This document provides practice exercises related to foundational concepts in statistics including: defining key terms; computing descriptive statistics like mean, median, mode, and range; generating frequency distributions and histograms; computing z-scores, percentiles, and confidence intervals; and defining relationships between statistical concepts. The exercises are intended to help students learn terminology and calculations involved in quantitative data analysis and drawing statistical inferences from samples.
The document discusses the power of recursion and induction in mathematics, modeling, and technology. It provides examples of how recursion appears in definitions of natural numbers and functions. Recursion can also be used to solve complex problems by breaking them down into simpler subproblems. Spreadsheets are an example of how recursion naturally occurs in technology. Teaching recursion enhances modeling skills and helps move from complex to simple problems.
The proposed method uses an online weighted ensemble of one-class SVMs for feature selection in background/foreground separation. It automatically selects the best features for different image regions. Multiple base classifiers are generated using weighted random subspaces. The best base classifiers are selected and combined based on error rates. Feature importance is computed adaptively based on classifier responses. The background model is updated incrementally using a heuristic approach. Experimental results on the MSVS dataset show the proposed method achieves higher precision, recall, and F-score than other methods compared.
This document provides the mark scheme for the October/November 2013 series of the Cambridge International Examinations 0580 MATHEMATICS exam. It shows the requirements for awarding marks to exam questions and serves as guidance for examiners. The mark scheme details the correct answers and working for each question and part, along with the maximum marks available and how partial marks can be awarded. It is intended to ensure consistent marking among examiners and provide transparency in the examination process.
The document proposes using random forests (RF), a machine learning tool, for approximate Bayesian computation (ABC) model choice rather than estimating model posterior probabilities. RF improves on existing ABC model choice methods by having greater discriminative power among models, being robust to the choice and number of summary statistics, requiring less computation, and providing an error rate to evaluate confidence in the model choice. The authors illustrate the power of the RF-based ABC methodology on controlled experiments and real population genetics datasets.
The document summarizes research on quantifying uncertainty in groundwater contamination modeling. It discusses using stochastic methods and surrogate models to estimate how uncertainties in geological parameters, like porosity, propagate and influence quantities of interest in groundwater flow and contaminant transport simulations. Numerical experiments were conducted in 2D and 3D domains using parallel computing to analyze mean concentrations, variances, and other statistics over time under different uncertainty scenarios.
This is the entrance exam paper for ISI MSQE Entrance Exam for the year 2013. Much more information on the ISI MSQE Entrance Exam and ISI MSQE Entrance preparation help available on http://crackdse.com
This is the entrance exam paper for ISI MSQE Entrance Exam for the year 2011. Much more information on the ISI MSQE Entrance Exam and ISI MSQE Entrance preparation help available on http://crackdse.com
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This is the entrance exam paper for ISI MSQE Entrance Exam for the year 2006. Much more information on the ISI MSQE Entrance Exam and ISI MSQE Entrance preparation help available on http://crackdse.com
This document provides instructions for a mathematics exam. It consists of 19 printed pages plus this cover page. Candidates are instructed to write their identification details on all work submitted. They are to write in dark blue or black pen, and may use a pencil for diagrams. Staples, paper clips, etc are not to be used. Calculators may be used. Answers should be given to three significant figures unless specified otherwise. The total marks for the exam is 130. The exam covers topics such as algebra, geometry, trigonometry, and statistics.
This document outlines 6 questions for a math assignment on various interpolation techniques:
1. Use a degree 3 polynomial to estimate life expectancies in 3 years for 2 countries.
2. Fit an exponential function to 5 data points to determine coefficients.
3. Compare accuracy of interpolating a function using cubic spline, pchip cubic, and degree 5 polynomial.
4. Generate and analyze cubic spline and pchip interpolants, with derivatives, for another data set.
5. Find the least squares solution to an overdetermined system of linear equations from altitude measurements.
6. Determine the best fitting function - quadratic, power, or exponential - for another data set. Instructions are provided for including
This is the entrance exam paper for ISI MSQE Entrance Exam for the year 2008. Much more information on the ISI MSQE Entrance Exam and ISI MSQE Entrance preparation help available on http://crackdse.com
This document contains 56 multiple choice questions about forecasting methods from the textbook "Quantitative Analysis for Management, 11e". The questions cover topics such as types of forecasts, time-series forecasting models, measures of forecast accuracy, and exponential smoothing. Correct answers are provided for each question along with the difficulty level and topic area.
This is the entrance exam paper for ISI MSQE Entrance Exam for the year 2005. Much more information on the ISI MSQE Entrance Exam and ISI MSQE Entrance preparation help available on http://crackdse.com
This is the entrance exam paper for ISI MSQE Entrance Exam for the year 2009. Much more information on the ISI MSQE Entrance Exam and ISI MSQE Entrance preparation help available on http://crackdse.com
Machine Learning and Data Mining - Decision Treeswebisslides
Benno Stein, Theo Lettmann
Machine Learning and Data Mining - Introduction - Organization & Literature
http://test.webis.de/lecturenotes/slides/slides.html#machine-learning
This document provides an overview of Support Vector Machines (SVMs). It discusses how SVMs find the optimal separating hyperplane between two classes that maximizes the margin between them. It describes how SVMs can handle non-linearly separable data using kernels to project the data into higher dimensions where it may be linearly separable. The document also discusses multi-class classification approaches for SVMs and how they can be used for anomaly detection.
