1) The document discusses estimation and sample size determination for finite populations, specifically for estimating the mean and proportion. It describes how to calculate confidence intervals and determine sample sizes when sampling without replacement using the finite population correction factor.
2) Formulas are provided for confidence intervals for the mean and proportion when sampling from a finite population without replacement. The finite population correction factor adjusts the standard error and sample size calculations.
3) Examples are given to illustrate how to set up confidence intervals for the mean and proportion and how to determine the necessary sample size using the finite population correction factor.
This document contains a chapter from a statistics textbook on the normal probability distribution. It includes objectives, examples, and explanations of key concepts regarding the normal distribution, such as:
- The properties of the normal distribution curve including that it is symmetric and bell-shaped.
- How to standardize a normal random variable and find areas under the standard normal curve using z-scores and tables of normal probabilities.
- Interpreting the area under the normal curve as probabilities or proportions for real-world examples like IQ scores and weights of animals.
The document discusses sample size determination and adjusting for non-response in marketing research. It provides definitions of key terms like population, parameter, statistic, and confidence interval. It presents methods for determining sample sizes needed to estimate means, proportions, and multiple characteristics based on desired precision levels and population variability. The document also reviews techniques for adjusting sample sizes based on incidence rates and improving response rates, as well as methods for adjusting estimates for non-response bias, like weighting and imputation. Finally, it provides an example of a company that bases its opinions on online surveys of 1,000 respondents.
This document provides an overview of the Poisson distribution and other special probability distributions. It discusses:
1) The Poisson distribution and its properties, including how it can model rare, independent events over time periods. Examples of how to calculate probabilities using the Poisson are provided.
2) Other discrete distributions like the binomial, negative binomial, and hypergeometric.
3) Continuous distributions like the uniform, exponential, gamma, chi-square, and Weibull distributions. Applications and properties of each are summarized.
4) Comparisons between distributions like how the Poisson approximates the binomial for large values. Overall, the document introduces several important probability distributions used in statistics.
The document discusses concepts related to the normal distribution, including:
1) It defines z-scores and how they are calculated by standardizing a value using the formula (x - μ) / σ, where x is the value, μ is the mean, and σ is the standard deviation.
2) It provides examples of calculating and interpreting z-scores, such as finding the z-score for a woman's height and interpreting what that z-score means in relation to the mean height.
3) It discusses how to use normal distribution tables to find the proportion of values above or below a given z-score, or to find the z-score that corresponds to a given proportion.
The document outlines rules for graphing and linear regression. It discusses key aspects of graphing such as properly labeling axes, selecting appropriate scales, and drawing lines smoothly through data points. It then covers linear regression, defining the linear equation and describing how to calculate the slope and y-intercept by minimizing the sum of the deviations between observed and calculated data points. Formulas for determining the slope and y-intercept using linear regression are provided.
This document provides an overview of quantitative analysis techniques for assessing relationships between variables. It discusses concepts related to relationships including presence, nature, direction, and strength of association. It also defines statistical techniques such as ANOVA, cluster analysis, conjoint analysis, discriminant analysis, factor analysis, logistic regression, and multiple regression. Examples are provided to demonstrate calculating explained and unexplained variance in regression, interpreting regression coefficients, and using dummy variables. Steps for conducting regression analysis are outlined including checking assumptions and interpreting residuals plots.
This document provides an introduction to the normal distribution and standard normal probability distribution. It defines key terms like mean, standard deviation, and z-scores. It explains that the normal distribution is a continuous, bell-shaped curve that is symmetrical about the mean. The document also shows how to find probabilities and areas under the normal curve using the z-table and standard normal distribution. Examples are provided to illustrate how to calculate areas between different z-values.
The document discusses various probability distributions including the binomial, Poisson, and normal distributions. It provides definitions and key properties of each distribution. It also discusses sampling with and without replacement as well as the Monte Carlo method for simulating physical systems using random sampling. The Monte Carlo method can be used to computationally estimate values like pi by simulating the throwing of darts at a circular target.
This document contains a chapter from a statistics textbook on the normal probability distribution. It includes objectives, examples, and explanations of key concepts regarding the normal distribution, such as:
- The properties of the normal distribution curve including that it is symmetric and bell-shaped.
- How to standardize a normal random variable and find areas under the standard normal curve using z-scores and tables of normal probabilities.
- Interpreting the area under the normal curve as probabilities or proportions for real-world examples like IQ scores and weights of animals.
The document discusses sample size determination and adjusting for non-response in marketing research. It provides definitions of key terms like population, parameter, statistic, and confidence interval. It presents methods for determining sample sizes needed to estimate means, proportions, and multiple characteristics based on desired precision levels and population variability. The document also reviews techniques for adjusting sample sizes based on incidence rates and improving response rates, as well as methods for adjusting estimates for non-response bias, like weighting and imputation. Finally, it provides an example of a company that bases its opinions on online surveys of 1,000 respondents.
This document provides an overview of the Poisson distribution and other special probability distributions. It discusses:
1) The Poisson distribution and its properties, including how it can model rare, independent events over time periods. Examples of how to calculate probabilities using the Poisson are provided.
2) Other discrete distributions like the binomial, negative binomial, and hypergeometric.
3) Continuous distributions like the uniform, exponential, gamma, chi-square, and Weibull distributions. Applications and properties of each are summarized.
