I can apply the properties **and** **formulas** of sampling distributions for proportions Quick Review The **formula** for the mean of a sampling distribution is: The **formula** for the standard deviation of the mean of a sampling distribution is: A parameter is: A **statistic** is: The symbol for population/Draw the sampling distribution of based on a random sample of 400. b) When n = 400, what is ? c) Is the **probability** in part (b.) larger or smaller than would be the case if n = 500? Think, don’t calculate. Example 4 A /

uniform prior (which is reasonable if all hypotheses are of the same complexity) ML is the standard (non-Bayesian) **statistical** learning method ML parameter learning Bag from a new manufacturer; fraction of cherry candies? Any is possible: continuum of/One more assumption **FORMULA** Learn_Naive_Bayes_Text (Examples, V) 1. Collect all words **and** other tokens that occur in Examples - Vocabulary all distinct words **and** other tokens in Examples 2. Calculate the required P(v j ) **and** P(w k |v j ) **probability** terms - For/

. (The sampling distribution of the mean is typically represented as a **probability** distribution in the format of a table, **probability** histogram, or **formula**.) Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Definition The value of a **statistic**, such as the sample mean x, depends on the particular values included in the sample, **and** generally varies from sample to sample. This variability of a/

CIVE2602 - Engineering Mathematics 2.2 (20 credits) **Statistics** **and** **Probability** Lecture 9 Hypothesis testing –Examples Undertaking experiments t-test for two sample means P-values F-tests Dr D Borman ©Claudio Nunez / with an unknown σ (population standard deviation) Use the t-test n x =5 n y =5 X =24,Y =27Find sample means: Find sample standard deviation: **Formula** B: **Formula** A: Degrees of freedom, v = n – 2 = 5+5 -2 = 8 (assume underlying population is Normally distributed) F-test – test on the variance So/

JULY Holiday Year 13 REVISION Year 12 Taxation: Income tax **and** National Insurance Taxation: Income tax **and** National Insurance Graphical representation Graphical representation **STATISTICAL** TECHNIQUES 2 **STATISTICAL** TECHNIQUES 2 Correlation **and** regression Correlation **and** regression **STATISTICAL** TECHNIQUES 3 **STATISTICAL** TECHNIQUES 3 **Probabilities** **and** estimation **Probabilities** **and** estimation CRITICAL PATH **AND** RISK ANALYSIS 3 CRITICAL PATH **AND** RISK ANALYSIS 3 Cost benefit analysis Cost benefit analysis CRITICAL/

**Statistics**: Bivariate Calculation **Formulae** for Pearson Product Moment Correlation Coefficient (r) Correlation Coefficient example using “calculation **formulae**” As we explore spatial **statistics**, we will see many analogies to the mean, the variance, **and** the correlation coefficient, **and** their various **formulae**/ assumption made regarding the type of sampling involved: Free (or normality) sampling assumes that the **probability** of a polygon having a particular value is not affected by the number or arrangement of the/

Simultaneous equations Simultaneous equations **Probability** Review **and** revision 9 **Statistics** recap **and** review Algebra: introduction to quadratics **and** rearranging **formulae** Algebra: introduction to quadratics **and** rearranging **formulae** Volume Algebra recap **and** review Algebra recap **and** review Linear **and** quadratic equations **and** their graphs Linear **and** quadratic equations **and** their graphs Sketching graphs Sketching graphs Geometry **and** measures recap **and** review Geometry **and** measures recap **and** review GCSE Mathematics/

l Addition rule l Conditional **probability** **formula** l Multiplication rule 4 - 23 © 1998 Prentice-Hall, Inc. **Statistics** for Managers Using Microsoft Excel, 1/e Event **Probability** Using Contingency Table Joint **Probability** Marginal (Simple) **Probability** 4 - 24 © 1998 Prentice-Hall, Inc. **Statistics** for Managers Using Microsoft Excel, 1/e Contingency Table Example Experiment: Draw 1 card. Note kind, color & suit. P(Ace) P(Ace **AND** Red)P(Red) 4/