This is the entrance exam paper for ISI MSQE Entrance Exam for the year 2013. Much more information on the ISI MSQE Entrance Exam and ISI MSQE Entrance preparation help available on http://crackdse.com
Developing visual material can help to recall memory and also be a quick way to show lots of information. Visualization helps us remember (like when we try to picture where we’ve parked our car, and what's in our cupboards when writing a shopping list). We can create diagrams and visual aids depicting module materials and put them up around the house so that we are constantly reminded of our learning
This document contains a series of exercises related to biostatistics and research methods. It includes exercises on categorizing variables, designing a questionnaire, different sampling methods like simple random sampling, systematic random sampling, stratified sampling and cluster sampling. It also includes exercises on ordering and presenting data in frequency distribution tables, calculating measures of central tendency and dispersion, the normal distribution and confidence intervals. The final exercises are on the chi-square test and t-test, including examples of applying these statistical tests to compare groups.
The document provides guidelines for writing a research protocol, including developing clear objectives, outlining the methodology, addressing ethical considerations, and formatting the protocol. An effective protocol clarifies the research question and plan, guides team-based research, and allows for critical review. Key sections include the introduction stating the problem and rationale, methods describing the design, participants, and analysis, and references supporting the information provided. Attention to detail in the protocol is important to properly plan and communicate the study.
The t test can be used to compare sample means to population means, compare means between independent samples, or compare readings within a single sample taken at different times. It involves testing a hypothesis about whether two means are statistically significantly different. The document provides examples of applying the t test to compare a sample mean to a population mean, compare means between independent male and female samples, and compare blood pressure readings within a single sample taken before and after treatment.
The chi-square test is used to determine if differences in frequencies observed in qualitative variable categories are statistically significant or likely due to chance. An example compares influenza rates in a vaccine vs placebo group in a clinical trial. The expected and observed frequencies are calculated. The chi-square test statistic is greater than the critical value, so the null hypothesis that the vaccine and placebo have the same influenza proportion is rejected. Therefore, the difference is likely due to the vaccine's effectiveness rather than chance.
This document provides practice exercises for an introduction to research in information studies course. It includes questions on defining statistical terms, computing descriptive statistics like mean, median and mode for sample data, generating frequency distributions and histograms, hypothesis testing, and constructing confidence intervals. The exercises cover topics like measures of central tendency and dispersion, probability distributions, sampling distributions, and both descriptive and inferential statistics.
The p-value represents the probability of obtaining a test statistic at least as extreme as the one that was actually observed, assuming the null hypothesis is true. A confidence interval provides a range within which the population mean is likely to fall. The variance measures how far data points are spread out from the mean and is calculated as the sum of squared deviations from the mean divided by the sample size minus one. The coefficient of variation allows for more meaningful comparison of distributions with different magnitudes than just comparing standard deviations. The standard error of the sample mean represents the variability of the sample mean as an estimate of the true population mean.
The document provides guidance on writing a research protocol, including its key components and characteristics. A well-written protocol should clearly state the research question/problem and aim, justify the need for the study, and outline the methodology in sufficient detail. Key sections include an introduction with objectives, methods, and ethical/gender considerations. The protocol guides the research plan and must be adhered to strictly.
This document contains 12 exercises related to statistical analysis and research methods. The exercises cover topics such as types of variables, designing a questionnaire, different sampling methods (simple random sampling, systematic random sampling, stratified sampling, cluster sampling), presenting data using frequency distribution tables, measures of central tendency and dispersion, the normal distribution, confidence intervals, the chi-square test, and the t-test. The goal of the exercises is to help students learn fundamental statistical concepts and analysis techniques.
1. The document discusses various hemodynamic disorders including edema, hyperemia, congestion, hemorrhage, thrombosis, embolism, infarction, and shock.
2. Edema results from fluid movement into tissues and can affect subcutaneous tissues, lungs, and brain. Congestion is the passive filling of tissues with blood due to impaired outflow.
3. Thrombosis is the formation of clots within vessels, which can then embolize and travel to other sites (embolism), potentially causing ischemic tissue damage or infarction if blood flow is not restored.
4. Shock represents a failure of circulation to maintain adequate tissue perfusion and oxygenation.
This document discusses measures used to describe the central tendency and dispersion of a frequency distribution. It describes the arithmetic mean, median, and mode as measures of central tendency and their advantages and disadvantages. Measures of dispersion discussed include range, variance, standard deviation, coefficient of variation, and standard error. The choice of central tendency measure depends on the distribution shape, and the mean is most useful for statistical tests while the median is unaffected by outliers.
Haemodynamic disorders , thromboembolism and shock by Dr Nadeem (RMC)Hassan Ahmad
The document discusses various haemodynamic disorders including thrombosis, embolism, shock, hyperemia, congestion, and edema. It provides details on the pathophysiology and morphological changes seen in these conditions.
Hyperemia is an active process resulting from increased blood flow due to arteriolar dilation, causing engorged tissue that appears red. Congestion is a passive process resulting from impaired outflow, causing tissue to appear bluish-red due to accumulation of deoxygenated blood.
Pulmonary congestion microscopically shows engorged alveolar capillaries and edema, while chronic pulmonary congestion shows thickened fibrotic septa and hemosiderin
This document summarizes key concepts related to hemodynamic disorders, thrombosis, and shock. It discusses edema, including the mechanisms and clinical significance of edema. It also covers hyperemia and congestion, hemorrhage, and thrombosis. For edema, it describes how fluid moves between vascular and interstitial spaces and the causes of increased interstitial fluid. It discusses the pathologic features and clinical significance of pulmonary, subcutaneous, and brain edema. For thrombosis, hemorrhage, hyperemia and congestion, it outlines the mechanisms, morphological changes, and clinical implications.
This document discusses different types of hypersensitivity and immunopathology. It covers four main types of hypersensitivity reactions (Type I-IV) that vary in severity from mild to life-threatening. Type I is an immediate reaction mediated by IgE antibodies and mast cells. Type II involves IgG and IgM antibodies against cell surface antigens. Type III reactions are caused by immune complexes circulating in the bloodstream. Type IV is a delayed hypersensitivity mediated by T cells. The document also discusses autoimmune disease, where the immune system attacks the body's own tissues, and immunodeficiencies that increase susceptibility to infection.