4) Comparisons between distributions like how the Poisson approximates the binomial for large values. Overall, the document introduces several important probability distributions used in statistics.
The document discusses concepts related to the normal distribution, including:
1) It defines z-scores and how they are calculated by standardizing a value using the formula (x - μ) / σ, where x is the value, μ is the mean, and σ is the standard deviation.
2) It provides examples of calculating and interpreting z-scores, such as finding the z-score for a woman's height and interpreting what that z-score means in relation to the mean height.
3) It discusses how to use normal distribution tables to find the proportion of values above or below a given z-score, or to find the z-score that corresponds to a given proportion.
The document outlines rules for graphing and linear regression. It discusses key aspects of graphing such as properly labeling axes, selecting appropriate scales, and drawing lines smoothly through data points. It then covers linear regression, defining the linear equation and describing how to calculate the slope and y-intercept by minimizing the sum of the deviations between observed and calculated data points. Formulas for determining the slope and y-intercept using linear regression are provided.
This document provides an overview of quantitative analysis techniques for assessing relationships between variables. It discusses concepts related to relationships including presence, nature, direction, and strength of association. It also defines statistical techniques such as ANOVA, cluster analysis, conjoint analysis, discriminant analysis, factor analysis, logistic regression, and multiple regression. Examples are provided to demonstrate calculating explained and unexplained variance in regression, interpreting regression coefficients, and using dummy variables. Steps for conducting regression analysis are outlined including checking assumptions and interpreting residuals plots.
This document provides an introduction to the normal distribution and standard normal probability distribution. It defines key terms like mean, standard deviation, and z-scores. It explains that the normal distribution is a continuous, bell-shaped curve that is symmetrical about the mean. The document also shows how to find probabilities and areas under the normal curve using the z-table and standard normal distribution. Examples are provided to illustrate how to calculate areas between different z-values.
The document discusses various probability distributions including the binomial, Poisson, and normal distributions. It provides definitions and key properties of each distribution. It also discusses sampling with and without replacement as well as the Monte Carlo method for simulating physical systems using random sampling. The Monte Carlo method can be used to computationally estimate values like pi by simulating the throwing of darts at a circular target.
The document discusses numerical solutions to physical problems described by Maxwell's equations. It explains that numerical solutions provide approximate solutions that converge to the exact solution as resolution increases. The document then uses the example of calculating the electrostatic potential of a uniformly charged square to demonstrate how results from different resolutions can be extrapolated to estimate the exact solution. Specifically, it shows that the midpoint integration rule converges quadratically while Simpson's rule converges quartically. Extrapolating the results allows obtaining eight-digit accuracy even when the individual computations only have three to four digits of accuracy. The document also discusses handling singular problems and practical procedures for non-uniform grids where extrapolation may be more difficult.
The document discusses the normal distribution and its key properties. It introduces the normal probability density function and how it is characterized by a mean and variance. Some key properties covered are that the sum of independent normally distributed variables is also normally distributed, with the mean being the sum of the individual means and the variance being the sum of the individual variances. It also discusses how to compute probabilities and find values for the standard normal distribution.
The document discusses several continuous probability distributions including the uniform, normal, and exponential distributions. It provides the probability density functions and key characteristics of each distribution. Examples are given to demonstrate calculating probabilities and parameter values for the uniform and normal distributions. Excel functions are also introduced for computing probabilities and values for normal distributions.
The document summarizes key concepts about the normal distribution and the standard normal distribution. It defines normal distributions as a continuous probability distribution with a bell curve shape defined by a mean and standard deviation. It then describes properties of normal distributions including that the mean, median and mode are equal. The rest of the document focuses on the standard normal distribution, defining it as a normal distribution with mean 0 and standard deviation 1. It provides examples of using standard normal tables to find probabilities associated with various z-scores.
CVPR2010: Advanced ITinCVPR in a Nutshell: part 4: Isocontours, Registrationzukun
This document discusses using isocontours and image registration. It proposes estimating densities and entropies from isocontour areas rather than samples to allow image-based density estimation without binning issues. The joint probability of two images can be estimated from the overlapping area of isocontours. Mutual information can then be used for registration by minimizing the joint entropy minus individual entropies estimated from isocontour densities. This approach is compared to standard histogramming.
- Statistical inference allows drawing conclusions about an entire population based on a sample. It includes estimation and hypothesis testing.
- Estimation involves using a statistic like the sample mean or proportion to estimate an unknown population parameter. Point estimates are single values while interval estimates specify a range of values with a given confidence level.
- Confidence intervals for a population mean are constructed using the sample mean as the point estimate and adding/subtracting a margin of error based on the standard error of the mean and reliability coefficient from the normal distribution table. This provides an interval that is expected to contain the true population mean a specified percentage of times.
Applied Business Statistics ,ken black , ch 3 part 1AbdelmonsifFadl
This document provides an overview of descriptive statistics concepts including measures of central tendency (mean, median, mode), measures of variability (range, standard deviation, variance), and how to calculate these measures from both ungrouped and grouped data. It defines key terms, explains how to compute various statistics, and includes example problems and solutions. The learning objectives are to understand and be able to compute different descriptive statistics and apply concepts like the empirical rule and Chebyshev's theorem.