Holds in state s iff **probability** is at least that holds over paths starting in s P < ( ) P ≥1– ( ) 7 Path **Formulas** Until: 1 U ≤T 2 Holds over path iff 2 becomes true in some state along before time T, **and** 1 is true in/sampling can be used to verify probabilistic properties of systems Sequential acceptance sampling adapts to the difficulty of the problem **Statistical** methods are easy to parallelize 28 Other Research Failure trace analysis “failure scenario” [Younes & Simmons 2004a] /

://www.pp.rhul.ac.uk/~cowan/stat_orsay.html G. Cowan **Statistics** for HEP / LAL Orsay, 3-5 January 2012 / Lecture 2 2 Outline Lecture 1: Introduction **and** basic formalism **Probability**, **statistical** tests, confidence intervals. Lecture 2: Tests based on likelihood ratios/ for fairly small samples. Median[q 0 |1] from Asimov data set; good agreement with MC. 37 Monte Carlo test of asymptotic **formulae** G. Cowan **Statistics** for HEP / LAL Orsay, 3-5 January 2012 / Lecture 2 Consider again n ~ Poisson ( s + b), m ~/

3 To include the cases for the teaching sample, we enter the selection criteria: "split = 1". After completing the **formula**, click on the Continue button to close the dialog box. Selecting the teaching sample - 4 To activate the selection, / to full model - 1 In the cross-validation analysis, the relationship between the independent variables **and** the dependent variable was **statistically** significant. The **probability** for the model chi-square (17.487) testing overall relationship was = 0.008. The significance/

**statistics** **and** p-values using DISPLAY NORMPROB **and** DISPLAY INVNORM note differences between these results **and** Page 668! display invnorm(.025) -1.959964 display invnorm(.975) 1.959964 * to verify that +/-1.96 are the z-scores you want display normprob(-1.96).0249979 display normprob(1.96).9750021 18 Notes about working with the normal curve The table for deriving **probabilities**/ error of a sample shrinks as n increases Recall the **formula** for a variance of a **probability** distribution: σ 2 = Σ((y – μ) 2 * /

3 To include the cases for the teaching sample, we enter the selection criteria: "split = 1". After completing the **formula**, click on the Continue button to close the dialog box. Selecting the teaching sample - 4 To activate the selection, / to full model - 1 In the cross-validation analysis, the relationship between the independent variables **and** the dependent variable was **statistically** significant. The **probability** for the model chi-square (17.487) testing overall relationship was = 0.008. The significance/

X to the standardized normal (the “Z” distribution) by subtracting the mean of X **and** dividing by its standard deviation: The Z distribution always has mean = 0 **and** standard deviation = 1 Basic Business **Statistics**, 11e © 2009 Prentice-Hall, Inc.. The Standardized Normal **Probability** Density Function The **formula** for the standardized normal **probability** density function is Where e = the mathematical constant approximated by 2.71828 π = the mathematical/

, select the Compute… command from the Transform menu. **Formula** for **probability** for Mahalanobis D² First, in the target variable text box, type the name "p_mah_1" as an acronym for the **probability** of the mah_1, the Mahalanobis D² score. Second,/variables at least 5 to 1? No Inappropriate application of a **statistic** Yes Run regression again using transformed variables **and** eliminating outliers Impact of assumptions **and** outliers - 4 Yes **Probability** of ANOVA test of regression less than/equal to level of /

Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-72 Chapter Summary Presented key continuous distributions normal, uniform, exponential Found **probabilities** using **formulas** **and** tables Recognized when to apply different distributions Applied distributions to decision problems **Statistics** for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-73 Chapter Summary Introduced sampling distributions Described the sampling distribution of the mean/

cover the true value of with **probability** ≥ 1 . Equivalent to confidence belt construction; confidence belt is acceptance region of a test. **Statistical** Data Analysis / Stat 4 6 Relation between confidence interval **and** p-value Equivalently we can consider a significance/use where cf. Cowan, Cranmer, Gross, Vitells, arXiv:1007.1727, EPJC 71 (2011) 1554. 63 Monte Carlo test of asymptotic **formulae** G. Cowan **Statistical** Data Analysis / Stat 4 Consider again n ~ Poisson ( s + b), m ~ Poisson( b) Use q /

