EMPIRICAL FREQUENCY DISTRIBUTION


 William York
 1 years ago
 Views:
Transcription
1 INTRODUCTION TO MEDICAL STATISTICS: Mirjana Kujundžić Tiljak EMPIRICAL FREQUENCY DISTRIBUTION observed data DISTRIBUTION  described by mathematical models 2 1
2 when some empirical distribution approximates a particular probability distribution theoretical knowledge of that distribution could be used answer questions about data evaluation of probabilities is required 3 PROBABILITY (P) measures uncertainty measures the chance of a given event occurring 0 P 1 P = 0 event cannot occur P = 1 event must occur Q = 1P probability of the complementary event (the event not occurring) 4 2
3 PROBABILITY (P) Various approaches in probability calculations: Subjective personal degree of belief that the event will occur (e.g. the world sill come to an end in the year 2050) Frequentist the proportion of times the event would occur if the experiment will be repeated a large number of times (e.g. the number of times we would get a head") A priori requires knowledge of the theoretical model probability distribution which describes the probabilities of all possible outcomes of the experiment (e.g. genetic theory allows us to describe the probability distribution for eye color in a baby born t a blueeyed women and browneyed man by initially specifying all possible genotypes of eye color in the baby and their probabilities) 5 PROBABILITY (P) The addition rule: if two events (A and B) are mutually exclusive the probability that either one or the other occurs (A or B) is equal to the sum of their probabilities Prob (A or B) = Prob (A) + Prob (B) The multiplication rule: if two events (A and B) are independent the probability that both events occur (A and B) is equal to the product of the probability of each Prob (A and B) = Prob (A) Prob (B) 6 3
4 RANDOM VARIABLES random variable a quantity that can take any one of a set of mutally excluseve values with a given probability discrete or discontinuous random variable = numerical values are integer E.g. number of children in family 0, 1, 2, 3, k continuus random variable = numerical values are real numbers E.g. body weight 72,35 kg, blood glucose level 7,2 mmol/l 7 PROBABILITY DISTRIBUTION Probability distribution shows the probabilities of all possible values of the random variable a theoretical distribution that is expressed mathematically has a mean and variance that are analogous to those of and empirical distribution parameters summary measures (e.g. mean, variance) characterizing that distribution are estimated in the sample by relevant statistics depending on whether the random variable is discrete or continuous the probability distribution can be either discrete or continuous 8 4
5 PROBABILITY DISCRETE (Binomial, Poisson) the probability can be derived corresponding to every possible value of the random variable the sum of all such probabilitis is 1 9 PROBABILITY CONTINUOUS (Normal, Chisquared, t, F) the probability of the random variable, x, taking values in certain ranges, could be derived if the horizontal axis represents the values of x the curve from the equation of the distribution could be drawn (= probability density function) Total area under the curve = 1 represents the probability of all possible events Probability that x lies between two limits is equal to the area under the curve between these values 10 5
6 PROBABILITY Probability that x lies between two limits? 11 PROBABILITY Probability that x lies between two limits? 12 6
7 THE NORMAL (GAUSSIAN) DISTRIBUTION one of the most important distributions in statistics german mathematician C.F. Gauss the most biological measurements follow normal distribution it is used in many analytical models 13 THE NORMAL (GAUSSIAN) DISTRIBUTION Probability density function: f (x) = (1/σ 2π) e a a = 1/2 ((xµ)/σ)
8 THE NORMAL (GAUSSIAN) DISTRIBUTION Completely described by two parameters:  mean (µ ) variance(σ 2 ) X~ N (µ,σ 2 ) 15 THE NORMAL (GAUSSIAN) DISTRIBUTION 16 8
9 THE NORMAL (GAUSSIAN) DISTRIBUTION normal distribution curve: area under curve = 1 bellshaped (unimodal= symmetrical about its mean apsolute maximum for x = µ shifted to the right if the mean is increased and to the left if the mean is decreased (assuming constant variance) flattened as the variance is increased but becomes more peaked as the variance is decreased (for a ficed mean) 17 THE NORMAL (GAUSSIAN) DISTRIBUTION the mean and median and mode of a Normal distribution are equal the probability (P) that a normally distributed random variable, x, with mean, µ, and standard deviation, σ, lies between: (µ  σ) and (µ + σ) = 0,68 (µ σ) and (µ σ) = 0.95 (µ 2.58σ) and (µ σ) = 0.99 these intervals may be used to define reference intervals 18 9