Intro to Quant Trading Strategies (Lecture 10 of 10)Adrian Aley
This document provides an overview of risk management strategies for algorithmic trading. It discusses various risk measurement techniques including Value at Risk (VaR), Extreme Value Theory (EVT), and the generalized extreme value distribution. Specific risks for different asset classes like bonds, stocks, derivatives and currencies are outlined. Monte Carlo simulation is presented as a technique for modeling rare events and fat tails in return distributions. The document emphasizes that risk is multifaceted and not fully captured by any single measure.
Relaxed Utility Maximization in Complete Marketsguasoni
This document provides an outline and summary of a presentation on relaxing the assumptions of utility maximization in complete markets. It discusses relaxing preferences to allow risk aversion to decline to zero as wealth increases, and relaxing payoffs to include more than just random variables. The presentation introduces a model that uses measures instead of densities to represent payoffs, and relaxes the utility functional. It presents results on an expected utility representation involving both absolutely continuous and singular components, and characterizes optimal solutions. The document provides context and motivation for considering cases where the asymptotic elasticity of the utility function approaches one.
This document provides an overview of discrete probability distributions and binomial distributions. It begins by defining discrete and continuous random variables, and how to construct a discrete probability distribution and calculate its mean, variance, and standard deviation. It then focuses on binomial distributions, defining binomial experiments and using the binomial probability formula to calculate probabilities. Examples are provided to illustrate key concepts such as determining if an experiment is binomial, finding binomial probabilities, and calculating measures of a discrete probability distribution.
The normal distribution is a continuous probability distribution defined by its probability density function. A random variable has a normal distribution if its density function is defined by a mean (μ) and standard deviation (σ). The normal distribution is symmetrical and bell-shaped. It is commonly used to approximate other distributions when the sample size is large.
The document discusses methods for efficiently and accurately estimating integrals, including Monte Carlo simulation, low-discrepancy sampling, and Bayesian cubature. It notes that product rules for estimating high-dimensional integrals become prohibitively expensive as dimension increases. Adaptive low-discrepancy sampling is proposed as a method that uses Sobol' or lattice points and normally doubles the number of points until a tolerance is reached.
Intro to Quant Trading Strategies (Lecture 2 of 10)Adrian Aley
This document provides an introduction to hidden Markov models for algorithmic trading strategies. It discusses key concepts like Bayes' theorem, Markov chains, and the Markov property. It then covers the three main problems in hidden Markov models: likelihood, decoding, and learning. It presents solutions to these problems, including the forward-backward, Viterbi, and Baum-Welch algorithms. It also discusses extensions to non-discrete distributions and trading ideas using hidden Markov models.
Chap05 continuous random variables and probability distributionsJudianto Nugroho
This chapter discusses continuous random variables and probability distributions, including the normal distribution. It introduces continuous random variables and their probability density functions. It describes the key characteristics and properties of the uniform and normal distributions. It also discusses how to calculate probabilities using the normal distribution, including how to standardize a normal distribution and use normal distribution tables.
1) The document discusses concepts related to probability distributions including uniform, normal, and binomial distributions.
2) It provides examples of calculating probabilities and values using the uniform, normal, and binomial distributions as well as the normal approximation to the binomial.
3) Key concepts covered include means, standard deviations, z-values, areas under the normal curve, and the continuity correction factor for approximating binomial with normal.
The document discusses the concept of elasticity, which measures the responsiveness of one variable to changes in another variable. It specifically addresses price elasticity of demand, which is the percentage change in quantity demanded divided by the percentage change in price. It defines elastic and inelastic demand based on whether the elasticity is greater than or less than one. The document also discusses how to calculate price elasticity between two points on a demand or supply curve and at a single point. It notes that elasticity is related to but not the same as slope, and can change along straight-line curves.
The document discusses different perspectives on simulating the mean of a function, including deterministic, randomized, and Bayesian approaches. It summarizes Monte Carlo methods using the central limit theorem and Berry-Esseen inequality to estimate error bounds. Low-discrepancy sampling and cubature methods are described which use Fourier coefficients to bound integration errors. Bayesian cubature is outlined, which assumes the function is drawn from a Gaussian process prior to perform optimal quadrature. Maximum likelihood is used to estimate the kernel hyperparameters.
This document discusses error analysis for quasi-Monte Carlo methods. It introduces the trio error identity that decomposes the error into three terms: the variation of the integrand, the discrepancy of the sampling measure from the probability measure, and the alignment between the integrand and the difference between the measures. Several examples are provided to illustrate the identity, including integration over a reproducing kernel Hilbert space. The discrepancy term can be evaluated in O(n^2) operations and converges at different rates depending on the sampling method and properties of the integrand.
McKesson's new Web-based staff scheduling and staffing solution called Web Staffing Insight (WSI) allows hospitals under 200 beds to remotely host their staff scheduling. WSI aims to reduce labor costs by ensuring units are fully staffed with no overtime or use of agency staff, increase staff satisfaction through web access to schedules, and decrease management time spent on scheduling. WSI features include self-scheduling guided by employee rules, schedule balancing tools, and automated messaging about open shifts.
Mundo Fit é uma academia de ginástica que oferece vários tipos de aulas para todos os níveis, como zumba, yoga, boxe e musculação. Localizada no centro da cidade, a academia possui equipamentos modernos e ambientes amplos e bem ventilados para proporcionar uma experiência agradável aos alunos. Os planos variam entre R$99,90 e R$149,90 por mês e oferecem acesso ilimitado às aulas e equipamentos.
