Showing papers on "Confidence interval published in 1973"
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01 Jan 1973TL;DR: In this article, the authors describe a decision-making process for making decisions in an uncertain environment in the context of time series data, and present a series of graphs and tables to describe the relationships between variables.
Abstract: CHAPTER 1 Describing Data: Graphical 1.1 Decision Making in an Uncertain Environment 1.2 Classification of Variables 1.3 Graphs to Describe Categorical Variables 1.4 Graphs to Describe Time-Series Data 1.5 Graphs to Describe Numerical Variables 1.6 Tables and Graphs to Describe Relationships Between Variables 1.7 Data Presentation Errors CHAPTER 2 Describing Data: Numerical 2.1 Measures of Central Tendency 2.2 Measures of Variability 2.3 Weighted Mean and Measures of Grouped Data 2.4 Measures of Relationships Between Variables CHAPTER 3 Probability 3.1 Random Experiment, Outcomes, Events 3.2 Probability and Its Postulates 3.3 Probability Rules 3.4 Bivariate Probabilities 3.5 Bayes' Theorem CHAPTER 4 Discrete Random Variables and Probability Distributions 4.1 Random Variables 4.2 Probability Distributions for Discrete Random Variables 4.3 Properties of Discrete Random Variables 4.4 Binomial Distribution 4.5 Hypergeometric Distribution 4.6 The Poisson Probability Distribution 4.7 Jointly Distributed Discrete Random Variables CHAPTER 5 Continuous Random Variables and Probability Distributions 5.1 Continuous Random Variables 5.2 Expectations for Continuous Random Variables 5.3 The Normal Distribution 5.4 Normal Distribution Approximation for Binomial Distribution 5.5 The Exponential Distribution 5.6 Jointly Distributed Continuous Random Variables CHAPTER 6 Sampling and Sampling Distributions 6.1 Sampling from a Population 6.2 Sampling Distributions of Sample Means 6.3 Sampling Distributions of Sample Proportions 6.4 Sampling Distributions of Sample Variances CHAPTER 7 Estimation: Single Population 7.1 Properties of Point Estimators 7.2 Confidence Interval Estimation of the Mean of a Normal Distribution: Population Variance Known 7.3 Confidence Interval Estimation of the Mean of a Normal Distribution: Population Variance Unknown 7.4 Confidence Interval Estimation of Population Proportion 7.5 Confidence Interval Estimation of the Variance of a Normal Distribution 7.6 Confidence Interval Estimation: Finite Populations CHAPTER 8 Estimation: Additional Topics 8.1 Confidence Interval Estimation of the Difference Between Two Normal Population Means: Dependent Samples 8.2 Confidence Interval Estimation of the Difference Between Two Normal Population Means: Independent Samples 8.3 Confidence Interval Estimation of the Difference Between Two Population Proportions 8.4 Sample Size Determination: Large Populations 8.5 Sample Size Determination: Finite Populations CHAPTER 9 Hypothesis Testing: Single Population 9.1 Concepts of Hypothesis Testing 9.2 Tests of the Mean of a Normal Distribution: Population Variance Known 9.3 Tests of the Mean of a Normal Distribution: Population Variance Unknown 9.4 Tests of the Population Proportion 9.5 Assessing the Power of a Test 9.6 Tests of the Variance of a Normal Distribution CHAPTER 10 Hypothesis Testing: Additional Topics 10.1 Tests of the Difference Between Two Population Means: Dependent Samples 10.2 Tests of the Difference Between Two Normal Population Means: Independent Samples 10.3 Tests of the Difference Between Two Population Proportions 10.4 Tests of the Equality of the Variances Between Two Normally Distributed Populations 10.5 Some Comments on Hypothesis Testing CHAPTER 11 Simple Regression 11.1 Overview of Linear Models 11.2 Linear Regression Model 11.3 Least Squares Coefficient Estimators 11.4 The Explanatory Power of a Linear Regression Equation 11.5 Statistical Inference: Hypothesis Tests and Confidence Intervals 11.6 Prediction 11.7 Correlation Analysis 11.8 Beta Measure of Financial Risk 11.9 Graphical Analysis CHAPTER 12 Multiple Regression 12.1 The Multiple Regression Model 12.2 Estimation of Coefficients 12.3 Explanatory Power of a Multiple Regression Equation 12.4 Confidence Intervals and Hypothesis Tests for Individual Regression Coefficients 12.5 Tests on Regression Coefficients 12.6 Prediction 12.7 Transformations for Nonlinear Regression Models 12.8 