10 THE NORMAL (GAUSSIAN) DISTRIBUTION changing µ, constant σ: 19 THE NORMAL (GAUSSIAN) DISTRIBUTION changing µ, constant σ: 20 10
11 THE NORMAL (GAUSSIAN) DISTRIBUTION changing σ, constant µ: 21 THE NORMAL (GAUSSIAN) DISTRIBUTION changing σ, constant µ: 22 11
12 THE NORMAL (GAUSSIAN) DISTRIBUTION changing σ, constant µ: 23 THE STANDARD NORMAL DISTRIBUTION transformation of original value (x) to Standardized Normal Deviate (SND) (z i ): z i = (x 1  µ)/σ sample: = random variable that has a Standard Normal distribution z i = (x 1  x)/s mean (µ) = 0; variance (σ 2 ) = 1; N (0,1) 24 12
13 THE STANDARD NORMAL DISTRIBUTION X 1 Z 1 X 2 Z 2 X 3 Z 3 X n Z n, s =?, s z =? 25 THE STANDARD NORMAL DISTRIBUTION X 1 Z 1 X 2 Z 2 X 3 Z 3 X n Z n, s =0, s z =
14 THE STANDARD NORMAL DISTRIBUTION X 1 Z 1 X 2 Z 2 X 3 Z 3 X n Z n, s =0, s z =1 Z~N(0,1) 27 THE STANDARD NORMAL DISTRIBUTION 28 14
15 THE STANDARD NORMAL DISTRIBUTION 29 THE STANDARD NORMAL DISTRIBUTION 30 15
16 THE STANDARD NORMAL DISTRIBUTION 31 THE STANDARD NORMAL DISTRIBUTION 32 16
17 THE STUDENT S tdistribution W.S. Gossett (pseudonym Student) parameter that characterizes the tdistribution = the degrees of freedom Similar shape as normal distribution (more spread out with longer tails) as the degrees of freedom increase its shape approaches Normality Useful for calculating confidence intervals for testing hypotheses about one or two means 33 THE STUDENT S tdistribution 34 17
18 THE CHISQUARE (χ 2 ) DISTRIBUTION a right skewed distribution taking positive values characterized by its degrees of freedom its shape depends on the degrees of freedom it becomes more symmetrical and approaches Normality as they increase useful for analysing categorical data 35 THE CHISQUARE (χ 2 ) DISTRIBUTION 36 18
19 THE FDISTRIBUTION skewed to the right defined by a ratio the distribution of a ratio of two estimated variances calculated from Normal dana approximates the Fdistritution characterized by degrees of freedom of the numerator and the denominator of the ratio useful for comparing two variances, and more than two means using the analysis of variance 37 THE LOGNORMAL DISTRIBUTION the probability distribution of a random variable whose log (to base 10 or e) follows the Normal distribution highly skewed to the right logs of row data skewed to the right an empirical distribution that is nearly Normal = data approximate Lognormal distribution geometric mean = a summary measure of location 38 19
20 THE LOGNORMAL DISTRIBUTION 39 THE BINOMIAL DISTRIBUTION theoretical distribution for discrete random variable definition: Jacob Bernuolli, two outcomes: success i failure n events E.g. n = 100 unrelated women undergoing IVF outcome = success (pregnancy) or failure 40 20
21 THE BINOMIAL DISTRIBUTION Two parameters that describe the Binomial distribution: n = number of indivudial in the sample (or repetitions of a trial) π = the true probability of success for each individual (or in each trial) X~B(n,p) 41 THE BINOMIAL DISTRIBUTION Mean = nπ (the value for the random variable that we expect if we look at n individuals, or repeat the trial n times) Variance = nπ (1 π) small n the distribution is skewed to the right if π <0.5 the distribution is skewed to dhe right if π >
22 THE BINOMIAL DISTRIBUTION the distribution becomes more symmetrical as the sample size increases and approximates to the Normal distribution if both nπ and nπ (1 π) are greater than 5 the properties of the Binomial distribution could be use when making inferences about proportions the Normal approximation of the Binomial distribution when analyzing proportions is often used 43 THE BINOMIAL DISTRIBUTION Example: gene recombination Chromosomal locus: 2 allels: A and a p = probability of A Q = 1 p = probability of a P(A) = p, P(a) = q, (p+q = 1) 44 22
23 THE BINOMIAL DISTRIBUTION conception outcame space:{aa, Aa, aa} P(AA) = P(A) * P(A)= p 2 P(aa) = P(a) * P(a) = q 2 P(Aa) = P(A) * P(a) = pq P(aA) = P(a) * P(A)= qp 1,0 2pq p 2 + 2pq + q 2 = (p+q) 2 = 1 2 = 1 45 THE BINOMIAL DISTRIBUTION 46 23
24 THE BINOMIAL DISTRIBUTION Example probability of genotypes: frequency of gene A = 0,33 frequency of gene a = 0,67 (p+q) 2 = (0,33 + 0,67) 2 = 0, * 0,33 * 0,67 + 0,67 2 P (AA)= 0,33 2 = P (Aa) = 0,33 * 0,67 = 0,2211 P (aa) = 0,67 * 0,33 = 0,2211 P (aa) = 0,67 2 = 0, THE BINOMIAL DISTRIBUTION Graphical presentatnion probabilities of different genotypes 0,5 0,45 0,4 0,35 0,3 P 0,25 0,2 0,15 0,1 0,05 0 AA Aa aa 48 24