The document outlines three offers from RASCOMSTAR-QAF (RSQ) for a telco to achieve pan-African connectivity and rural coverage using RSQ's infrastructure: 1) A standalone R*BCS solution for national and international networking; 2) A standalone R*TES solution to connect isolated areas; 3) A mixed R*BCS and R*TES solution allowing both connectivity types. Pricing and equipment details are provided for the gateway systems and terminals under preferential early taker conditions. The roles and highlights of a potential contract between the telco and RSQ are also summarized.
The document discusses numerical solutions to physical problems described by Maxwell's equations. It explains that numerical solutions provide approximate solutions that converge to the exact solution as resolution increases. The document then uses the example of calculating the electrostatic potential of a uniformly charged square to demonstrate how results from different resolutions can be extrapolated to estimate the exact solution. Specifically, it shows that the midpoint integration rule converges quadratically while Simpson's rule converges quartically. Extrapolating the results allows obtaining eight-digit accuracy even when the individual computations only have three to four digits of accuracy. The document also discusses handling singular problems and practical procedures for non-uniform grids where extrapolation may be more difficult.
The document discusses the normal distribution and its key properties. It introduces the normal probability density function and how it is characterized by a mean and variance. Some key properties covered are that the sum of independent normally distributed variables is also normally distributed, with the mean being the sum of the individual means and the variance being the sum of the individual variances. It also discusses how to compute probabilities and find values for the standard normal distribution.
The document discusses several continuous probability distributions including the uniform, normal, and exponential distributions. It provides the probability density functions and key characteristics of each distribution. Examples are given to demonstrate calculating probabilities and parameter values for the uniform and normal distributions. Excel functions are also introduced for computing probabilities and values for normal distributions.
The document summarizes key concepts about the normal distribution and the standard normal distribution. It defines normal distributions as a continuous probability distribution with a bell curve shape defined by a mean and standard deviation. It then describes properties of normal distributions including that the mean, median and mode are equal. The rest of the document focuses on the standard normal distribution, defining it as a normal distribution with mean 0 and standard deviation 1. It provides examples of using standard normal tables to find probabilities associated with various z-scores.
CVPR2010: Advanced ITinCVPR in a Nutshell: part 4: Isocontours, Registrationzukun
This document discusses using isocontours and image registration. It proposes estimating densities and entropies from isocontour areas rather than samples to allow image-based density estimation without binning issues. The joint probability of two images can be estimated from the overlapping area of isocontours. Mutual information can then be used for registration by minimizing the joint entropy minus individual entropies estimated from isocontour densities. This approach is compared to standard histogramming.
- Statistical inference allows drawing conclusions about an entire population based on a sample. It includes estimation and hypothesis testing.
- Estimation involves using a statistic like the sample mean or proportion to estimate an unknown population parameter. Point estimates are single values while interval estimates specify a range of values with a given confidence level.
- Confidence intervals for a population mean are constructed using the sample mean as the point estimate and adding/subtracting a margin of error based on the standard error of the mean and reliability coefficient from the normal distribution table. This provides an interval that is expected to contain the true population mean a specified percentage of times.
Applied Business Statistics ,ken black , ch 3 part 1AbdelmonsifFadl
This document provides an overview of descriptive statistics concepts including measures of central tendency (mean, median, mode), measures of variability (range, standard deviation, variance), and how to calculate these measures from both ungrouped and grouped data. It defines key terms, explains how to compute various statistics, and includes example problems and solutions. The learning objectives are to understand and be able to compute different descriptive statistics and apply concepts like the empirical rule and Chebyshev's theorem.
Intro to Quant Trading Strategies (Lecture 10 of 10)Adrian Aley
This document provides an overview of risk management strategies for algorithmic trading. It discusses various risk measurement techniques including Value at Risk (VaR), Extreme Value Theory (EVT), and the generalized extreme value distribution. Specific risks for different asset classes like bonds, stocks, derivatives and currencies are outlined. Monte Carlo simulation is presented as a technique for modeling rare events and fat tails in return distributions. The document emphasizes that risk is multifaceted and not fully captured by any single measure.
Relaxed Utility Maximization in Complete Marketsguasoni
This document provides an outline and summary of a presentation on relaxing the assumptions of utility maximization in complete markets. It discusses relaxing preferences to allow risk aversion to decline to zero as wealth increases, and relaxing payoffs to include more than just random variables. The presentation introduces a model that uses measures instead of densities to represent payoffs, and relaxes the utility functional. It presents results on an expected utility representation involving both absolutely continuous and singular components, and characterizes optimal solutions. The document provides context and motivation for considering cases where the asymptotic elasticity of the utility function approaches one.
This document provides an overview of discrete probability distributions and binomial distributions. It begins by defining discrete and continuous random variables, and how to construct a discrete probability distribution and calculate its mean, variance, and standard deviation. It then focuses on binomial distributions, defining binomial experiments and using the binomial probability formula to calculate probabilities. Examples are provided to illustrate key concepts such as determining if an experiment is binomial, finding binomial probabilities, and calculating measures of a discrete probability distribution.
The normal distribution is a continuous probability distribution defined by its probability density function. A random variable has a normal distribution if its density function is defined by a mean (μ) and standard deviation (σ). The normal distribution is symmetrical and bell-shaped. It is commonly used to approximate other distributions when the sample size is large.