Hemodynamic disorders, thrombosis and shock (practical pathology)Mohaned Lehya
This document discusses hemodynamic disorders and thrombosis. It covers several topics including edema, congestion, hemorrhage, thrombosis, embolism, and infarction. Edema is an accumulation of fluid in tissues and organs, and can occur in the lungs (pulmonary edema), abdomen (ascites), and brain. Congestion and hyperemia involve increased blood volume in organs and tissues, seen in conditions like heart failure and liver disease. Thrombosis is the formation of a clot (thrombus) in a blood vessel. Key factors in thrombosis are described by Virchow's triad. Thrombi can embolize and block vessels in other organs, potentially leading to infarction or tissue death.
The document discusses various hemodynamic disorders including hyperemia, congestion, thrombosis, embolism, and infarction. Hyperemia is an increased blood volume in tissue from vasodilation. Congestion is increased blood volume from impaired venous return. Thrombosis is the formation of a blood clot within vessels. An embolism occurs when a piece of thrombus or other material blocks a vessel. Infarction is tissue death from blocked arteries or veins.
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ISM_Session_5 _ 23rd and 24th December.pptxssuser1eba67
The document discusses random variables and their probability distributions. It defines discrete and continuous random variables and their key characteristics. Discrete random variables can take on countable values while continuous can take any value in an interval. Probability distributions describe the probabilities of a random variable taking on different values. The mean and variance are discussed as measures of central tendency and variability. Joint probability distributions are introduced for two random variables. Examples and homework problems are also provided.
Paper Summary of Disentangling by Factorising (Factor-VAE)준식 최
The paper proposes Factor-VAE, which aims to learn disentangled representations in an unsupervised manner. Factor-VAE enhances disentanglement over the β-VAE by encouraging the latent distribution to be factorial (independent across dimensions) using a total correlation penalty. This penalty is optimized using a discriminator network. Experiments on various datasets show that Factor-VAE achieves better disentanglement than β-VAE, as measured by a proposed disentanglement metric, while maintaining good reconstruction quality. Latent traversals qualitatively demonstrate disentangled factors of variation.
The document introduces factor of safety and probability of failure in engineering design. It discusses using sensitivity studies to systematically vary parameters over their credible ranges to determine the influence on factor of safety. This allows a more rational assessment of design risks than relying on a single calculated factor of safety. The document then provides an introduction to probability theory and statistical concepts used in probabilistic analyses, including random variables, probability distributions, sampling techniques, and calculating the probability of failure for a slope design example.
When modeling a system, encountering missing data is common.
What shall a modeler do in the case of unknown or missing information?
When dealing with missing data, it is critical to make correct assumptions to ensure that the system is accurate.
One must common strategy for handling such situations is calculate the average of available data for the similar existing systems (i.e., creating sampling data).
Use this average as a reasonable estimate for the missing value.
This document provides an overview of key concepts in probability and statistics including:
1) Definitions of random variables, discrete and continuous distributions. Discrete variables can take countable values while continuous can take any value in an interval.
2) Common probability distributions like the binomial, Poisson, uniform, and normal distributions. Formulas are provided for the probability mass/density functions and calculating mean, variance, and probability.
3) The exponential distribution with applications like waiting times. Its probability density function and formulas for mean and variance are defined.
Bayesian inference for mixed-effects models driven by SDEs and other stochast...Umberto Picchini
An important, and well studied, class of stochastic models is given by stochastic differential equations (SDEs). In this talk, we consider Bayesian inference based on measurements from several individuals, to provide inference at the "population level" using mixed-effects modelling. We consider the case where dynamics are expressed via SDEs or other stochastic (Markovian) models. Stochastic differential equation mixed-effects models (SDEMEMs) are flexible hierarchical models that account for (i) the intrinsic random variability in the latent states dynamics, as well as (ii) the variability between individuals, and also (iii) account for measurement error. This flexibility gives rise to methodological and computational difficulties.
Fully Bayesian inference for nonlinear SDEMEMs is complicated by the typical intractability of the observed data likelihood which motivates the use of sampling-based approaches such as Markov chain Monte Carlo. A Gibbs sampler is proposed to target the marginal posterior of all parameters of interest. The algorithm is made computationally efficient through careful use of blocking strategies, particle filters (sequential Monte Carlo) and correlated pseudo-marginal approaches. The resulting methodology is is flexible, general and is able to deal with a large class of nonlinear SDEMEMs [1]. In a more recent work [2], we also explored ways to make inference even more scalable to an increasing number of individuals, while also dealing with state-space models driven by other stochastic dynamic models than SDEs, eg Markov jump processes and nonlinear solvers typically used in systems biology.
[1] S. Wiqvist, A. Golightly, AT McLean, U. Picchini (2020). Efficient inference for stochastic differential mixed-effects models using correlated particle pseudo-marginal algorithms, CSDA, https://doi.org/10.1016/j.csda.2020.107151
[2] S. Persson, N. Welkenhuysen, S. Shashkova, S. Wiqvist, P. Reith, G. W. Schmidt, U. Picchini, M. Cvijovic (2021). PEPSDI: Scalable and flexible inference framework for stochastic dynamic single-cell models, bioRxiv doi:10.1101/2021.07.01.450748.
The document discusses sampling distributions and standard errors. It provides:
1) An explanation of sampling distributions as the set of values a statistic can take when calculated from all possible samples of a given size.
2) Formulas for calculating the mean and variance of sampling distributions.
3) A definition of standard error as the standard deviation of a sampling distribution.
4) Common standard errors formulas for statistics like the sample mean, proportion, and difference between means.
5) An example problem demonstrating calculation of the mean and standard error of a sampling distribution of sample means.