25 THE BINOMIAL DISTRIBUTION Example death outcome as binomial distribution: Letality od neke bolesti = 0,30..(30/100) Survival probability = 0,70 n = 5 Binom: (0,30 + 0,70) 5 Number of death examinees Binom Probability 5 (everybody) (nobody) P 5 5p 4 q 10p 3 q 2 10p 2 q 3 5pq 4 q 5 0, , , , , ,16807 Total 1, THE POISSON DISTRIBUTION Poisson (begining of XIX century) the Poisson random variable = the count or the number of events that occur independently and randomly in time or space at some average rate, µ (0 and all positive integers) example: the number of hospital admissions per day typically follows the Poisson distribution use of the Poisson cistribution to calculate the probability of a certain number of admissions on any particular day 50 25
26 THE POISSON DISTRIBUTION Mean (average rate, µ) = the parameter that describes the Poisson distribution The mean equals the variance in the Poisson distribution Unimodal curve, right skewed if the mean is small, but becomes more symmetrical as the mean increases, when it approximates n Normal distribution 51 26
Exploratory Data Analysis
Exploratory Data Analysis Johannes Schauer johannes.schauer@tugraz.at Institute of Statistics Graz University of Technology Steyrergasse 17/IV, 8010 Graz www.statistics.tugraz.at February 12, 2008 Introduction
More informationAn Introduction to Basic Statistics and Probability
An Introduction to Basic Statistics and Probability Shenek Heyward NCSU An Introduction to Basic Statistics and Probability p. 1/4 Outline Basic probability concepts Conditional probability Discrete Random
More informationPROBABILITY AND SAMPLING DISTRIBUTIONS
PROBABILITY AND SAMPLING DISTRIBUTIONS SEEMA JAGGI AND P.K. BATRA Indian Agricultural Statistics Research Institute Library Avenue, New Delhi  0 0 seema@iasri.res.in. Introduction The concept of probability
More informationLecture 8. Confidence intervals and the central limit theorem
Lecture 8. Confidence intervals and the central limit theorem Mathematical Statistics and Discrete Mathematics November 25th, 2015 1 / 15 Central limit theorem Let X 1, X 2,... X n be a random sample of
More informationConfidence intervals, t tests, P values
Confidence intervals, t tests, P values Joe Felsenstein Department of Genome Sciences and Department of Biology Confidence intervals, t tests, P values p.1/31 Normality Everybody believes in the normal
More informationHYPOTHESIS TESTING (ONE SAMPLE)  CHAPTER 7 1. used confidence intervals to answer questions such as...
HYPOTHESIS TESTING (ONE SAMPLE)  CHAPTER 7 1 PREVIOUSLY used confidence intervals to answer questions such as... You know that 0.25% of women have red/green color blindness. You conduct a study of men
More informationSection 5 Part 2. Probability Distributions for Discrete Random Variables
Section 5 Part 2 Probability Distributions for Discrete Random Variables Review and Overview So far we ve covered the following probability and probability distribution topics Probability rules Probability
More informationChapter 4. Probability and Probability Distributions
Chapter 4. robability and robability Distributions Importance of Knowing robability To know whether a sample is not identical to the population from which it was selected, it is necessary to assess the
More informationSTT315 Chapter 4 Random Variables & Probability Distributions KM. Chapter 4.5, 6, 8 Probability Distributions for Continuous Random Variables
Chapter 4.5, 6, 8 Probability Distributions for Continuous Random Variables Discrete vs. continuous random variables Examples of continuous distributions o Uniform o Exponential o Normal Recall: A random
More information12.5: CHISQUARE GOODNESS OF FIT TESTS
125: ChiSquare Goodness of Fit Tests CD121 125: CHISQUARE GOODNESS OF FIT TESTS In this section, the χ 2 distribution is used for testing the goodness of fit of a set of data to a specific probability
More informationCA200 Quantitative Analysis for Business Decisions. File name: CA200_Section_04A_StatisticsIntroduction
CA200 Quantitative Analysis for Business Decisions File name: CA200_Section_04A_StatisticsIntroduction Table of Contents 4. Introduction to Statistics... 1 4.1 Overview... 3 4.2 Discrete or continuous
More informationSection 7.2 Confidence Intervals for Population Proportions
Section 7.2 Confidence Intervals for Population Proportions 2012 Pearson Education, Inc. All rights reserved. 1 of 83 Section 7.2 Objectives Find a point estimate for the population proportion Construct
More informationSTAT 155 Introductory Statistics. Lecture 5: Density Curves and Normal Distributions (I)
The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL STAT 155 Introductory Statistics Lecture 5: Density Curves and Normal Distributions (I) 9/12/06 Lecture 5 1 A problem about Standard Deviation A variable
