost 30% of the sample variation in log(salary . This is certainly not “most” of the variation.(iii Adding ceoten to the equation givesn log( 4.56.162log(.102log(.000029.012salary sales mktval profits ceoten =++++2177,.318.n R ==This means that one more year as CEO increases predicted salary by about 1.2%. (iv The sample correlation between log(mktval and profits is about .78, which is fairly high. As we know, this causes no bias in the OLS estimators, although it can cause their variances to be large. Given the fairly substantial correlation between market value andfirm profits, it is not too surprising that the latter adds nothing to explaining CEO salaries. Also, profits is a short term measure of how the firm is doing while mktval is based on past, current, and expected future profitability.C3.4 (i The minimum, maximum, and average values for these three variables are given in the table below:Variable Average Minimum Maximum atndrte priGPA ACT 81.71 2.59 22.516.25 .86131003.93 32(ii The estimated equation isn 75.7017.26 1.72atndrtepriGPA ACT =+− n = 680, R 2 = .291.The intercept means that, for a student whose prior GPA is zero and ACT score is zero, the predicted attendance rate is 75.7%. But this is clearly not an interesting segment of thepopulation. (In fact, there are no students in the college population with priGPA = 0 and ACT = 0, or with values even close to zero. (iii The coefficient on priGPA means that, if a student’s prior GPA is one point higher (say, from 2.0 to 3.0, the attendance rate is about 17.3 percentage points higher. This holds ACT fixed. The negative coefficient on ACT is, perhaps initially a bit surprising. Five more points on the ACT is predicted to lower attendance by 8.6 percentage points at a given level of priGPA . As priGPAmeasures performance in college (and, at least partially, could reflect, past attendance rates, while ACT is a measure of potential in college, it appears that students that had more promise (which could mean more innate ability think they can get by with missing lectures. 课后答案网ww w.kh d aw .c om(iv We have atndrte = 75.70 + 17.267(3.65 –1.72(20 ≈ 104.3. Of course, a student cannot have higher than a 100% attendance rate. Getting predictions like this is always possible when using regression methods for dependent variables with natural upper or lower bounds. In practice, we would predict a 100% attendance rate for this student. (In fact, this student had an actual attendance rate of 87.5%. (v The difference in predicted attendance rates for A and B is 17.26(3.1 − 2.1 − (21 − 26 = 25.86. C3.5 The regression of educ on exper and tenure yields n = 526, R2 = .101. ˆ Now, when we regres s log(wage on r1 we obtain ˆ log( wage = 1.62 + .092 r1 n = 526, R2 = .207. (ii The slope coefficientfrom log(wage on educ is β1 = .05984. ˆ ˆ (iv We have β1 + δ 1 β 2 = .03912 +3.53383(.00586 ≈ .05983, which is very close to .05984; the small difference is due to rounding error. C3.7 (i The results of the regression are math10 = −20.36 + 6.23log(expend − .305 lnchprg 课 (iii The slope coefficients from log(wage on educ and IQ are ˆ = .03912 and β = .00586, respectively. ˆ β1 2 后答案 C3.6 (i The slope coefficient from the regression IQ on educ is (rounded to five decimal places δ1 = 3.53383. n = 408, R2 = .180. 25 This edition is intended for use outside of the U.S. only, with content that may be different from the U.S. Edition. This may not be resold, copied, or distributed without the prior consent of the publisher. 网ˆ As expected, the coefficient on r1 in the second regression is identical to the coefficient on educ in equation (3.19. Notice that the R-squared from the above regression is below that in (3.19. ˆ In effect, the regression of log(wage on r1 explains log(wage using only the part of educ that is uncorrelated with exper and tenure; separate effects of exper and tenure are not included. ww w. kh da w. co m ˆ educ = 13.57 − .074 exper + .048 ten ure + r1 .The signs of the estimated slopes imply that more spending increases the pass rate (holding lnchprg fixed and a higher poverty rate (proxied well by lnchprg decreases the pass rate (holding spending fixed. These are what we expect. (ii As usual, the estimated intercept is the predicted value of the dependent variable when all regressors are set to zero. Setting lnchprg = 0 makes sense, as there are schools with low poverty rates. Setting log(expend = 0 does not make sense, because it is the same as setting expend = 1, and spending is measured