. (The sampling distribution of the mean is typically represented as a **probability** distribution in the format of a table, **probability** histogram, or **formula**.) Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Definition The value of a **statistic**, such as the sample mean x, depends on the particular values included in the sample, **and** generally varies from sample to sample. This variability of a/

**probability**, assuming μ, to find data at least as “extreme” as the data observed. The critical region of a test of size α can be defined from the set of data outcomes with p μ < α. Often use, e.g., α = 0.05. If observe x ∈ w μ, reject μ. G. Cowan Shandong seminar / 1 September 2014 4 Test **statistics** **and**/ 71 (2011) 1554 G. Cowan Shandong seminar / 1 September 2014 14 Monte Carlo test of asymptotic **formula** Here take = 1. Asymptotic **formula** is good approximation to 5 level (q 0 = 25) already for b ~ 20. G./

with Fractions Decimals **and** Percentages **Statistical** Measures Summer Examinations **and** Revision Summer Examinations **and** Revision Year 11 AQA GCSE Mathematics (4365) Route Map – Foundation Tier Year 11 **Probability** 1 Representing Data Volume Fractions **and** Decimals Holiday Inequalities Enlargements Trial **and** Improvement Mock Examinations **and** Revision Mock Examinations **and** Revision Holiday Holiday Percentages Ratios Scatter Graphs Maps **and** Scale Drawings Algebra Recap Holiday **Formulae** Constructions Loci Quadratic/

Prentice-Hall, Inc. Chap 6-56 Chapter Summary Presented key continuous distributions normal, uniform, exponential Found **probabilities** using **formulas** **and** tables Recognized when to apply different distributions Applied distributions to decision problems Basic Business **Statistics**, 10e © 2006 Prentice-Hall, Inc. Chap 6-57 Mean **and** Standard Deviation of the Uniform Distribution Basic Business **Statistics**, 10e © 2006 Prentice-Hall, Inc. Chap 6-58 指數分數 ─ 林惠玲、陳正倉「應用統計學」 p.226 Basic Business/

(ii) the queuing system is described by an ergodic process (time averages are equal to the corresponding **statistical** averages). The Little **formula** relates T **and** N quantities for a queue ( denotes the ‘mean rate of requests accepted into the system’): J. C./ the same derivations made in the M/M/S case, we can obtain the following state **probability** distribution: © 2013 Queuing Theory **and** Telecommunications: Networks **and** Applications – All rights reserved The M/M/S/S Queue (cont’d) Since the mean/

not reject H0 Type II Error (β) Review Question 1 If we have a p-value of 0.03 **and** so decide that our effect is **statistically** significant, what is the **probability** that we’re wrong (i.e., that the hypothesis test gave us a false positive)? .03 .06/power to detect a reduction of 10 points or more in the treatment group relative to placebo. What is 10 in your sample size **formula**? a. Standard deviation b. mean change c. Effect size d. Standard error e. Significance level Homework Problem Set 3 Reading: continue/

. To develop the notion of independent events. To develop Bayes’s **formula**. Chapter 8: Introduction to **Probability** **and** **Statistics** Chapter Objectives 2007 Pearson Education Asia Basic Counting Principle **and** Permutations Combinations **and** Other Counting Principles Sample Spaces **and** Events **Probability** Conditional **Probability** **and** Stochastic Processes Independent Events Bayes’ **Formula** 8.1) 8.2) 8.3) 8.4) Chapter 8: Introduction to **Probability** **and** **Statistics** Chapter Outline 8.5) 8.6) 8.7) 2007 Pearson/