The document discusses methods for efficiently and accurately estimating integrals, including Monte Carlo simulation, low-discrepancy sampling, and Bayesian cubature. It notes that product rules for estimating high-dimensional integrals become prohibitively expensive as dimension increases. Adaptive low-discrepancy sampling is proposed as a method that uses Sobol' or lattice points and normally doubles the number of points until a tolerance is reached.
Intro to Quant Trading Strategies (Lecture 2 of 10)Adrian Aley
This document provides an introduction to hidden Markov models for algorithmic trading strategies. It discusses key concepts like Bayes' theorem, Markov chains, and the Markov property. It then covers the three main problems in hidden Markov models: likelihood, decoding, and learning. It presents solutions to these problems, including the forward-backward, Viterbi, and Baum-Welch algorithms. It also discusses extensions to non-discrete distributions and trading ideas using hidden Markov models.
Chap05 continuous random variables and probability distributionsJudianto Nugroho
This chapter discusses continuous random variables and probability distributions, including the normal distribution. It introduces continuous random variables and their probability density functions. It describes the key characteristics and properties of the uniform and normal distributions. It also discusses how to calculate probabilities using the normal distribution, including how to standardize a normal distribution and use normal distribution tables.
1) The document discusses concepts related to probability distributions including uniform, normal, and binomial distributions.
2) It provides examples of calculating probabilities and values using the uniform, normal, and binomial distributions as well as the normal approximation to the binomial.
3) Key concepts covered include means, standard deviations, z-values, areas under the normal curve, and the continuity correction factor for approximating binomial with normal.
The document discusses the concept of elasticity, which measures the responsiveness of one variable to changes in another variable. It specifically addresses price elasticity of demand, which is the percentage change in quantity demanded divided by the percentage change in price. It defines elastic and inelastic demand based on whether the elasticity is greater than or less than one. The document also discusses how to calculate price elasticity between two points on a demand or supply curve and at a single point. It notes that elasticity is related to but not the same as slope, and can change along straight-line curves.
The document discusses different perspectives on simulating the mean of a function, including deterministic, randomized, and Bayesian approaches. It summarizes Monte Carlo methods using the central limit theorem and Berry-Esseen inequality to estimate error bounds. Low-discrepancy sampling and cubature methods are described which use Fourier coefficients to bound integration errors. Bayesian cubature is outlined, which assumes the function is drawn from a Gaussian process prior to perform optimal quadrature. Maximum likelihood is used to estimate the kernel hyperparameters.
This document discusses error analysis for quasi-Monte Carlo methods. It introduces the trio error identity that decomposes the error into three terms: the variation of the integrand, the discrepancy of the sampling measure from the probability measure, and the alignment between the integrand and the difference between the measures. Several examples are provided to illustrate the identity, including integration over a reproducing kernel Hilbert space. The discrepancy term can be evaluated in O(n^2) operations and converges at different rates depending on the sampling method and properties of the integrand.
McKesson's new Web-based staff scheduling and staffing solution called Web Staffing Insight (WSI) allows hospitals under 200 beds to remotely host their staff scheduling. WSI aims to reduce labor costs by ensuring units are fully staffed with no overtime or use of agency staff, increase staff satisfaction through web access to schedules, and decrease management time spent on scheduling. WSI features include self-scheduling guided by employee rules, schedule balancing tools, and automated messaging about open shifts.
Mundo Fit é uma academia de ginástica que oferece vários tipos de aulas para todos os níveis, como zumba, yoga, boxe e musculação. Localizada no centro da cidade, a academia possui equipamentos modernos e ambientes amplos e bem ventilados para proporcionar uma experiência agradável aos alunos. Os planos variam entre R$99,90 e R$149,90 por mês e oferecem acesso ilimitado às aulas e equipamentos.
The document outlines three offers from RASCOMSTAR-QAF (RSQ) for a telco to achieve pan-African connectivity and rural coverage using RSQ's infrastructure: 1) A standalone R*BCS solution for national and international networking; 2) A standalone R*TES solution to connect isolated areas; 3) A mixed R*BCS and R*TES solution allowing both connectivity types. Pricing and equipment details are provided for the gateway systems and terminals under preferential early taker conditions. The roles and highlights of a potential contract between the telco and RSQ are also summarized.
The document appears to be a resume or portfolio of graphic design and production work by Daryl A. Hamilton including logo designs for various organizations, brochure and advertisement designs, wedding invitation designs, and examples of retouched and silhouetted photography work. It provides visual examples and descriptions of the types of design and production projects handled by Daryl A. Hamilton including logo design, brochure design, advertisement design, and photographic retouching and styling.
Deal Structures for Early Stage Financinglerchearly
The document discusses common structures for early stage financing, including common stock issued to founders and friends/family, convertible notes often used in friends/family rounds and by angels, and preferred stock. It provides details on these structures such as rights conveyed, benefits of convertible notes, liquidation preferences for preferred stock, and rights/protections included in financing agreements. The document is intended to educate startups and investors on typical deal terms for early financing rounds.
Lerch Early volunteers at special olympics inspiration walklerchearly
Lerch Early family law attorneys organized the first annual Special Olympics Inspiration Walk to benefit the Special Olympics of Montgomery County. Attorneys and staff from Lerch Early helped register participants, distribute chips to runners, and make opening remarks at the event, which included a 5K run and walk. Neivy Richardson, a paralegal from Lerch Early, sprinted to finish the Inspiration Walk & Run.