in dollars per student. Presumably this is well outside any sensible range. Not surprisingly, the prediction of a −20 pass rate is nonsensical. (iii The simple regression results are failing to account for the poverty rate leads to an overestimate of the effect of spending. C3.8 (i The average of prpblck is .113 with standarddeviation .182; the average of income is 47,053.78 with standard deviation 13,179.29. It is evident that prpblck is a proportion and that income is measured in dollars. (ii The results from the OLS regression are psoda = .956 + .115 prpblck + .0000016 income 后 If, say, prpblck increases by .10 (ten percentage points, the price of soda is estimated toincrease by .0115 dollars, or about 1.2 cents. While this does not seem large, there are communities with no black population and others that are almost all black, in which case the difference in psoda is estimated to be almost 11.5 cents. (iii The simple regression estimate on prpblck is .065, so the simple regression estimate is actually lower. This is because prpblck and income are negatively correlated (-.43 and income has a positive coefficient in the multiple regression. (iv To get a constant elasticity, income should be in logarithmic form. I estimate the constant elasticity model: 26 This edition is intended for use outside of the U.S. only, with content that may be different from the U.S. Edition. This may not be resold, copied, or distributed without the prior consent of the publisher. 课答案 n = 401, R2 = .064. 网ww ˆ (v We can use equation (3.23. Because Corr(x1,x2 < 0, which means δ1 < 0 , and β 2 < 0 , ˆ the simple regression estimate, β , is larger than the multiple regression estimate, β . Intuitively, 1 w. kh (iv The sample correl ation between lexpend and lnchprg is about −.19 , which means that, on average, high schools with poorer students spent less per student. This makes sense, especially in 1993 in Michigan, where school funding was essentially determined by local property tax collections. da w. n = 408, R2 = .030 and the estimated spending effect is larger than it was in part (i –almost double. co 1 m math10 = −69.34 + 11.16 log(expendlog( psoda = −.794 + .122 prpblck + .077 log(income n = 401, R2 = .068. If prpblck increases by .20, log(psoda is estimated to increase by .20(.122 = .0244, or about 2.44 percent. ˆ (v β prpblck falls to about .073 when prppov is added to the regression. (vi The correlation is about −.84 , which makes sense because poverty rates are determined by income (but not directly in terms of median income. (vii There is no argument that they are highly correlated, but we are using them simply as controls to determine if the is price discrimination against blacks. In order to isolate the pure discrimination effect, we need to control for as many measures of income as we can; including both variables makes sense. C3.9 (i The estimated equation is (iv The estimated equation is gift = −7.33 + 1.20 mailsyear − .261 giftlast + 16.20 propresp + .527 avggift Aft er controlling for the average past gift level, the effect of mailings becomes even smaller: 1.20 guilders, or less thanhalf the effect estimated by simple regression. (v After controlling for the average of past gifts – which we can view as measuring the “typical” generosity of the person and is positively related to the current gift level – we find that the current gift amount is negatively related to the most recent gift. A negative relationship makes some sense, as people might follow a large donation with a smaller one. 27 This edition is intended for use outside of the U.S. only, with content that may be different from the U.S. Edition. This may not be resold, copied, or distributed without the prior consent of the publisher. 课 n = 4,268, R 2 = .2005 后 (iii Because propresp is a proportion, it makes little sense to increase it by one. Such an increase can happen only if propresp goes from zero to one. Instead, consider a .10 increase in propresp, which means a 10 percentage point increase. Then, gift i s estimated to be 15.36(.1 ≈ 1.54 guilders higher. 答案 (ii Holding giftlast and propresp fixed, one more mailing per year is estimated to increase gifts by 2.17 guilders. The simple regression estimate is 2.65, so the multiple regression estimate is somewhat smaller. Remember, the simple regression estimate holds no other factors fixed. 网 ww The R-squared is now about .083, compared with about .014 for the simple regression case. Therefore, the variables giftlast and propresp help to explain significantly more variation in gifts in the sample (although still just over eight percent. w. n = 4,268, R 2= .0834 kh gift = −4.55 + 2.17 mailsyear + .0059 giftlast + 15.36 propresp da w. co m。