More information4. Continuous Random Variables, the Pareto and Normal Distributions
4. Continuous Random Variables, the Pareto and Normal Distributions A continuous random variable X can take any value in a given range (e.g. height, weight, age). The distribution of a continuous random
More informationDescriptive Statistics. Understanding Data: Categorical Variables. Descriptive Statistics. Dataset: Shellfish Contamination
Descriptive Statistics Understanding Data: Dataset: Shellfish Contamination Location Year Species Species2 Method Metals Cadmium (mg kg  ) Chromium (mg kg  ) Copper (mg kg  ) Lead (mg kg  ) Mercury
More informationChapter 2, part 2. Petter Mostad
Chapter 2, part 2 Petter Mostad mostad@chalmers.se Parametrical families of probability distributions How can we solve the problem of learning about the population distribution from the sample? Usual procedure:
More informationLecture 8: More Continuous Random Variables
Lecture 8: More Continuous Random Variables 26 September 2005 Last time: the eponential. Going from saying the density e λ, to f() λe λ, to the CDF F () e λ. Pictures of the pdf and CDF. Today: the Gaussian
More informationProbability and Statistics Prof. Dr. Somesh Kumar Department of Mathematics Indian Institute of Technology, Kharagpur
Probability and Statistics Prof. Dr. Somesh Kumar Department of Mathematics Indian Institute of Technology, Kharagpur Module No. #01 Lecture No. #15 Special DistributionsVI Today, I am going to introduce
More informationThe Normal distribution
The Normal distribution The normal probability distribution is the most common model for relative frequencies of a quantitative variable. Bellshaped and described by the function f(y) = 1 2σ π e{ 1 2σ
More information6.4 Normal Distribution
Contents 6.4 Normal Distribution....................... 381 6.4.1 Characteristics of the Normal Distribution....... 381 6.4.2 The Standardized Normal Distribution......... 385 6.4.3 Meaning of Areas under
More informationInferential Statistics
Inferential Statistics Sampling and the normal distribution Zscores Confidence levels and intervals Hypothesis testing Commonly used statistical methods Inferential Statistics Descriptive statistics are
More informationMAT140: Applied Statistical Methods Summary of Calculating Confidence Intervals and Sample Sizes for Estimating Parameters
MAT140: Applied Statistical Methods Summary of Calculating Confidence Intervals and Sample Sizes for Estimating Parameters Inferences about a population parameter can be made using sample statistics for
More informationThe basics of probability theory. Distribution of variables, some important distributions
The basics of probability theory. Distribution of variables, some important distributions 1 Random experiment The outcome is not determined uniquely by the considered conditions. For example, tossing a
More informationCHISQUARE: TESTING FOR GOODNESS OF FIT
CHISQUARE: TESTING FOR GOODNESS OF FIT In the previous chapter we discussed procedures for fitting a hypothesized function to a set of experimental data points. Such procedures involve minimizing a quantity
More informationChapter Additional: Standard Deviation and Chi Square
Chapter Additional: Standard Deviation and Chi Square Chapter Outline: 6.4 Confidence Intervals for the Standard Deviation 7.5 Hypothesis testing for Standard Deviation Section 6.4 Objectives Interpret
More informationDescriptive Statistics. Purpose of descriptive statistics Frequency distributions Measures of central tendency Measures of dispersion
Descriptive Statistics Purpose of descriptive statistics Frequency distributions Measures of central tendency Measures of dispersion Statistics as a Tool for LIS Research Importance of statistics in research
More informationSummary of Probability
Summary of Probability Mathematical Physics I Rules of Probability The probability of an event is called P(A), which is a positive number less than or equal to 1. The total probability for all possible
More informationUsing pivots to construct confidence intervals. In Example 41 we used the fact that
Using pivots to construct confidence intervals In Example 41 we used the fact that Q( X, µ) = X µ σ/ n N(0, 1) for all µ. We then said Q( X, µ) z α/2 with probability 1 α, and converted this into a statement
More informationComment on the Tree Diagrams Section
Comment on the Tree Diagrams Section The reversal of conditional probabilities when using tree diagrams (calculating P (B A) from P (A B) and P (A B c )) is an example of Bayes formula, named after the
More information5/31/2013. 6.1 Normal Distributions. Normal Distributions. Chapter 6. Distribution. The Normal Distribution. Outline. Objectives.