Documenting, closing and funding the sba loan lerch early may 2012lerchearly
The document discusses the fundamentals of SBA lending, including documenting, closing, and funding SBA loans. It provides an overview of 7(a) loan programs, eligibility requirements, loan terms, processing types, permissible uses of loan proceeds, and requirements for credit standards, collateral, appraisals, environmental policies, loan conditions, and reviewing third-party reports. The document is intended to guide lenders through the key considerations and steps involved in SBA lending.
Chapter 6 part1- Introduction to Inference-Estimating with Confidence (Introd...nszakir
Introduction to Inference, Estimating with Confidence, Inference, Statistical Confidence, Confidence Intervals, Confidence Interval for a Population Mean, Choosing the Sample Size
The document discusses confidence intervals, which provide a range of values that are likely to contain an unknown population parameter based on a sample statistic. It covers how to construct confidence intervals for a population mean when the population standard deviation is known and unknown, using the t-distribution when the standard deviation is unknown. The document also discusses how confidence intervals provide more information than point estimates and how their width and confidence level are determined.
This document outlines key concepts related to estimation and confidence intervals. It defines point estimates as single values used to estimate population parameters and interval estimates as ranges of values within which the population parameter is expected to occur. Confidence intervals provide an interval range based on sample observations within which the population parameter is expected to fall at a specified confidence level, such as 95% or 99%. The document discusses how to construct confidence intervals for the population mean when the population standard deviation is known or unknown.
A Note on Confidence Bands for Linear Regression Means-07-24-2015Junfeng Liu
This document discusses confidence bands for estimating linear regression means across a domain. It begins by introducing the linear regression model and ordinary least squares estimation. It then discusses how to construct confidence bands that provide simultaneous coverage of the response means across the domain. Both Bonferroni adjustments and C-scaling adjustments are considered. The coverage probabilities of the different band constructions are compared through simulations. Hypothesis testing using the bands is also explored. The consequences of model mis-specification are demonstrated through some examples.
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http://www.youtube.com/onlineteaching
Chapter 7: Estimating Parameters and Determining Sample Sizes
7.3: Estimating a Population Standard Deviation or Variance
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 7: Estimating Parameters and Determining Sample Sizes
7.3: Estimating a Population Standard Deviation or Variance
Shape restrictions such as monotonicity often naturally arise. In this talk, we consider a Bayesian approach to monotone nonparametric regression with a normal error. We assign a prior through piecewise constant functions and impose a conjugate normal prior on the coefficient. Since the resulting functions need not be monotone, we project samples from the posterior on the allowed parameter space to construct a “projection posterior”. We obtain the limit posterior distribution of a suitably centered and scaled posterior distribution for the function value at a point. The limit distribution has some interesting similarity and difference with the corresponding limit distribution for the maximum likelihood estimator. By comparing the quantiles of these two distributions, we observe an interesting new phenomenon that coverage of a credible interval can be more than the credibility level, the exact opposite of a phenomenon observed by Cox for smooth regression. We describe a recalibration strategy to modify the credible interval to meet the correct level of coverage.
This talk is based on joint work with Moumita Chakraborty, a doctoral student at North Carolina State University.
I am Hannah Lucy. Currently associated with statisticshomeworkhelper.com as statistics homework helper. After completing my master's from Kean University, USA, I was in search of an opportunity that expands my area of knowledge hence I decided to help students with their homework. I have written several statistics homework till date to help students overcome numerous difficulties they face.
This document provides procedures for estimating precision error and determining confidence intervals of measurement means and differences between data sets. It begins by explaining how to calculate a confidence interval of the mean using the central limit theorem when sample size is large (>30), which gives a Gaussian distribution of mean values. When sample size is small (<30), it recommends using the Student's t-distribution. Examples are given of calculating confidence intervals at 95% probability levels. The document also covers the Student's t-test for determining if two data sets are significantly different based on calculating the t-value and comparing to t-distribution tables. In general, it advises using a 95% probability level for uncertainty calculations.
This document provides an overview of key statistical concepts including point estimation, confidence intervals, hypothesis testing, and sample size determination. It discusses how to calculate point estimates like the sample mean. It explains how to construct confidence intervals using the normal and t-distributions. It outlines how to perform lower tail, upper tail, and two-tailed hypothesis tests on means and proportions. It also provides formulas for determining required sample sizes.
The document discusses optimal reinsurance strategies to meet a target ruin probability. It describes proportional and nonproportional reinsurance contracts. For proportional reinsurance, the ruin probability can always be decreased by increasing the cession ratio. For nonproportional reinsurance, ruin is possible even if it did not occur without reinsurance. The document models different claim size distributions and analyzes how the optimal deductible varies to meet the target ruin probability.
1) The document describes methods for constructing confidence intervals for the difference between two population means. It provides formulas and examples for when the population variances are known, unknown but assumed equal, and unknown and not assumed equal.
2) It also describes how to construct a confidence interval for the mean difference between two dependent or matched pair samples. An example is given of estimating the difference in effectiveness of two drugs using a 99% confidence interval from data on cholesterol reductions in paired patients.
3) Key steps shown in examples include calculating sample means and variances, determining the test statistic and degrees of freedom, and reporting the confidence interval range.