Dummy Variables for Regression Models 12.9 Multiple Regression Analysis Application Procedure CHAPTER 13 Additional Topics in Regression Analysis 13.1 Model-Building Methodology 13.2 Dummy Variables and Experimental Design 13.3 Lagged Values of the Dependent Variables as Regressors 13.4 Specification Bias 13.5 Multicollinearity 13.6 Heteroscedasticity 13.7 Autocorrelated Errors CHAPTER 14 ANALYSIS OF CATEGORICAL DATA 14.1 Goodness-of-Fit Tests: Specified Probabilities 14.2 Goodness-of-Fit Tests: Population Parameters Unknown 14.3 Contingency Tables 14.4 Sign Test and Confidence Interval 14.5 Wilcoxon Signed Rank Test 14.6 Mann--Whitney U Test 14.7 Wilcoxon Rank Sum Test 14.7 Spearman Rank Correlation CHAPTER 15 Analysis of Variance 15.1 Comparison of Several Population Means 15.2 One-Way Analysis of Variance 15.3 The Kruskal--Wallis Test 15.4 Two-Way Analysis of Variance: One Observation per Cell, Randomized Blocks 15.5 Two-Way Analysis of Variance: More Than One Observation per Cell CHAPTER 16 Time-Series Analysis and Forecasting 16.1 Index Numbers 16.2 A Nonparametric Test for Randomness 16.3 Components of a Time Series 16.4 Moving Averages 16.5 Exponential Smoothing 16.6 Autoregressive Models 16.7 Autoregressive Integrated Moving Average Models CHAPTER 17 Sampling: Additional Topics 17.1 Stratified Sampling 17.2 Other Sampling Methods CHAPTER 18 Statistical Decision Theory 18.1 Decision Making Under Uncertainty 18.2 Solutions Not Involving Specification of Probabilities 18.3 Expected Monetary Value TreePlan 18.4 Sample Information: Bayesian Analysis and Value 18.5 Allowing for Risk: Utility Analysis APPENDIX TABLES 1. Cumulative Distribution Function of the Standard Normal Distribution 2. Probability Function of the Binomial Distribution 3. Cumulative Binomial Probabilities 4. Values of e --lambda 5. Individual Poisson Probabilities 6. Cumulative Poisson Probabilities 7. Cutoff Points of the Chi-Square Distribution Function 8. Cutoff Points for the Student's t Distribution 9. Cutoff Points for the F Distribution 10. Cutoff Points for the Distribution of the Wilcoxon Test Statistic 11. Cutoff Points for the Distribution of Spearman Rank Correlation Coefficient 12. Cutoff Points for the Distribution of the Durbin--Watson Test Statistic 13 Critical Values of the Studentized Range Q (page 964 965 Applied Statistical Methods Carlson, Thorne Prentice Hall 1997) 14. Cumulative Distribution Function of the Runs Test Statistic ANSWERS TO SELECTED EVEN-NUMBERED EXERCISES INDEX I-1
991 citations
TL;DR: In this article, the conditional probability of simultaneous coverage is shown to be smaller than the unconditional probability for sufficiently large critical values and at least two degrees of freedom for error, and Monte Carlo studies of the discrepancy and interpretation are presented.
Abstract: Suppose that in regression problems the simultaneous confidence intervals of the S-method are used only when a preliminary F-test rejects the null hypothesis that the regression parameters are zero. (Some of Scheffe's work emphasizes this usage [12, p. 87; 13, p. 66].) The probability of coverage should then be conditioned on the rejection. That for sufficiently large critical values and at least two degrees of freedom for error the conditional probability of simultaneous coverage is always smaller than the unconditional probability is established in this article. Also included are Monte Carlo studies of the discrepancy and interpretation.
73 citations
TL;DR: This paper presents a method for estimating the variances of sample performance measures in queueing simulations, for removing the bias in these sample measures due to initial conditions and for deriving approximating confidence intervals for the true performance measures.
Abstract: This paper presents a method for estimating the variances of sample performance measures in queueing simulations, for removing the bias in these sample measures due to initial conditions and for deriving approximating confidence intervals for the true performance measures. It also describes how a variance reduction can be achieved using the method when comparing performance measures for two queue disciplines. The approach requires only that every time the system passes through the empty and idle condition future behavior is independent of past behavior.