《孟德尔随机化研究指南》中英文版全文共3篇示例，供读者参考篇1Randomized research is a vital component of scientific studies, allowing researchers to investigate causal relationships between variables and make accurate inferences about the effects of interventions. One of the most renowned guides for conducting randomized research is the "Mendel Randomization Research Guide," which provides detailed instructions and best practices for designing and implementing randomized controlled trials.The Mendel Randomization Research Guide offers comprehensive guidance on all aspects of randomized research, from study design and sample selection to data analysis and interpretation of results. It emphasizes the importance of randomization in reducing bias and confounding effects, thus ensuring the validity and reliability of study findings. With clear and practical recommendations, researchers can feel confident in the quality and rigor of their randomized research studies.The guide highlights the key principles of randomization, such as the use of random assignment to treatment groups, blinding of participants and researchers, and intent-to-treat analysis. It also discusses strategies for achieving balance in sample characteristics and minimizing the risk of selection bias. By following these principles and guidelines, researchers can maximize the internal validity of their studies and draw accurate conclusions about the causal effects of interventions.In addition to the technical aspects of randomized research, the Mendel Randomization Research Guide also addresses ethical considerations and practical challenges that researchers may face. It emphasizes the importance of obtaining informed consent from participants, protecting their privacy and confidentiality, and ensuring the safety and well-being of study subjects. The guide also discusses strategies for overcoming common obstacles in randomized research, such as recruitment and retention issues, data collection problems, and statistical challenges.Overall, the Mendel Randomization Research Guide is a valuable resource for researchers looking to improve the quality and validity of their randomized research studies. By following its recommendations and best practices, researchers can conductstudies that produce reliable and actionable findings, advancing scientific knowledge and contributing to evidence-based decision making in various fields.篇2Mendel Randomization Study GuideIntroductionMendel Randomization Study Guide is a comprehensive and informative resource for researchers and students interested in the field of Mendel randomization. This guide provides anin-depth overview of the principles and methods of Mendel randomization, as well as practical advice on how to design and conduct Mendel randomization studies.The guide is divided into several sections, each covering a different aspect of Mendel randomization. The first section provides a brief introduction to the history and background of Mendel randomization, tracing its origins to the work of Gregor Mendel, the father of modern genetics. It also discusses the theoretical foundations of Mendel randomization and its potential applications in causal inference.The second section of the guide focuses on the methods and techniques used in Mendel randomization studies. This includesa detailed explanation of how Mendel randomization works, as well as guidelines on how to select instrumental variables and control for potential confounders. It also discusses the strengths and limitations of Mendel randomization, and provides practical tips on how to deal with common challenges in Mendel randomization studies.The third section of the guide is dedicated to practical considerations in Mendel randomization studies. This includes advice on how to design a Mendel randomization study, collect and analyze data, and interpret the results. It also provides recommendations on how to report Mendel randomization studies and publish research findings in scientific journals.In addition, the guide includes a glossary of key terms and concepts related to Mendel randomization, as well as a list of recommended readings for further study. It also includes case studies and examples of Mendel randomization studies in practice, to illustrate the principles and techniques discussed in the guide.ConclusionIn conclusion, the Mendel Randomization Study Guide is a valuable resource for researchers and students interested in Mendel randomization. It provides a comprehensive overview ofthe principles and methods of Mendel randomization, as well as practical advice on how to design and conduct Mendel randomization studies. Whether you are new to Mendel randomization or looking to deepen your understanding of the field, this guide is an essential reference for anyone interested in causal inference and genetic epidemiology.