The Normal Distribution C H 6A P T E R The Normal Distribution Outline 6 1 6 2 Applications of the Normal Distribution 6 3 The Central Limit Theorem 6 4 The Normal Approximation to the Binomial Distribution
More information
An interval estimate (confidence interval) is an interval, or range of values, used to estimate a population parameter. For example 0.476
Lecture #7 Chapter 7: Estimates and sample sizes In this chapter, we will learn an important technique of statistical inference to use sample statistics to estimate the value of an unknown population parameter.
More informationSampling and Hypothesis Testing
Population and sample Sampling and Hypothesis Testing Allin Cottrell Population : an entire set of objects or units of observation of one sort or another. Sample : subset of a population. Parameter versus
More informationWeek 4: Standard Error and Confidence Intervals
Health Sciences M.Sc. Programme Applied Biostatistics Week 4: Standard Error and Confidence Intervals Sampling Most research data come from subjects we think of as samples drawn from a larger population.
More informationPROBABILITIES AND PROBABILITY DISTRIBUTIONS
Published in "Random Walks in Biology", 1983, Princeton University Press PROBABILITIES AND PROBABILITY DISTRIBUTIONS Howard C. Berg Table of Contents PROBABILITIES PROBABILITY DISTRIBUTIONS THE BINOMIAL
More informationBinomial Distribution n = 20, p = 0.3
This document will describe how to use R to calculate probabilities associated with common distributions as well as to graph probability distributions. R has a number of built in functions for calculations
More informationGoodness of Fit. Proportional Model. Probability Models & Frequency Data
Probability Models & Frequency Data Goodness of Fit Proportional Model Chisquare Statistic Example R Distribution Assumptions Example R 1 Goodness of Fit Goodness of fit tests are used to compare any
More informationBASIC STATISTICAL METHODS FOR GENOMIC DATA ANALYSIS
BASIC STATISTICAL METHODS FOR GENOMIC DATA ANALYSIS SEEMA JAGGI Indian Agricultural Statistics Research Institute Library Avenue, New Delhi110 012 seema@iasri.res.in Genomics A genome is an organism s
More informationMBA 611 STATISTICS AND QUANTITATIVE METHODS
MBA 611 STATISTICS AND QUANTITATIVE METHODS Part I. Review of Basic Statistics (Chapters 111) A. Introduction (Chapter 1) Uncertainty: Decisions are often based on incomplete information from uncertain
More informationThe Normal Curve. The Normal Curve and The Sampling Distribution
Discrete vs Continuous Data The Normal Curve and The Sampling Distribution We have seen examples of probability distributions for discrete variables X, such as the binomial distribution. We could use it
More informationChiSquare Test. Contingency Tables. Contingency Tables. ChiSquare Test for Independence. ChiSquare Tests for GoodnessofFit
ChiSquare Tests 15 Chapter ChiSquare Test for Independence ChiSquare Tests for Goodness Uniform Goodness Poisson Goodness Goodness Test ECDF Tests (Optional) McGrawHill/Irwin Copyright 2009 by The
More informationHYPOTHESIS TESTING (ONE SAMPLE)  CHAPTER 7 1. used confidence intervals to answer questions such as...
HYPOTHESIS TESTING (ONE SAMPLE)  CHAPTER 7 1 PREVIOUSLY used confidence intervals to answer questions such as... You know that 0.25% of women have red/green color blindness. You conduct a study of men
More informationStatistics I for QBIC. Contents and Objectives. Chapters 1 7. Revised: August 2013
Statistics I for QBIC Text Book: Biostatistics, 10 th edition, by Daniel & Cross Contents and Objectives Chapters 1 7 Revised: August 2013 Chapter 1: Nature of Statistics (sections 1.11.6) Objectives
More informationA frequency distribution is a table used to describe a data set. A frequency table lists intervals or ranges of data values called data classes
A frequency distribution is a table used to describe a data set. A frequency table lists intervals or ranges of data values called data classes together with the number of data values from the set that
More informationMT426 Notebook 3 Fall 2012 prepared by Professor Jenny Baglivo. 3 MT426 Notebook 3 3. 3.1 Definitions... 3. 3.2 Joint Discrete Distributions...
MT426 Notebook 3 Fall 2012 prepared by Professor Jenny Baglivo c Copyright 20042012 by Jenny A. Baglivo. All Rights Reserved. Contents 3 MT426 Notebook 3 3 3.1 Definitions............................................
More informationH + T = 1. p(h + T) = p(h) x p(t)
Probability and Statistics Random Chance A tossed penny can land either heads up or tails up. These are mutually exclusive events, i.e. if the coin lands heads up, it cannot also land tails up on the same
More informationFairfield Public Schools
Mathematics Fairfield Public Schools AP Statistics AP Statistics BOE Approved 04/08/2014 1 AP STATISTICS Critical Areas of Focus AP Statistics is a rigorous course that offers advanced students an opportunity
More informationLecture 5 : The Poisson Distribution
Lecture 5 : The Poisson Distribution Jonathan Marchini November 10, 2008 1 Introduction Many experimental situations occur in which we observe the counts of events within a set unit of time, area, volume,
More information4. Introduction to Statistics
Statistics for Engineers 41 4. Introduction to Statistics Descriptive Statistics Types of data A variate or random variable is a quantity or attribute whose value may vary from one unit of investigation
More informationA review of the portions of probability useful for understanding experimental design and analysis.