1, 2, 3,.... Like binomial random variables, it is important to be able to distinguish situations in which the geometric distribution does **and** doesn’t apply! The Practice of **Statistics**, 5 th Edition30 The Practice of **Statistics**, 5 th Edition31 Geometric **Probability** **Formula** The Lucky Day Game. The random variable of interest in this game is Y = the number of guesses it takes to correctly/

it is used to draw conclusions or inferences about characteristics of populations based on data from a sample. We use **statistics** to make inferences about parameters. Therefore, we can make an estimate, prediction, or decision about a population based on/two chess players played 12 games, what is the **probability** that Player A would win 7 games, Player B would win 2 games, **and** the remaining 3 games would be drawn?" The following **formula** gives the **probability** of obtaining a specific set of outcomes when there /

. Order (high to low) of precedence of operators in a structure **formula** is ‘ ’, ‘/’, ‘*’ **and** ‘+’. For example, the structure **formula** A * B / C is the same as A * ( B / C ). **Statistical** Modelling Chapter VI 23 Example VI.1 Calf diets (continued) The factors/levels for the factor can be described using a **probability** distribution function. Definition VI.11: A factor will be designated as fixed if it is anticipated that a **probability** distribution function will not provide a satisfactory description the/

Wallis test A paired sample t test Wilcoxin rank sum test Friedman test Correlation Regression F- test 2 **Statistics** in Business research **Statistics** is an area that most management **and** marketing students find difficult. The **formulae** are often complicated, the calculations tedious, degrees of freedom mysterious, **and** **probability** tables confusing. But in fact students need no longer grapple with any of these. In real life, business/

, the mean or average quiz score is determined by summing all the scores **and** dividing by the number of students taking the exam. Descriptive **Statistics** The Mean or average is **probably** the most commonly used method of describing central tendency. For example, consider the/ The bottom part is a measure of the variability or dispersion of the scores. T-Test **Statistical** Analysis of the t-test This **formula** is essentially another example of the signal-to- noise metaphor in research: the difference between the/

**and** National Insurance Taxation: Income tax **and** National Insurance Graphical representation Graphical representation **STATISTICAL** TECHNIQUES 2 **STATISTICAL** TECHNIQUES 2 Correlation **and** regression Correlation **and** regression **STATISTICAL** TECHNIQUES 3 **STATISTICAL** TECHNIQUES 3 **Probabilities** **and** estimation **Probabilities** **and** estimation CRITICAL PATH **AND** RISK ANALYSIS 3 CRITICAL PATH **AND**/ Nuffield Foundation Savings Facts **and** **Formulae** - Nuffield Foundation Savings Facts **and** **Formulae** - Nuffield Foundation Student /

than that of the Ho , Type I error **and** p-value?? is the **probability** of a Type I error (e.g., =.05) a Type I error refers to when we mistakenly reject Ho. p-value is the **probability** of obtaining the sample **statistic** actually obtained, if Ho is true Testing the **statistical** significance of correlation coefficients definition **formula**: r - t = s r ; s r = √ (1-r/

**and** Methodology for the Health Sciences86 Chapter 3 **Probability** The Basis of the **Statistical** inference Text Book : Basic Concepts **and** Methodology for the Health Sciences 88 Key words: Key words: **Probability**, objective **Probability**, **Probability**, objective **Probability**, subjective **Probability**, equally likely Mutually exclusive, multiplicative rule Conditional **Probability**, independent events, Bayes theorem Text Book : Basic Concepts **and**/ by the following **formula** Text Book : Basic Concepts **and** Methodology for the /

**and** mass at unit temperature **and** density 0.001) **STATISTICAL** AVERAGE: **PROBABILITY** MEASURE Ergodicity (Boltzmann 1871, 1884): time average = phase-space average stationary **probability** density representing the invariant **probability** measure Spectrum of unitary time evolution: Ergodicity: The stationary **probability**/ & THEIR HELFAND MOMENT Transport coefficients: Green-Kubo **formula**: microscopic current: Einstein **formula**: Helfand moment: Transport property: moment: self-diffusion: shear viscosity: bulk /