Pattern learning and recognition on statistical manifolds: An information-geo...Frank Nielsen
This document provides an overview of Frank Nielsen's talk on pattern learning and recognition using information geometry and statistical manifolds. The talk focuses on departing from vector space representations and dealing with (dis)similarities that do not have Euclidean or metric properties. This poses new theoretical and computational challenges for pattern recognition. The talk describes using exponential family mixture models defined on dually flat statistical manifolds induced by convex functions. On these manifolds, dual coordinate systems and dual affine geodesics allow for computing-friendly representations of divergences and similarities between probabilistic patterns. The techniques aim to achieve statistical invariance and enable algorithmic approaches to problems like Gaussian mixture modeling, shape retrieval, and diffusion tensor imaging analysis.
HW1_STAT206.pdfStatistical Inference II J. Lee Assignment.docxwilcockiris
HW1_STAT206.pdf
Statistical Inference II: J. Lee Assignment 1
Problem 1. Suppose the day after the Drexel-Northeastern basketball game, a poll of 1000 Drexel students
was conducted and it was determined that 850 out of the 1000 watched the game (live or on television).
Assume that this was a simple random sample and that the Drexel undergraduate population is 20000.
(a) Generate an unbiased estimate of the true proportion of Drexel undergraduate students who watched
the game.
(b) What is your estimated standard error for the proportion estimate in (a)?
(c) Give a 95% confidence interval for the true proportion of Drexel undergraduate students who watched
the game.
Problem 2. (Exercise 18 in Chapter 7 of Rice) From independent surveys of two populations, 90% con-
fidence intervals for the population means are conducted. What is the probability that neither interval
contains the respective population mean? That both do?
Problem 3. (Exercise 23 in Chapter 7 of Rice)
(a) Show that the standard error of an estimated proportion is largest when p = 1/2.
(b) Use this result and Corollary B of Section 7.3.2 (also, on Page 17 of the lecture notes) to conclude that
the quantity
1
2
√
N − n
N(n − 1)
is a conservative estimate of the standard error of p̂ no matter what the value of p may be.
(c) Use the central limit theorem to conclude that the interval
p̂ ±
√
N − n
N(n − 1)
contains p with probability at least .95.
HW2_STAT206.pdf
Statistical Inference II: J. Lee Assignment 2
Problem 1. The following data set represents the number of NBA games in January 2016, watched by 10
randomly selected student in STAT 206.
7, 0, 4, 2, 2, 1, 0, 1, 2, 3
(a) What is the sample mean?
(b) Calculate sample variance.
(c) Estimate the mean number of NBA games watched by a student in January 2016.
(d) Estimate the standard error of the estimated mean.
Problem 2. True or false? Tell me why for the false statements.
(a) The center of a 95% confidence interval for the population mean is a random variable.
(b) A 95% confidence interval for µ contains the sample mean with probability .95.
(c) A 95% confidence interval contains 95% of the population.
(d) Out of one hundred 95% confidence intervals for µ, 95 will contain µ.
Problem 3. An investigator quantifies her uncertainty about the estimate of a population mean by reporting
X ± sX . What size confidence interval is?
Problem 4. For a random sample of size n from a population of size N, consider the following as an
estimate of µ:
Xc =
n∑
i=1
ciXi,
where the ci are fixed numbers and X1, . . . ,Xn are the sample. Find a condition on the ci such that the
estimate is unbiased.
Problem 5. A sample of size 100 has the sample mean X = 10. Suppose the we know that the population
standard deviation σ = 5. Find a 95% confidence interval for the population mean µ.
Problem 6. Suppose the we know that the population standard deviation σ = 5. Then how large should a
sample be to estimate the popula.
Possible applications of low-rank tensors in statistics and UQ (my talk in Bo...Alexander Litvinenko
Just some ideas how low-rank matrices/tensors can be useful in spatial and environmental statistics, where one usually has to deal with very large data
I am Driss Fumio. I am a Multivariate Methods Assignment Expert at statisticsassignmentexperts.com. I hold a Master’s Degree in Statistics, from New Brunswick University, Canada. I have been helping students with their assignments for the past 14 years. I solve assignments related to Multivariate Methods. Visit statisticsassignmentexperts.com or email info@statisticsassignmentexperts.com. You can also call on +1 678 648 4277 for any assistance with Multivariate Methods Assignments.
Multiple estimators for Monte Carlo approximationsChristian Robert
This document discusses multiple estimators that can be used to approximate integrals using Monte Carlo simulations. It begins by introducing concepts like multiple importance sampling, Rao-Blackwellisation, and delayed acceptance that allow combining multiple estimators to improve accuracy. It then discusses approaches like mixtures as proposals, global adaptation, and nonparametric maximum likelihood estimation (NPMLE) that frame Monte Carlo estimation as a statistical estimation problem. The document notes various advantages of the statistical formulation, like the ability to directly estimate simulation error from the Fisher information. Overall, the document presents an overview of different techniques for combining Monte Carlo simulations to obtain more accurate integral approximations.
The document discusses distributed online convex optimization algorithms for coordinating multiple agents. It presents a coordination algorithm where each agent performs proportional-integral feedback to minimize local objectives while sharing information with neighbors over noisy communication channels. The algorithm is proven to achieve exponential convergence of second moments to the optimal solution and an ultimate bound on the error that depends on the noise level. Simulation results on a medical diagnosis example are also presented to illustrate the algorithm's behavior.
An investigation of inference of the generalized extreme value distribution b...Alexander Decker
This document presents an investigation of parameter estimation for the generalized extreme value distribution based on record values. Maximum likelihood estimation is used to estimate the parameters β (scale parameter) and ξ (shape parameter). Likelihood equations are derived and solved numerically. Bootstrap and Markov chain Monte Carlo methods are proposed to construct confidence intervals for the parameters since intervals based on asymptotic normality may not perform well due to small sample sizes of records. Bayesian estimation of the parameters using MCMC is also investigated. An illustrative example involving simulated records is provided.