71 citations
TL;DR: In many practical problems in industry, it is desired to use the results of a previous sample to predict the result of a future sample as discussed by the authors, which can be used for planning purposes.
Abstract: In many practical problems in industry, it is desired to use the results of a previous sample to predict the results of a future sample. For example, data on warranty costs on large motors over the past three years are to be used for planning purposes t..
69 citations
TL;DR: In this paper, a more general relationship is derived and then compared with the simplified assumptions, and the results show the importance of using the most representative formulas, such as Pearson type 3 and gamma distributions.
Abstract: Estimation of events with given return periods and the sampling variance of these events are of great importance in hydrology. In the case of Pearson type 3 and gamma distributions that efficiently describe the hydrologic extreme values, floods, for example, the usual formulas are based on simplified assumptions such as neglecting the variability of the skewness coefficient. More general relationships are derived and then compared with the simplified ones. Great differences may happen: for instance, for large return periods the confidence interval is underestimated, specially when C8 ≠ 2CV, whereas for return periods T < 20 the confidence interval given by usual methods is overestimated. These errors in estimating the confidence intervals may come out with economical implications, particularly in works constitution. This result shows the importance of using the most representative formulas.
62 citations
TL;DR: In this paper, the problem of estimating Y (1), the smallest of a future sample of k observations from the Weibull distribution, based on an observed sample from the same distribution, is considered.
Abstract: The problem of estimating Y (1), the smallest of a future sample of k observations from the Weibull distribution, based on an observed sample from the same distribution, is considered. A conditional confidence interval solution is proposed for the estimation of Y (1). The results have direct application in reliability theory, where the time until the first failure in a group of k items in service provides a measure of assurance regarding the operation of the items.
58 citations
41 citations
36 citations
TL;DR: In this article, a Monte Carlo study of confidence limits for the slope of the major axis of a bivariate normal distribution was conducted, and it was shown that when imaginary limits are interpreted as corresponding to an infinite interval covering all possible values of the parameter, the confidence interval behaves satisfactorily under repeated sampling.
Abstract: A Monte Carlo study of confidence limits for the slope of the major axis of a bivariate normal distribution confirms that, when imaginary limits are interpreted as corresponding to an infinite interval covering all possible values of the parameter, the confidence interval behaves satisfactorily under repeated sampling. The excess of the actual over the nominal significance level is negligible even if samples are small and correlation is moderate. The probability of detecting a relationship correctly is never smaller for the major axis than for ordinary regression. Imaginary and exclusive confidence limits do not create problems in practice.
30 citations
TL;DR: In this article, a family of confidence intervals with endpoints that are simple averages of sample order statistics is defined, and the interval with shortest expected length is selected and compared to other confidence intervals for the same parameter.
Abstract: A family of confidence intervals with endpoints that are simple averages of sample order statistics is defined. In this family the interval with shortest expected length is selected and compared to other confidence intervals for the same parameter.
29 citations
TL;DR: Analogous conditional and unconditional confidence interval procedures for the two parameter Weibull distribution are discussed in this article, which suggest that, in many practical situations, the two procedures considered lead to near-equivalent results.
Abstract: Analogous conditional and unconditional confidence interval procedures for the two parameter Weibull distribution are discussed. Results of an empirical study are given which suggest that, in many practical situations, the two procedures considered lead to near-equivalent results. Ramifications of the empirical results are also considered.
TL;DR: In this article, a table of standard confidence limits for linear functions of the mean and variance of a normal distribution is presented, where confidence limits can be obtained using the tables given observed values of Y distributed as N(μ,σ 2 /γ 2) and S 2 distributed independently as σ 2/v times χ 2 with v degrees of freedom, where μ and σ2 are unknown parameters and v, γ, and λ are known but arbitrary numbers.
Abstract: Tables of standard confidence limits for linear functions of the mean and variance of a normal distribution are presented. Algorithms are presented by which confidence limits for an arbitrary linear function μ + λσ 2 can be obtained using the tables given observed values of Y distributed as N(μ,σ 2 /γ 2) and S 2 distributed independently as σ2/v times χ2 with v degrees of freedom, where μ and σ2 are unknown parameters and v, γ, and λ are known but arbitrary numbers.
TL;DR: In this article, it was shown that the same confidence procedure is at least almost admissible when the criterion is probability of not covering the true value and probability of covering false values.