Confidence Intervals––Exact Intervals, Jackknife, and BootstrapFrancesco Casalegno
The document discusses confidence intervals and different methods for computing them, including exact, asymptotic, jackknife, and bootstrap methods. The exact method computes an interval based on the known distribution of the estimator, but this is often impossible. The asymptotic method uses the asymptotic normality of maximum likelihood estimators, but requires large sample sizes. The jackknife method uses leave-one-out resampling to estimate bias and variance up to O(1/n^2), while bootstrap resamples with replacement to estimate the full distribution and compute confidence intervals.
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The document provides information and examples to study for the week's quiz on normal distributions and the central limit theorem. Key points to remember include: the total area under the standard normal curve is 1; the mean and variance of any normal distribution do not depend on sample size; and the central limit theorem states that as sample size increases, the sampling distribution of the mean approaches a normal distribution, regardless of the original data's distribution. Examples are provided to practice calculating probabilities and confidence intervals using the normal distribution.
This document outlines key concepts related to constructing confidence intervals for estimating population means and proportions. It discusses how to calculate confidence intervals when the population standard deviation is known or unknown. Specifically, it provides the formulas and assumptions for constructing confidence intervals for a population mean using the normal and t-distributions. It also outlines how to calculate confidence intervals for a population proportion using the normal approximation. Examples are provided to demonstrate how to construct 95% confidence intervals for a mean and proportion based on sample data.
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1. 8.7: Estimation and Sample Size Determination for Finite Populations CD8-1
8.7: ESTIMATION AND SAMPLE SIZE DETERMINATION FOR FINITE POPULATIONS
Estimating the Mean
In section 7.3 the finite population correction (fpc) factor is used to reduce the standard error
by a value equal to (N − n )/(N − 1) . When developing confidence interval estimates for popu-
lation parameters, the fpc factor is used when samples are selected without replacement. Thus,
the (1 Ϫ ␣) × 100% confidence interval estimate for the mean is calculated as in Equation
(8.12).
CONFIDENCE INTERVAL FOR A MEAN (σ UNKNOWN) FOR A FINITE POPULATION
S N −n
X ± t n −1 (8.12)
n N −1
To illustrate the finite population correction factor, refer to the confidence interval estimate for
the mean developed for Saxon Plumbing Company in section 8.2. Suppose that in this month there
are 5,000 sales invoices. Using X = $110.27, S = $28.95, N = 5,000, n = 100, and with 95% confi-
dence, t99 = 1.9842. From equation (8.12)
5,000 − 100
28.95
110.27 ± (1.9842)
100 5,000 − 1
= 110.27 ± 5.744(0.99)
110.27 ± 5.69
$104.58 ≤ µ ≤ $115.96
Figure 8.29 illustrates the PHStat output for this example.
FIGURE 8.29
Confidence interval estimate for
the mean sales invoice amount
with the finite population
correction for Saxon Plumbing
Company
In this case, because the sample is a very small fraction of the population, the correction factor has a
minimal effect on the width of the confidence interval. To examine the effect of the correction factor
when the sample size is more than 5% of the population size, see Example 8.10.
2. CD8-2 CD MATERIAL
Example 8.10 In Example 8.3, a sample of 30 insulators was selected. Suppose a population of 300 insulators were
produced by the company. Set up a 95% confidence interval estimate of the population mean.
ESTIMATING THE MEAN FORCE
FOR INSULATORS SOLUTION Using the finite population correction factor, with X = 1,723.4 pounds, S = 89.55,
n = 30, N = 300, and t29 = 2.0452 (for 95% confidence):
S N −n
X ± t n −1
n N −1
89.55 300 − 30
= 1,723.4 ± (2.0452)
30 300 − 1
= 1,723.4 ± 33.44(0.9503)
= 1,723.4 ± 31.776
1,691.62 ≤ µ ≤ 1,755.18
Here, because 10% of the population is to be sampled, the fpc factor has a small effect on the confi-
dence interval estimate.
Estimating the Proportion
In sampling without replacement, the (1 Ϫ ␣) × 100% confidence interval estimate of the propor-
tion is defined in Equation (8.13).
CONFIDENCE INTERVAL ESTIMATE FOR THE PROPORTION USING THE FINITE
POPULATION CORRECTION FACTOR
ps (1 − ps ) N − n
ps ± Z (8.13)
n N −1
To illustrate the use of the finite population correction factor when developing a confidence interval
estimate of the population proportion, consider again the estimate developed for Saxon Home
Improvement Company in section 8.3. For these data, N = 5,000, n = 100, ps = 10/100 = 0.10, and
with 95% confidence, Z = 1.96. Using Equation (8.13),
ps (1 − ps ) N − n
ps ± Z
n N −1
(0.10)(0.90) 5,000 − 100
= 0.10 ± (1.96)
100 5,000 − 1
= 0.10 ± (1.96)(0.03)(0.99)
= 0.10 ± .0582
0.0418 ≤ p ≤ 0.1582
In this case, because the sample is a very small fraction of the population, the fpc factor has virtually
no effect on the confidence interval estimate.