This document provides a summary of Bayesian phylogenetic inference and Markov chain Monte Carlo (MCMC) methods. It begins with an introduction to probability distributions and stochastic processes relevant to phylogenetic modeling. It then discusses how Bayesian inference is applied to phylogenetics by combining prior distributions on tree topologies and other model parameters with the likelihood of the data to obtain posterior distributions. MCMC methods like the Metropolis-Hastings algorithm are introduced as a way to sample from these posterior distributions. Issues around convergence, mixing, and tuning MCMC proposals are also covered.
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.
This document provides information on statistics and grouped data. It defines key terms related to frequency distribution tables, measures of central tendency, measures of dispersion, measures of position, and grouped data. For frequency distribution tables, it discusses variables, frequency, and ways to represent the data through graphs. For measures of central tendency, it defines mean, mode, median, harmonic mean and geometric mean. Measures of dispersion include variance, standard deviation, and mean deviation. Measures of position are quartiles, deciles, and percentiles. The document also discusses terms related to grouped data such as class intervals, class marks, and ways to represent grouped data.
Talk presented on GAMM 2019 Conference in Vienna, Austria.
Parallel algorithm for uncertainty quantification in the density driven subsurface flow. Estimate risks of subsurface flow pollution.
This document contains examples of various statistical analyses exercises involving:
1. Categorizing variables as numerical or categorical.
2. Designing a questionnaire and explaining how to enter data.
3. Different sampling methods like simple random sampling, systematic random sampling, stratified random sampling, and cluster sampling.
4. Presenting data in order arrays, frequency distribution tables, and calculating measures of central tendency and dispersion.
5. Explaining the normal distribution and calculating percentages.
6. Calculating confidence intervals and using the chi-square and t-tests to analyze data.
One-way analysis of variance (ANOVA) compares the means of multiple groups, such as patients with different types of sickle cell disease. ANOVA assesses how much of the overall variation in the data is explained by differences in group means versus differences within groups. If the between-groups variation is large compared to the within-groups variation, then the group means are likely different. ANOVA extends the two-sample t-test to compare more than two groups and provides an F-statistic to test the hypothesis that all group means are equal. Key assumptions are normality of data and equal variances across groups.
This document discusses measures of central tendency and dispersion used to describe frequency distributions. It describes the four key properties: central tendency, dispersion, skewness, and kurtosis. For measures of central tendency, it defines and compares the arithmetic mean, median, and mode. For measures of dispersion, it explains the range, variance, standard deviation, coefficient of variation, and standard error of the sample mean. It provides advantages and disadvantages of each measure.
Stat 4 the normal distribution & steps of testing hypothesisForensic Pathology
The document discusses the normal distribution and statistical hypothesis testing. It notes that the normal distribution is also called the Gaussian distribution, and has equal mean, median and mode. It then discusses how much of the data falls within standard deviations of the mean for the normal distribution. The document also covers confidence intervals for means, the steps of statistical hypothesis testing including assumptions, hypotheses, significance levels and tests, and different statistical tests used for numerical and categorical data like t-tests, ANOVA, regression and correlation.
This document discusses methods for organizing and summarizing data, including ordered arrays, frequency distributions, and frequency polygons. An ordered array lists values from smallest to largest. A frequency distribution groups observations into class intervals to summarize the data, with 6-15 intervals typically used. Sturge's rule provides a formula for calculating the number of intervals. The width of intervals should be equal if possible. A example frequency distribution table with age data in intervals is provided. Figures also demonstrate a success rate by college graph and frequency polygon displaying age data.
This document discusses correlation and the Pearson correlation coefficient (r). It investigates the linear association between body weight and plasma volume in 8 subjects. The correlation coefficient (r) between weight and plasma volume is calculated to be 0.76, indicating a strong positive correlation. A t-test shows this correlation is statistically significant. Values of r range from -1 to 1, where higher positive or negative values indicate stronger linear relationships.
Statistics is the science of collecting, organizing, and analyzing data to draw conclusions. There are two main types of data: data from measurements and data from counts. Data can come from various sources like records, surveys, experiments, and external reports. Biostatistics analyzes data from biological sciences and medicine. Variables are characteristics that can take different values and are either quantitative (measured) or qualitative (categorical). Variables can be random, continuous, discrete, independent, or dependent. Samples are subsets of populations used for statistical analysis. Common random sampling methods include simple random sampling, systematic sampling, stratified sampling, and cluster sampling.
1. The document discusses various types of pancreatic cysts including pseudocysts, congenital cysts, and neoplastic cystic tumors.
2. It outlines benign cystic neoplasms like serous cystadenomas and malignant mucinous cystic neoplasms.
3. Pancreatic ductal adenocarcinoma is discussed as the fourth leading cause of cancer death which often has KRAS and p16 mutations and a desmoplastic response.
The document describes the anatomy, histology, embryology, congenital anomalies, and types of pancreatitis of the pancreas. It notes that the pancreas has exocrine and endocrine components. It also lists the main congenital anomalies as agenesis, pancreas divisum, annular pancreas, and ectopic pancreas. The document provides details on the pathogenesis, morphology, clinical features, diagnosis, and treatment of both acute and chronic pancreatitis.
This document summarizes liver diseases including α1-antitrypsin deficiency, a genetic disorder causing liver and lung disease. It also describes intrahepatic biliary tract diseases like primary and secondary biliary cirrhosis. Benign liver tumors like hemangiomas and adenomas are outlined as well as primary malignant tumors such as hepatoblastoma and angiosarcoma. Hepatocellular carcinoma is discussed in depth, including risk factors, morphology, clinical features, and prognosis. Metastatic liver tumors from other primary cancers are also noted.