篇3"Guide to Mendelian Randomization Studies" English VersionIntroductionMendelian randomization (MR) is a method that uses genetic variants to investigate the causal relationship between an exposure and an outcome. It is a powerful tool that can help researchers to better understand the underlying mechanisms of complex traits and diseases. The "Guide to Mendelian Randomization Studies" provides a comprehensive overview of MR studies and offers practical guidance on how to design and carry out these studies effectively.Chapter 1: Introduction to Mendelian RandomizationThis chapter provides an overview of the principles of Mendelian randomization, including the assumptions andlimitations of the method. It explains how genetic variants can be used as instrumental variables to estimate the causal effect of an exposure on an outcome, and outlines the key steps involved in conducting an MR study.Chapter 2: Choosing Genetic InstrumentsIn this chapter, the guide discusses the criteria for selecting appropriate genetic instruments for Mendelian randomization. It covers issues such as the relevance of the genetic variant to the exposure of interest, the strength of the instrument, and the potential for pleiotropy. The chapter also provides practical tips on how to search for suitable genetic variants in public databases.Chapter 3: Data Sources and ValidationThis chapter highlights the importance of using high-quality data sources for Mendelian randomization studies. It discusses the different types of data that can be used, such asgenome-wide association studies and biobanks, and offers advice on how to validate genetic instruments and ensure the reliability of the data.Chapter 4: Statistical MethodsIn this chapter, the guide explains the various statistical methods that can be used to analyze Mendelian randomization data. It covers techniques such as inverse variance weighting, MR-Egger regression, and bi-directional Mendelian randomization, and provides guidance on how to choose the most appropriate method for a given study.Chapter 5: Interpretation and ReportingThe final chapter of the guide focuses on the interpretation and reporting of Mendelian randomization results. It discusses how to assess the strength of causal inference, consider potential biases, and communicate findings effectively in research papers and presentations.ConclusionThe "Guide to Mendelian Randomization Studies" is a valuable resource for researchers who are interested in using genetic data to investigate causal relationships in epidemiological studies. By following the guidance provided in the guide, researchers can enhance the rigor and validity of their Mendelian randomization studies and contribute to a better understanding of the determinants of complex traits and diseases.。

israel(1992)提出的计算样本大小的公式英文版The Formula for Calculating Sample Size Proposed by Israel (1992)In statistical research, sample size is a crucial element that determines the validity and reliability of the research findings. Selecting an appropriate sample size is essential to ensure that the results of a study are generalizable and accurate. Among the various methods for calculating sample size, the formula proposed by Israel in 1992 stands out as a widely used and reliable approach.The Israel (1992) formula for calculating sample size is based on several key considerations, including the population size, the desired level of confidence, and the margin of error. The formula takes into account these factors to determine the minimum number of samples required to achieve statistically significant results.The formula can be expressed as follows:n = N / (1 + N * e^2)where:n represents the sample sizeN is the population sizee is the margin of error, expressed as a decimal (e.g., 0.05 for a 5% margin of error)This formula allows researchers to calculate the sample size based on their specific research requirements and constraints. By plugging in the values