Chapter 3 Review of Probability A review of the portions of probability useful for understanding experimental design and analysis. The material in this section is intended as a review of the topic of probability
More informationInstitute of Actuaries of India Subject CT3 Probability and Mathematical Statistics
Institute of Actuaries of India Subject CT3 Probability and Mathematical Statistics For 2015 Examinations Aim The aim of the Probability and Mathematical Statistics subject is to provide a grounding in
More informationChapter 10 Monte Carlo Methods
411 There is no result in nature without a cause; understand the cause and you will have no need for the experiment. Leonardo da Vinci (14521519) Chapter 10 Monte Carlo Methods In very broad terms one
More informationIntroduction to Hypothesis Testing. Point estimation and confidence intervals are useful statistical inference procedures.
Introduction to Hypothesis Testing Point estimation and confidence intervals are useful statistical inference procedures. Another type of inference is used frequently used concerns tests of hypotheses.
More informationDATA INTERPRETATION AND STATISTICS
PholC60 September 001 DATA INTERPRETATION AND STATISTICS Books A easy and systematic introductory text is Essentials of Medical Statistics by Betty Kirkwood, published by Blackwell at about 14. DESCRIPTIVE
More informationfifty Fathoms Statistics Demonstrations for Deeper Understanding Tim Erickson
fifty Fathoms Statistics Demonstrations for Deeper Understanding Tim Erickson Contents What Are These Demos About? How to Use These Demos If This Is Your First Time Using Fathom Tutorial: An Extended Example
More informationMendelian Inheritance & Probability
Mendelian Inheritance & Probability (CHAPTER 2 Brooker Text) January 31 & Feb 2, 2006 BIO 184 Dr. Tom Peavy Problem Solving TtYy x ttyy What is the expected phenotypic ratio among offspring? Tt RR x Tt
More informationWhy Taking This Course? Course Introduction, Descriptive Statistics and Data Visualization. Learning Goals. GENOME 560, Spring 2012
Why Taking This Course? Course Introduction, Descriptive Statistics and Data Visualization GENOME 560, Spring 2012 Data are interesting because they help us understand the world Genomics: Massive Amounts
More informationChiSquare Distribution. is distributed according to the chisquare distribution. This is usually written
ChiSquare Distribution If X i are k independent, normally distributed random variables with mean 0 and variance 1, then the random variable is distributed according to the chisquare distribution. This
More informationSampling Distributions
Sampling Distributions You have seen probability distributions of various types. The normal distribution is an example of a continuous distribution that is often used for quantitative measures such as
More information4: Probability. What is probability? Random variables (RVs)
4: Probability b binomial µ expected value [parameter] n number of trials [parameter] N normal p probability of success [parameter] pdf probability density function pmf probability mass function RV random
More informationUNIT I: RANDOM VARIABLES PART A TWO MARKS
UNIT I: RANDOM VARIABLES PART A TWO MARKS 1. Given the probability density function of a continuous random variable X as follows f(x) = 6x (1x) 0
More information16. THE NORMAL APPROXIMATION TO THE BINOMIAL DISTRIBUTION
6. THE NORMAL APPROXIMATION TO THE BINOMIAL DISTRIBUTION It is sometimes difficult to directly compute probabilities for a binomial (n, p) random variable, X. We need a different table for each value of
More informationF. Farrokhyar, MPhil, PhD, PDoc
Learning objectives Descriptive Statistics F. Farrokhyar, MPhil, PhD, PDoc To recognize different types of variables To learn how to appropriately explore your data How to display data using graphs How
More informationHow to Conduct a Hypothesis Test
How to Conduct a Hypothesis Test The idea of hypothesis testing is relatively straightforward. In various studies we observe certain events. We must ask, is the event due to chance alone, or is there some
More informationSYSM 6304: Risk and Decision Analysis Lecture 3 Monte Carlo Simulation
SYSM 6304: Risk and Decision Analysis Lecture 3 Monte Carlo Simulation M. Vidyasagar Cecil & Ida Green Chair The University of Texas at Dallas Email: M.Vidyasagar@utdallas.edu September 19, 2015 Outline
More informationChapter 3 RANDOM VARIATE GENERATION
Chapter 3 RANDOM VARIATE GENERATION In order to do a Monte Carlo simulation either by hand or by computer, techniques must be developed for generating values of random variables having known distributions.