1. Liver cirrhosis is the end stage of many chronic liver diseases and is characterized by diffuse hepatic fibrosis and parenchymal nodule formation.
2. Liver abscesses can be caused by parasitic or pyogenic infections and present as solitary or multiple lesions on gross and microscopic examination.
3. Alcoholic liver diseases include fatty liver, alcoholic steatohepatitis (ASH), and alcoholic cirrhosis, progressing from steatosis to necroinflammation and fibrosis.
1. Liver lies in right hypochondrium and divided in to right and left lobes.
2. Microarchitecture : liver is divided into 1 to 2 mm hexagonal lobules.
3. There are four methods for liver biopsy.
4. Most hepatic infections are viral in origin.
5. In fulminant hepatitis hepatic insufficiency progresses from onset of symptoms to hepatic encephalopathy within2 to 3 weeks.
1. The gallbladder can be divided into three parts and lacks two layers. It may also have congenital anomalies.
2. Common gallbladder disorders include cholelithiasis and cholecystitis.
3. Extrahepatic bile duct disorders include stones, infections, and atresia in infants.
4. Tumors of the biliary tract include both benign and malignant types such as adenomas and adenocarcinomas.
This document discusses various types of valvular heart diseases including stenosis, insufficiency, and combinations of the two. Specific conditions covered include calcific aortic stenosis caused by age-related degeneration, myxomatous mitral valve prolapse, rheumatic fever and rheumatic heart disease, infective endocarditis, and nonbacterial thrombotic endocarditis. The morphology, pathogenesis, clinical features, and diagnostic criteria for each condition are described in detail.
This document discusses pericardial diseases and tumors of the heart. It describes different types of pericardial effusions and pericarditis such as serous, fibrinous, purulent, hemorrhagic, and caseous pericarditis. The causes, morphology, and characteristics of each type are provided. It also discusses primary and metastatic tumors of the heart, describing the most common types like myxomas, lipomas, and rhabdomyomas, and how they present and are diagnosed. Metastatic tumors commonly spread to the heart from the lungs, breast, or lymphomas.
1) Ischemic heart disease results from an imbalance between the heart's demand for oxygenated blood and the supply delivered by the coronary arteries, usually due to atherosclerotic plaque buildup.
2) It manifests as stable angina, unstable angina, myocardial infarction, or sudden cardiac death.
3) Myocardial infarction occurs when a blockage in a coronary artery results in prolonged ischemia and cell death in the heart muscle.
This document discusses various types of heart disease including hypertensive heart disease, cardiomyopathies, valvular heart disease, and infective endocarditis. It provides details on the criteria, morphology, causes, and clinical features of each condition. Specifically, it describes how hypertensive heart disease can cause left or right ventricular hypertrophy and heart failure. It also explains the differences between dilated, hypertrophic, and restrictive cardiomyopathies and their causes and features.
This document discusses congenital heart diseases (CHD), which are the most common type of heart disease in children. CHD can be caused by genetic factors like chromosomal abnormalities or environmental factors like infections during pregnancy. They are classified as defects causing left-to-right shunts, right-to-left shunts, or obstructions. Common examples of each type are described along with their typical presentations, morphologies, and clinical outcomes. Surgical correction or intervention is often needed for severe defects.
Arteriosclerosis is a thickening and loss of elasticity of arterial walls that can occur in three patterns: atherosclerosis, Mönckeberg medial calcific sclerosis, and arteriolosclerosis. Atherosclerosis involves the buildup of fatty plaques in arteries due to risk factors like hyperlipidemia, hypertension, smoking, and diabetes. Over time, plaques can rupture, limiting blood flow and risking heart attack or stroke. Arteriolosclerosis affects small arteries and arterioles, presenting as either hyaline or hyperplastic thickening of vessel walls that reduces blood flow and can cause organ damage.
An aneurysm is an abnormal dilation of a blood vessel or heart wall. True aneurysms involve all layers of the vessel wall, while false aneurysms involve only some layers. The most common causes of aneurysms are atherosclerosis and cystic medial degeneration. Abdominal aortic aneurysms usually occur below the renal arteries and are more common in men over 50. Aortic dissections involve blood entering the vessel wall and dissecting between layers, sometimes rupturing outward and causing hemorrhage. Hypertension is a major risk factor.
1. PRACTICE EXERCISES
LIS 397C
Introduction to Research in Information Studies
School of Information
University of Texas at Austin
Dr. Philip Doty
Version 3.5
Copyright Philip Doty, University of Texas at Austin, August 2004 1
2. LIS 397C
Doty
Practice Exercises 3.5
1. Define the following terms and symbols:
n
∑x
x
x
s
s2
N
µ
σ
σ 2
coefficient of variation (CV)
mode
median
arithmetic mean
range
variance
interquartile range (IQR)
standard deviation
sample
statistic
parameter
frequency distribution
2. A sample of the variable x assumes the following values:
9 11 13 3 7 2 8 9 6 10
Compute:
(a) n
(b) ∑ x
(c) x
(d) s
(e) s 2
(f) median
(g) mode
(h) range
Copyright Philip Doty, University of Texas at Austin, August 2004 2
4. 3. A sample of the variable x assumes the following values:
57 51 58 52 50 59 57 51 59 56
50 53 54 50 57 51 53 55 52 54
Generate a frequency distribution indicating x, frequency of x, cumulative
frequency of x, relative frequency of x, and cumulative relative frequency
of x.