for the population size and the desired margin of error, the formula provides a scientifically sound estimate of the minimum number of samples needed for the study.It is important to note that while the Israel (1992) formula provides a useful starting point for sample size calculation, it may not be applicable in all scenarios. The formula assumes a simple random sampling without replacement, and it may needto be adjusted for more complex sampling designs or specific research contexts.Nevertheless, the Israel (1992) formula remains a valuable tool for researchers seeking to determine an appropriate sample size for their studies. By carefully considering the relevant factors and using this formula, researchers can ensure that their sample size is adequate to support statistically valid and reliable research findings.中文版以色列（1992）提出的计算样本大小的公式在统计研究中，样本大小是决定研究结果的有效性和可靠性的关键因素。

Sample Selection Bias as a Specification ErrorJames J.HeckmanEconometrica,Vol.47,No.1.(Jan.,1979),pp.153-161.Stable URL:/sici?sici=0012-9682%28197901%2947%3A1%3C153%3ASSBAAS%3E2.0.CO%3B2-J Econometrica is currently published by The Econometric Society.Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use,available at/about/terms.html.JSTOR's Terms and Conditions of Use provides,in part,that unless you have obtained prior permission,you may not download an entire issue of a journal or multiple copies of articles,and you may use content in the JSTOR archive only for your personal,non-commercial use.Please contact the publisher regarding any further use of this work.Publisher contact information may be obtained at/journals/econosoc.html.Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission.JSTOR is an independent not-for-profit organization dedicated to and preserving a digital archive of scholarly journals.For more information regarding JSTOR,please contact support@.Wed Apr1811:20:522007You have printed the following article:Sample Selection Bias as a Specification ErrorJames J.HeckmanEconometrica ,Vol.47,No.1.(Jan.,1979),pp.153-161.Stable URL:/sici?sici=0012-9682%28197901%2947%3A1%3C153%3ASSBAAS%3E2.0.CO%3B2-JThis article references the following linked citations.If you are trying to access articles from anoff-campus location,you may be required to first logon via your library web site to access JSTOR.Please visit your library's website or contact a librarian to learn about options for remote access to JSTOR.[Footnotes]2Wage Comparisons--A Selectivity BiasReuben GronauThe Journal of Political Economy ,Vol.82,No.6.(Nov.-Dec.,1974),pp.1119-1143.Stable URL:/sici?sici=0022-3808%28197411%2F12%2982%3A6%3C1119%3AWCSB%3E2.0.CO%3B2-L2Comments on Selectivity Biases in Wage ComparisonsH.Gregg LewisThe Journal of Political Economy ,Vol.82,No.6.(Nov.-Dec.,1974),pp.1145-1155.Stable URL:/sici?sici=0022-3808%28197411%2F12%2982%3A6%3C1145%3ACOSBIW%3E2.0.CO%3B2-YReferences1Regression Analysis when the Dependent Variable Is Truncated NormalTakeshi AmemiyaEconometrica ,Vol.41,No.6.(Nov.,1973),pp.997-1016.Stable URL:/sici?sici=0012-9682%28197311%2941%3A6%3C997%3ARAWTDV%3E2.0.CO%3B2-FLINKED CITATIONS -Page 1of 2-2Specification Bias in Estimates of Production FunctionsZvi GrilichesJournal of Farm Economics ,Vol.39,No.1.(Feb.,1957),pp.8-20.Stable URL:/sici?sici=1071-1031%28195702%2939%3A1%3C8%3ASBIEOP%3E2.0.CO%3B2-V 4Wage Comparisons--A Selectivity BiasReuben GronauThe Journal of Political Economy ,Vol.82,No.6.(Nov.-Dec.,1974),pp.1119-1143.Stable URL:/sici?sici=0022-3808%28197411%2F12%2982%3A6%3C1119%3AWCSB%3E2.0.CO%3B2-L8Dummy Endogenous Variables in a Simultaneous Equation SystemJames J.HeckmanEconometrica ,Vol.46,No.4.(Jul.,1978),pp.931-959.Stable URL:/sici?sici=0012-9682%28197807%2946%3A4%3C931%3ADEVIAS%3E2.0.CO%3B2-F9Asymptotic Properties of Non-Linear Least Squares EstimatorsRobert I.JennrichThe Annals of Mathematical Statistics ,Vol.40,No.2.(Apr.,1969),pp.633-643.Stable URL:/sici?sici=0003-4851%28196904%2940%3A2%3C633%3AAPONLS%3E2.0.CO%3B2-311Comments on Selectivity Biases in Wage ComparisonsH.Gregg LewisThe Journal of Political Economy ,Vol.82,No.6.(Nov.-Dec.,1974),pp.1145-1155.Stable URL:/sici?sici=0022-3808%28197411%2F12%2982%3A6%3C1145%3ACOSBIW%3E2.0.CO%3B2-Y 12Specification Errors and the Estimation of Economic RelationshipsH.TheilRevue de l'Institut International de Statistique /Review of the International Statistical Institute ,Vol.25,No.1/3.(1957),pp.41-51.Stable URL:/sici?sici=0373-1138%281957%2925%3A1%2F3%3C41%3ASEATEO%3E2.0.CO%3B2-BLINKED CITATIONS -Page 2of 2-。