Abstract: It will be shown that if a confidence procedure is admissible when the criterion is probability of not covering the true value and expected length (or something more general than length), then the same confidence procedure is at least almost admissible when the criterion is probability of not covering the true value and probability of covering false values. The result is true (under mild conditions) for virtually all confidence region estimation problems.
TL;DR: In this article, the authors considered the problem of estimating the largest mean of a population with a common variance and showed that the confidence interval obtained is unsymmetric for k > 2 and behaves asymptotically as well as the optimal interval.
Abstract: Interval estimation of the largest mean of $k$ normal populations $(k \geqq 1)$ with a common variance $\sigma^2$ is considered. When $\sigma^2$ is known the optimal fixed-width interval is given so that, to have the probability of coverage uniformly lower bounded by $\gamma$ (preassigned), the sample size needed is minimized. This optimal interval is unsymmetric for $k > 2$. When $\sigma^2$ is unknown a sequential procedure is proposed and its behavior is studied. It is shown that the confidence interval obtained, which is also unsymmetric for $k > 2$, behaves asymptotically as well as the optimal interval. This represents an improvement of the procedure of symmetric intervals considered by the author previously; the improvement is significant, especially when $k$ is large.
TL;DR: The Kafue lechwe population of the Kafue Flats, Zambia, was counted four times within a 26-month period (April 1970-June 1972) by aerial stratified random sampling which was based on calculation of animal densities on a linear basis, instead of on densities per unit area.
Abstract: Summary
The Kafue lechwe population of the Kafue Flats, Zambia, was counted four times within a 26-month period (April 1970-June 1972) by aerial stratified random sampling. The censuses gave good agreement which did not differ statistically. By combining the data for the three most precise surveys, the estimated population is 93 975 with 95% confidence limits of ± 8563 (or ± 9-1%). A method of sampling was used which is based on calculation of animal densities on a linear basis, instead of on densities per unit area. This method has several advantages and is discussed in relation to the problems involved in aerial census. The past and present status of the Kafue lechwe population is reviewed.
TL;DR: In this paper, the confidence coefficients associated with the confidence regions for variance ratios of balanced random models are derived in terms of the upper tail of the probability integrals of the inverted Dirichlet distribution.
Abstract: Exact confidence coefficients associated with the confidence regions for variance ratios of balanced random models [3] are derived in terms of the upper tail of the probability integrals of the inverted Dirichlet distribution. Numerical calculations show that the conservative confidence coefficients given in [3] are very good approximations to the exact ones.
TL;DR: Two fundamental ways of analyzing a multivariate analysis of variance (MANOVA) problem in more detail are the stepdown analysis (Roy, 1958), and the use of simultaneous confidence intervals (Gabriel, 1968; Morrison, 1967).
Abstract: Two fundamental ways of analyzing a multivariate analysis of variance (MANOVA) problem in more detail are the stepdown analysis (Roy, 1958; Bock, 1963), and the use of simultaneous confidence intervals (Gabriel, 1968; Morrison, 1967). Assuming a global test of significance, such as Wilk's A, has indicated non- ohance association between the independent variable and the set of dependent variables (this paper will limit discussion to the one- way design), the aforementioned methods may be used to deter- mine which of the dependent variables and/or groups are responsible for the global significance.
TL;DR: Effect of index age on the probable size of random error in site index was evaluated by comparing the 95% confidence interval for site index by curves based on index age 50 years at breast height (BH) with that by curve age 100 years at BH.
Abstract: Effect of index age on the probable size of random error in site index was evaluated by comparing the 95% confidence interval for site index by curves based on index age 50 years at breast height (...
TL;DR: In this article, a class of confidence intervals and a subclass of optimal confidence intervals for θ* are given for a general class of distributions, and it is shown that the coverage probability of the confidence intervals is minimized for a specified configuration of the set of admissible values of θ.
Abstract: There are given k univariate distributions, indexed by a real-valued parameter θ, and k independent observations, one from each distribution. Let θ* denote the largest among the values of θ associated with the given distributions. This article is concerned with the estimation of θ*. A class of confidence intervals and a subclass of “optimal” confidence intervals for θ* are given. For a general class of distributions it is shown that the coverage probability of the confidence intervals is minimized for a specified configuration of the set of admissible values of θ.