4. For the frequency distribution in problem 3, compute:
(a) n
(b) ∑ x
(c) x
(d) s
(e) s 2
(f) median
(g) mode
(h) range
(i) CV
5. Generate a histogram for the data in problem 3.
6. Generate a frequency polygon for the data in problem 3.
7. Define the following terms:
skewness
ordinate
abscissa
central tendency
bimodal
ordered pair
Cartesian plane
stem-and-leaf plot
outlier
reliability
dispersion or variability
negatively skewed
positively skewed
validity
box plot
whiskers
Copyright Philip Doty, University of Texas at Austin, August 2004 4
5. 8. What is the relationship between or among the terms?
(a) sample/population
(b) x /µ
(c) s/ σ
(d) variance/standard deviation
(e) n/N
(f) statistic/parameter
(g) mean, median, mode of the normal curve
(h) coefficient of variation/IQR
9. Graph the following sample distributions (histogram and frequency
polygon) using three different pairs of axes.
Distribution 1 Distribution 2 Distribution 3
Score | Frequency Score | Frequency Score | Frequency
25 4 1 25 2 8
26 10 2 31 4 20
27 6 3 40 6 25
28 3 4 44 8 35
29 3 5 51 10 40
30 1 6 19 12 45
53 1 7 10 13 24
85 1 14 20
10. For each of the distributions in problem 9, answer the following questions.
(a) Is the curve of the distribution positively or negatively skewed?
(b) What is n?
(c) Is the mode > median? Compute the answer and also answer it
graphically, i.e., label the position of the mode and the median on
the curve.
(d) Is the mean > mode? Compute the answer and also answer it
graphically as in part (c) of this question.
(e) What is the variance of the distribution?
(f) What is the standard deviation of the distribution?
(g) What is the range of the distribution?
(h) What is the coefficient of variation of the distribution?
(i) Which measure of central tendency, mode, median, or (arithmetic)
mean, is the fairest and clearest description of the distribution?
Copyright Philip Doty, University of Texas at Austin, August 2004 5
7. 11. Define:
error model
quartile
percentile
Q1
Q2
Q3
freq (x)
cf (x)
rel freq (x)
cum rel freq (x)
PR
z-scores
χ
deviation score
s of z-scores
x of z-scores
∑x
centile
Interquartile Range (IQR)
non-response bias
self-selection
12. Define the relationship(s) between or among the terms:
Q 2 /median
median/N or n
z-score/raw score
z-score/deviation score/standard deviation
Q 1/ Q 2/ Q 3
z-score/ χ /s or σ
median/fifth centile
cf (x)/freq (x)/PR
Literary Digest poll/bias
13. The observations of the values of variable x can be summarized in the
population frequency distribution below.
x freq (x)
9 3
8 9
6 5
Copyright Philip Doty, University of Texas at Austin, August 2004 7
8. 3 2
2 6
Copyright Philip Doty, University of Texas at Austin, August 2004 8
9. For this distribution of x, calculate:
(a) Cumulative frequency, relative frequency, and cumulative relative
frequency for each value
(b) N
(c) the range
(d) median
(e) mode
(f) µ
(g) σ
(h) Q 1 , Q 2 , and Q 3
(i) CV (coefficient of variation)
(j) IQR
(k) the percentile rank of x = 8, x = 2, x = 3
(l) z-scores for x = 6, x = 8, x = 2, x = 3, x = 9
14. Generate a box plot for the data in problem 13.
15. Generate a box plot for the data in problem 3.
16. For a normally distributed distribution of variable x, where µ = 50 and σ
= 2.5 [ND (50, 2.5)], calculate:
(a) the percentile rank of x = 45
(b) the z-score of x = 52.6
(c) the percentile rank of x = 58
(d) the 29.12th percentile
(e) the 89.74th percentile
(f) the z-score of x = 45
(g) the percentile rank of x = 49
17. Define:
α
sampling distribution
Central Limit Theorem
Standard Error (SE)
ND(µ, ) σ
decile
descriptive statistics
inferential statistics
effect size
confidence interval (C.I.) on µ
Copyright Philip Doty, University of Texas at Austin, August 2004 9
10. Student's t
degrees of freedom (df)
random sampling
stratified random sample
Copyright Philip Doty, University of Texas at Austin, August 2004 10
11. 18. Define the relationship(s) between or among the terms:
E( x )/µ
α /df/t
Central Limit Theorem/C.I. on µ
α /C.I. on µ when σ
is known
α /C.I. on µ when σ
is not known
z/confidence interval on µ
t/C.I. on µ
t/z
19. The following values indicate the number of microcomputer applications
available to a sample of 10 computer users.
2, 5, 9, 5, 3, 6, 6, 3, 1, 13
For the population from which the sample was drawn, µ = 4.1 and
σ= 2.93.
Calculate:
a. The expected value of the mean of the sampling distribution of means
b. The standard deviation of the sampling distribution of means.
20. From previous research, we know that the standard deviation of the ages
of public library users is 3.9 years. If the "average" age of a sample of 90
public library users is 20.3 years, construct:
a. a 95% C.I. around µ
b. a 90% C.I. around µ
c. a 99% C.I. around µ.
What does a 95% interval around µ mean?
What is our best estimate of µ?
21. The sample size in problem 20 was increased to 145, while the "average"
age of the sample remained at 20.3 years. Construct three confidence
intervals around µ with the same levels of confidence as in Question 20.
22. Determine t for a C.I. of 95% around µ when n equals 20, 9, ∞ , and 1.
Copyright Philip Doty, University of Texas at Austin, August 2004 11
12. 23. Determine t on the same values of n (20, 9, ∞ , and 1) as in Question 22,
but for a C.I. of 99% around µ. Should this new interval be narrower or
wider than a 95% confidence interval on µ? Why? Answer the question
both conceptually and algebraically.