More information1) Write the following as an algebraic expression using x as the variable: Triple a number subtracted from the number
1) Write the following as an algebraic expression using x as the variable: Triple a number subtracted from the number A. 3(x  x) B. x 3 x C. 3x  x D. x  3x 2) Write the following as an algebraic expression
More informationLecture 7: Binomial Test, Chisquare
Lecture 7: Binomial Test, Chisquare Test, and ANOVA May, 01 GENOME 560, Spring 01 Goals ANOVA Binomial test Chi square test Fisher s exact test Su In Lee, CSE & GS suinlee@uw.edu 1 Whirlwind Tour of One/Two
More information2.0 Lesson Plan. Answer Questions. Summary Statistics. Histograms. The Normal Distribution. Using the Standard Normal Table
2.0 Lesson Plan Answer Questions 1 Summary Statistics Histograms The Normal Distribution Using the Standard Normal Table 2. Summary Statistics Given a collection of data, one needs to find representations
More informationBasic Statistics. Probability and Confidence Intervals
Basic Statistics Probability and Confidence Intervals Probability and Confidence Intervals Learning Intentions Today we will understand: Interpreting the meaning of a confidence interval Calculating the
More informationProbability and Statistics Vocabulary List (Definitions for Middle School Teachers)
Probability and Statistics Vocabulary List (Definitions for Middle School Teachers) B Bar graph a diagram representing the frequency distribution for nominal or discrete data. It consists of a sequence
More informationVariables. Exploratory Data Analysis
Exploratory Data Analysis Exploratory Data Analysis involves both graphical displays of data and numerical summaries of data. A common situation is for a data set to be represented as a matrix. There is
More information15.0 More Hypothesis Testing
15.0 More Hypothesis Testing 1 Answer Questions Type I and Type II Error Power Calculation Bayesian Hypothesis Testing 15.1 Type I and Type II Error In the philosophy of hypothesis testing, the null hypothesis
More informationSimple Linear Regression Inference
Simple Linear Regression Inference 1 Inference requirements The Normality assumption of the stochastic term e is needed for inference even if it is not a OLS requirement. Therefore we have: Interpretation
More informationStatistics. Measurement. Scales of Measurement 7/18/2012
Statistics Measurement Measurement is defined as a set of rules for assigning numbers to represent objects, traits, attributes, or behaviors A variableis something that varies (eye color), a constant does
More informationTechnology StepbyStep Using StatCrunch
Technology StepbyStep Using StatCrunch Section 1.3 Simple Random Sampling 1. Select Data, highlight Simulate Data, then highlight Discrete Uniform. 2. Fill in the following window with the appropriate
More information2 GENETIC DATA ANALYSIS
2.1 Strategies for learning genetics 2 GENETIC DATA ANALYSIS We will begin this lecture by discussing some strategies for learning genetics. Genetics is different from most other biology courses you have
More informationReview of Random Variables
Chapter 1 Review of Random Variables Updated: January 16, 2015 This chapter reviews basic probability concepts that are necessary for the modeling and statistical analysis of financial data. 1.1 Random
More informationChi Square Tests. Chapter 10. 10.1 Introduction
Contents 10 Chi Square Tests 703 10.1 Introduction............................ 703 10.2 The Chi Square Distribution.................. 704 10.3 Goodness of Fit Test....................... 709 10.4 Chi Square
More informationQuantitative Methods for Finance
Quantitative Methods for Finance Module 1: The Time Value of Money 1 Learning how to interpret interest rates as required rates of return, discount rates, or opportunity costs. 2 Learning how to explain
More informationThe Standard Normal distribution
The Standard Normal distribution 21.2 Introduction Massproduced items should conform to a specification. Usually, a mean is aimed for but due to random errors in the production process we set a tolerance
More informationProbability Distributions
Learning Objectives Probability Distributions Section 1: How Can We Summarize Possible Outcomes and Their Probabilities? 1. Random variable 2. Probability distributions for discrete random variables 3.
More informationSummary of Formulas and Concepts. Descriptive Statistics (Ch. 14)
Summary of Formulas and Concepts Descriptive Statistics (Ch. 14) Definitions Population: The complete set of numerical information on a particular quantity in which an investigator is interested. We assume
More informationBusiness Statistics. Successful completion of Introductory and/or Intermediate Algebra courses is recommended before taking Business Statistics.