TL;DR: In this article, a procedure for choosing a power transformation so that data from a replicated two-way classification better satisfy the usual analysis of variance assumptions is described, and the statistic proposed in this article is compared with the Box and Cox [7] likelihood procedure by empirical sampling.
Abstract: A procedure for choosing a power transformation so that data from a replicated two-way classification better satisfy the usual analysis of variance assumptions is described. The statistic proposed in this article is compared with the Box and Cox [7] likelihood procedure by empirical sampling. Its 95 percent confidence intervals performed at their nominal level, which was not true for the likelihood method. Point estimates for both techniques were the same on the average, although those based upon the likelihood statistic were somewhat less variable.
TL;DR: The Bonferroni inequality can be used to derive a value of Student's t (or a unit normal z) for confidence intervals, and guarantees that the probability is at least 1 -a that all intervals for m contrasts, chosen in advance, cover the true values of their contrasts as discussed by the authors.
Abstract: The Bonferroni inequality can be used to derive a value of Student's t (or a unit normal z) for confidence intervals, and guarantees that the probability is at least 1 — a that all intervals for m contrasts, chosen in advance, cover the true values of their contrasts. An attempt has been made to extend this technique to contrasts chosen post hoc. It is shown that this extension is not generally valid.
TL;DR: The wave-interval procedure as discussed by the authors is a procedure for obtaining a 95% confidence interval for the center (mean, median) of a symmetric distribution that is not only highly efficient when the data have a Normal distribution but also performs well when some or all of the data come from a long-tailed distribution such as the Cauchy.
Abstract: A procedure called the wave-interval is presented for obtaining a 95% confidence interval for the center (mean, median) of a symmetric distribution that is not only highly efficient when the data have a Normal distribution but also performs well when some or all of the data come from a long-tailed distribution such as the Cauchy. Use of the wave-interval greatly reduces the risk of asserting much less than one's data will support. The only table required is the usual t-table. The wave-interval procedure is definitely recommended for samples of ten or more, and appears satisfactory for samples of nine or eight.
TL;DR: Heritability estimates of viability of turkeys from 0 to 4 weeks of age averaged .080 ± .014 with a confidence interval of .014, finding that the lower the number the better, the more likely the turkey to survive.
TL;DR: In this paper, two tables of modest size and three relatively simple formulas provide all that is required to obtain percentage points of the X2 distribution at 25 significance levels with relative accuracy of at least.99995; and to obtain Poisson confidence limits at 13 confidence levels.
Abstract: Two tables of modest size and three relatively simple formulas provide all that is required to obtain percentage points of the X2 distribution at 25 significance levels with relative accuracy of at least .99995; and to obtain Poisson confidence limits of such accuracy at 13 confidence levels. For a small number of degrees of freedom or observed number of events, where the formulas are weak, the tables give exact X2 values or Poisson limits.
TL;DR: The fixed-width confidence intervals for a population regression line over a finite interval of x have been derived by Gafarian as mentioned in this paper for the difference between two population regression lines, resulting in a simple procedure analogous to the Johnson-Neyman technique.
Abstract: Fixed-width confidence intervals for a population regression line over a finite interval of x have recently been derived by Gafarian. The method is extended to provide fixed-width confidence intervals for the difference between two population regression lines, resulting in a simple procedure analogous to the Johnson-Neyman technique.
TL;DR: In this article, uniform confidence bands are developed for a quadratic regression model, in which it is assumed the sum of the cubed deviations of the independent variable about its mean is equal to zero.
Abstract: In this paper uniform confidence bands are developed for a quadratic regression model. Tables are supplied to determine a uniform confidence band for the quadratic model in which it is assumed the sum of the cubed deviations of the independent variable about its mean is equal to zero. The uniform confidence band is then compared with the classical competitor, the Scheffit confidence band. A table summarizing the results of this comparison is presented. Finally, an example is examined.
TL;DR: A method is described for the determination of bromides in biological samples using neutron activation analysis that eliminates sources of interference from other halogens and contamination of analytical agents with extraneous bromide.
Abstract: SUMMARY A method is described for the determination of bromides in biological samples using neutron activation analysis. The specificity of this method eliminates sources of interference from other halogens and contamination of analytical agents with extraneous bromide. In duplicate samples of blood, at varying concentrations of potassium bromide, the mean difference was 4.5% ±1.95 and the mean of the SD of twelve duplicate estimations was ±0.87. The 95% confidence limit of a single estimation was ±1.6%.