3. 8.7: Estimation and Sample Size Determination for Finite Populations CD8-3
Determining the Sample Size
Just as the fpc factor is used to develop confidence interval estimates, it also is used to determine
sample size when sampling without replacement. For example, in estimating the mean, the sampling
error is
σ N −n
e =Z
n N −1
and in estimating the proportion, the sampling error is
p (1 − p ) N − n
e =Z
n N −1
To determine the sample size in estimating the mean or the proportion from Equations (8.4) and (8.5),
Z 2 σ2 Z 2 p (1 − p )
n0 = and n0 =
e2 e2
where n0 is the sample size without considering the finite population correction factor. Applying the
fpc factor results in the actual sample size n, computed as in Equation (8.14).
SAMPLE SIZE DETERMINATION USING THE FINITE POPULATION CORRECTION FACTOR
n 0N
n = (8.14)
n 0 + (N − 1)
In determining the sample size for Saxon Home Improvement Company, a sample size of 97 was
needed (rounded up from 96.04) for the mean and a sample of 100 (rounded up from 99.96) was
needed for the proportion. Using the fpc factor in Equation (8.14) for the mean, with N = 5,000,
e = $5, S = $25, and Z = 1.96 (for 95% confidence), leads to
(96.04)(5,000)
n = = 94.24
96.04 + (5,000 − 1)
Thus, n = 95.
Using the fpc factor in Equation (8.14) for the proportion, with N = 5,000, e = 0.07, p = 0.15,
and Z = 1.96 (for 95% confidence),
(99.96)(5,000)
n = = 98.02
99.96 + (5,000 − 1)
Thus, n = 99.
To satisfy both requirements simultaneously with one sample, the larger sample size of 99 is
needed. PHStat output is displayed in Figure 8.30.
FIGURE 8.30
Sample size for estimating the
mean sales invoice amount with
the finite population correction
for the Saxon Home
Improvement Company
4. CD8-4 CD MATERIAL
PROBLEMS FOR SECTION 8.7
Learning the Basics depositors who have more than one account at the bank.
What sample size is needed if sampling is done without
8.88 If X = 75, S = 24, n = 36, and N = 200, set up a 95% confi- replacement?
dence interval estimate of the population mean µ if sam- c. What are your answers to (a) and (b) if the bank has
pling is done without replacement. 2,000 depositors?
8.89 Consider a population of 1,000 where the standard deviation 8.93 An automobile dealer wants to estimate the proportion of cus-
is assumed equal to 20. What sample size would be required tomers who still own the cars they purchased 5 years earlier.
if sampling is done without replacement if you desire 95% Sales records indicate that the population of owners is 4,000.
confidence and a sampling error of ±5? a. Set up a 95% confidence interval estimate of the popula-
tion proportion of all customers who still own their cars
Applying the Concepts 5 years after they were purchased if a random sample of
200 customers selected without replacement from the
8.90 The quality control manager at a lightbulb factory needs to automobile dealer’s records indicate that 82 still own cars
estimate the mean life of a large shipment of lightbulbs. The that were purchased 5 years earlier.
process standard deviation is known to be 100 hours. b. What sample size is necessary to estimate the true pro-
Assume that the shipment contains a total of 2,000 light- portion to within ±0.025 with 95% confidence?
bulbs and that sampling is done without replacement. c. What are your answers to (a) and (b) if the population
a. Set up a 95% confidence interval estimate of the popula- consists of 6,000 owners?
tion mean life of lightbulbs in this shipment if a random
8.94 The inspection division of the Lee County Weights and
sample of 50 lightbulbs selected from the shipment indi-
Measures Department is interested in estimating the actual
cates a sample average life of 350 hours.
amount of soft drink that is placed in 2-liter bottles at the
b. Determine the sample size needed to estimate the average
local bottling plant of a large nationally known soft-drink
life to within ±20 hours with 95% confidence.
company. The population consists of 2,000 bottles. The bot-
c. What are your answers to (a) and (b) if the shipment
tling plant has informed the inspection division that the
contains 1,000 lightbulbs?
standard deviation for 2-liter bottles is 0.05 liter.
8.91 A survey is planned to determine the mean annual family a. Set up a 95% confidence interval estimate of the popula-
medical expenses of employees of a large company. The tion mean amount of soft drink per bottle if a random
management of the company wishes to be 95% confident sample of one hundred 2-liter bottles obtained without
that the sample average is correct to within ±$50 of the true replacement from this bottling plant indicates a sample
average annual family medical expenses. A pilot study indi- average of 1.99 liters.
cates that the standard deviation is estimated as $400. How b. Determine the sample size necessary to estimate the pop-
large a sample size is necessary if the company has 3,000 ulation mean amount to within ±0.01 liter with 95%
employees and if sampling is done without replacement? confidence.
c. What are your answers to (a) and (b) if the population
8.92 The manager of a bank that has 1,000 depositors in a small
consists of 1,000 bottles?
city wants to determine the proportion of its depositors with
more than one account at the bank. 8.95 A stationery store wants to estimate the mean retail value of
a. Set up a 90% confidence interval estimate of the popu- greeting cards that it has in its 300-card inventory.
lation proportion of the bank’s depositors who have a. Set up a 95% confidence interval estimate of the popula-
more than one account at the bank if a random sample tion mean value of all greeting cards that are in its inven-
of 100 depositors is selected without replacement and tory if a random sample of 20 greeting cards selected
30 state that they have more than one account at the without replacement indicates an average value of $1.67
bank. and a standard deviation of $0.32.
b. A bank manager wants to be 90% confident of being cor- b. What is your answer to (a) if the store has 500 greeting
rect to within ±0.05 of the true population proportion of cards in its inventory?