Business Course Text Bowerman, Bruce L., Richard T. O'Connell, J. B. Orris, and Dawn C. Porter. Essentials of Business, 2nd edition, McGrawHill/Irwin, 2008, ISBN: 9780073319889. Required Computing
More informationGCSE HIGHER Statistics Key Facts
GCSE HIGHER Statistics Key Facts Collecting Data When writing questions for questionnaires, always ensure that: 1. the question is worded so that it will allow the recipient to give you the information
More information8. THE NORMAL DISTRIBUTION
8. THE NORMAL DISTRIBUTION The normal distribution with mean μ and variance σ 2 has the following density function: The normal distribution is sometimes called a Gaussian Distribution, after its inventor,
More informationMATHEMATICS FOR ENGINEERS STATISTICS TUTORIAL 4 PROBABILITY DISTRIBUTIONS
MATHEMATICS FOR ENGINEERS STATISTICS TUTORIAL 4 PROBABILITY DISTRIBUTIONS CONTENTS Sample Space Accumulative Probability Probability Distributions Binomial Distribution Normal Distribution Poisson Distribution
More informationMathematics. Probability and Statistics Curriculum Guide. Revised 2010
Mathematics Probability and Statistics Curriculum Guide Revised 2010 This page is intentionally left blank. Introduction The Mathematics Curriculum Guide serves as a guide for teachers when planning instruction
More informationThe Binomial Probability Distribution
The Binomial Probability Distribution MATH 130, Elements of Statistics I J. Robert Buchanan Department of Mathematics Fall 2015 Objectives After this lesson we will be able to: determine whether a probability
More informationResearch Variables. Measurement. Scales of Measurement. Chapter 4: Data & the Nature of Measurement
Chapter 4: Data & the Nature of Graziano, Raulin. Research Methods, a Process of Inquiry Presented by Dustin Adams Research Variables Variable Any characteristic that can take more than one form or value.
More informationMATH4427 Notebook 2 Spring 2016. 2 MATH4427 Notebook 2 3. 2.1 Definitions and Examples... 3. 2.2 Performance Measures for Estimators...
MATH4427 Notebook 2 Spring 2016 prepared by Professor Jenny Baglivo c Copyright 20092016 by Jenny A. Baglivo. All Rights Reserved. Contents 2 MATH4427 Notebook 2 3 2.1 Definitions and Examples...................................
More informationREPEATED TRIALS. The probability of winning those k chosen times and losing the other times is then p k q n k.
REPEATED TRIALS Suppose you toss a fair coin one time. Let E be the event that the coin lands heads. We know from basic counting that p(e) = 1 since n(e) = 1 and 2 n(s) = 2. Now suppose we play a game
More informationDescriptive Statistics
Descriptive Statistics Suppose following data have been collected (heights of 99 fiveyearold boys) 117.9 11.2 112.9 115.9 18. 14.6 17.1 117.9 111.8 16.3 111. 1.4 112.1 19.2 11. 15.4 99.4 11.1 13.3 16.9
More informationDescriptive Statistics
Y520 Robert S Michael Goal: Learn to calculate indicators and construct graphs that summarize and describe a large quantity of values. Using the textbook readings and other resources listed on the web
More informationChapter Five: Paired Samples Methods 1/38
Chapter Five: Paired Samples Methods 1/38 5.1 Introduction 2/38 Introduction Paired data arise with some frequency in a variety of research contexts. Patients might have a particular type of laser surgery
More informationTHE FIRST SET OF EXAMPLES USE SUMMARY DATA... EXAMPLE 7.2, PAGE 227 DESCRIBES A PROBLEM AND A HYPOTHESIS TEST IS PERFORMED IN EXAMPLE 7.
THERE ARE TWO WAYS TO DO HYPOTHESIS TESTING WITH STATCRUNCH: WITH SUMMARY DATA (AS IN EXAMPLE 7.17, PAGE 236, IN ROSNER); WITH THE ORIGINAL DATA (AS IN EXAMPLE 8.5, PAGE 301 IN ROSNER THAT USES DATA FROM
More informationSenior Secondary Australian Curriculum
Senior Secondary Australian Curriculum Mathematical Methods Glossary Unit 1 Functions and graphs Asymptote A line is an asymptote to a curve if the distance between the line and the curve approaches zero
More informationConsider a system that consists of a finite number of equivalent states. The chance that a given state will occur is given by the equation.
Probability and the ChiSquare Test written by J. D. Hendrix Learning Objectives Upon completing the exercise, each student should be able: to determine the chance that a given state will occur in a system
More informationAP Statistics Solutions to Packet 2
AP Statistics Solutions to Packet 2 The Normal Distributions Density Curves and the Normal Distribution Standard Normal Calculations HW #9 1, 2, 4, 68 2.1 DENSITY CURVES (a) Sketch a density curve that
More informationData Modeling & Analysis Techniques. Probability & Statistics. Manfred Huber 2011 1
Data Modeling & Analysis Techniques Probability & Statistics Manfred Huber 2011 1 Probability and Statistics Probability and statistics are often used interchangeably but are different, related fields
More information
Beta Distribution. Paul Johnson and Matt Beverlin June 10, 2013
Beta Distribution Paul Johnson and Matt Beverlin June 10, 2013 1 Description How likely is it that the Communist Party will win the net elections in Russia? In my view,
More information