24. The following frequency distribution gives the values for variable x in a
sample drawn from a larger population.
x freq(x)
43 6
42 11
39 3
36 7
35 7
34 15
Calculate:
(a) x
(b) s
(c) SEµ = σx
(d) E( x )
(e) Q 1 , Q 2 , and Q 3
(f) mode
(g) CV
(h) IQR
(i) a 95% C.I. on µ
(j) the width of the confidence interval in part (i)
(k) a 99% C.I. on µ
(l) the width of the confidence interval in part (k)
(m) PR (percentile rank) of x = 42
(n) our best estimates of µ and σ
from the data.
25. Generate a box plot for the data in problem 24.
26. Construct a stem-and-leaf plot for the following data set; indicate Q 1 , Q 2
, and Q 3 on the plot; and generate the six-figure summary. Be sure that
you are able to identify the stems and the leaves and to identify their units
of measurement.
The heights of members of an extended family were measured in inches.
The observations were: 62, 48, 56, 37, 37, 26, 74, 66, 29, 49, 72, 77, 69, 62,
Copyright Philip Doty, University of Texas at Austin, August 2004 12
14. 27. H 0 : There is no relationship between computer expertise and minutes
spent doing known-item searches in an OPAC at = 0.10. α
Should we reject the H0 given the following data? Remember that the
acceptable error rate is 0.10.
TIME (MINS)
EXPERTISE ≤5 > 5, ≤ 10 > 10
Novice 14 20 19
Intermediate 15 16 9
Expert 22 11 2
28. Answer Question 27 at an acceptable error rate of 0.05.
29. Define:
statistical hypothesis
Ho
H1
p
Type I error
Type II error
χ2
nonparametric
contingency table
statistically significant
E (expected value) in χ 2
O (observed value) in χ 2
30. Discuss the relationship(s) between or among the terms:
α/p
df/R/C [in χ 2 situation]
H o / H1
α
χ 2 / /df
α/Type I error
E/O/ χ 2
Type I/Type II error
Copyright Philip Doty, University of Texas at Austin, August 2004 14
15. SELECTED ANSWERS 3.4
2. (a) n = 10
(b) ∑ = 78
x
∑78
x
(c) x = = = 7.8
n 10
(d) s =
∑ 714 −
x −
nx2
=
608.4
2
= =
11.73 3.43
n−
1 9
2
(e) s = 11.73
n +1 10 +1
(f) P(med) = th score = th score = 5.5th score
2 2
5thscore + 6thscore 8+9
med = 5.5th score = = = 8.5
2 2
(g) mode = 9
(h) range = highest value - lowest value = 13 - 2 = 11
s 3.43
(i) coefficient of variation (CV) = = = 0.44
x 7.8
4. (a) n = 20
(b) ∑ = 1079
x
∑ x 1079
(c) x = = = 53.95
n 20
(d) s =
∑ 58395− = =
x −
nx
=
2
58212
2
9.63 3.1
n−
1 19
2
(e) s = 9.63
n +1 20 +1
(f) P(med) = th score = th score = 10.5th score
2 2
Copyright Philip Doty, University of Texas at Austin, August 2004 15
16. 53 + 54
median = =53.5
2
(g) mode = 50, 51, 57 (trimodal)
(h) range = 59 - 50 = 9
s 3.1
(i) CV = = =
0.06
x 53.95
10. (a) Pos, Pos, Neg
(b) n = 28, 221, 217
(c) mode = 26, 5, 12
P(median) = 14.5th, 111th, 109th observations
median = 26.5, 4, 10
mode1 < median1 ; mode2 > median2; mode3 > median3
∑ ; 2058 =
(d) mean = x = =
x 776 907
; 27.7,4.1,9.5
n 28 221 217
mean1 > mode1; mode2 > mean2; mode3 > mean3
(e) s 2 ∑
=
x − 22218 −
nx2
=
2
(28)(27.7) 733.88
= =
2
27.18
1
n−
1 28 −
1 27
2 10887 −
(221)(4.1)2 7171.99
s2 = = =
32.60
220 220
2 21948 −
(217)(9.5) 2 2363.75
s3 = = =
10.94
216 216
(f) s1,s2 ,s3 =
s2 =
27.18, 32.60 , 10.94 =
5.21,5.71,3.31
(g) Range = highest observation - lowest observation = H - L = 53 -25 = 28;
85 - 1 = 84; 14 - 2 = 12
s 5.21 5.71 3.31
(h) CV = ,
= , =
0.19,1.39,0.35
x 27.7 4.1 9.5
Copyright Philip Doty, University of Texas at Austin, August 2004 16
17. 13. (a) x freq x cum freq x rel freq x cum rel freq x
9 3 25 0.12 1.00
8 9 22 0.36 0.88
6 5 13 0.20 0.52
3 2 8 0.08 0.32
2 6 6 0.24 0.24
25 1.00
(b) N = 25
(c) Range = H - L = 9 - 2 = 7
(d) P(median) = N + 1 th position = 13th observation; median = 6
2
(e) mode = 8
(f) µ
=
∑ =
x 147
= 5.88
N 25
Nµ
∑ 1041−
σ x −= (25)(5.88)
2 2 2
(g) 174.64
= = ==
7.07 2.66
N 25 25
(h) P(Q2) = 13th observation; Q2 = 6
↓
1+ 2 ) 1+
P(Q 13
P(Q1 ) = = = 7th observation from the beginning; Q1=
2 2
3
↓
1+ 2 ) 1+
P(Q 13
P(Q 3 ) = = = 7th observation from the end; Q3= 8
2 2
σ =
2.66
(i) CV == 0.45
µ5.88
(j) IQR = Q3 - Q1 = 8 - 3 = 5
(k) PR = % lower + 1/2 % of that score
PR(8) = 0.52 + (0.36) = 0.70
2
Copyright Philip Doty, University of Texas at Austin, August 2004 17
18. PR(2) = 0 + (0.24) = 0.12
2
PR(3) = 0.24 + (0.08) = 0.28
2
Copyright Philip Doty, University of Texas at Austin, August 2004 18