. Prior: Posterior: Prior: Posterior: SECTION 2.5 Binomial Distribution Model **Probability** Model: Mathematical equation or **formula** used to generate **probabilities** based on certain assumptions about the process. Very important for **statistical** inference. Binomial Model: Two possible outcomes – often labeled as “success”/holds for samples that: Minimum[np, n(1-p)] > 5, where n is the sample size **and** p is **probability** of the outcome in the population. Central Limit Theorem (Practice) CharacteristicNµσµXµX σXσX Age /

G. Cowan Aachen 2014 / **Statistics** for Particle Physics, Lecture 42 Outline 1 **Probability** Definition, Bayes’ theorem, **probability** densities **and** their properties, catalogue of pdfs, Monte Carlo 2 **Statistical** tests general concepts, test **statistics**, multivariate methods, goodness-of/Cowan, Cranmer, Gross, Vitells, arXiv:1007.1727, EPJC 71 (2011) 1554. 51 Monte Carlo test of asymptotic **formulae** G. Cowan Aachen 2014 / **Statistics** for Particle Physics, Lecture 4 Consider again n ~ Poisson ( s + b), m ~ Poisson( /

(2014) G. Cowan Interdisciplinary Cluster **Statistics** Workshop, Munich, 17,18 Feb 2014 32 Bayesian vs. Frequentist Approaches Frequentist: **Probability** only assigned to (repeatable) data outcomes. Results depend on **probability** for data found **and** not found, e.g., p-value/1007.1727, EPJC 71 (2011) 1554 40 Monte Carlo test of asymptotic **formula** G. Cowan Interdisciplinary Cluster **Statistics** Workshop, Munich, 17,18 Feb 2014 Here take = 1. Asymptotic **formula** is good approximation to 5 level (q 0 = 25) already/

Hypergeometric **Formula** Since the hypergeometric **formula** **and** tables are tedious **and** impractical, use Excel’s hypergeometric function to find **probabilities**. Hypergeometric Distribution Using Software: Excel Figure 6.27 6-97 6-98 Module 4, the **probabilities** are given below the graph. Copy **and** paste graph as a bitmap; copy **and** paste **probabilities** into Excel. “Spin” N **and** n; overlay a normal or binomial curve. Hypergeometric Distribution Using Software: Visual **Statistics** Figure/

€35,000 - €40,000 Some Mathematical Jargon: The **formula** for the normal distribution is formally called the normal **probability** density function (pdf) The Shaded portion of the Histogram is the Proportion of interest Can visualise this using the histogram of salaries. Since the histogram of salaries is symmetric **and** bell shaped, we model this in **statistics** with a Normal distribution curve. Proportion = the proportion/

© 1998, Triola, Elementary **Statistics** Addison Wesley Longman 7 Figure 5-24 Solving Binomial **Probability** Problems Using a Normal Approximation Can the problem be easily solved with the binomial **probability** **formula** ? Use binomial **probability** **formula** Yes P( x ) =/ p x q (n – x ) !x! n!n! 7 6 5 4 Are np 5 **and** nq 5 both true ? No Yes Compute µ = np **and** = npq Draw the normal curve, **and** identify the region representing the **probability**/

Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-81 Chapter Summary Presented key continuous distributions normal, uniform, exponential Found **probabilities** using **formulas** **and** tables Recognized when to apply different distributions Applied distributions to decision problems **Statistics** for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-82 Chapter Summary Introduced sampling distributions Described the sampling distribution of the mean/

of the mean is typically represented as a **probability** distribution in the format of a table, **probability** histogram, or **formula**.) Definition The value of a **statistic**, such as the sample mean x, depends on the particular values included in the sample, **and** generally varies from sample to sample. This variability of a **statistic** is called sampling variability. Estimators Some **statistics** work much better than others as estimators/

To include the cases for the teaching sample, we enter the selection criteria: "split = 1". After completing the **formula**, click on the Continue button to close the dialog box. SW388R7 Data Analysis & Computers II Slide 74 Selecting the/model to full model - 1 In the cross-validation analysis, the relationship between the independent variables **and** the dependent variable was **statistically** significant. The **probability** for the model chi- square (25.513) testing overall relationship was = 0.003. The significance/

is equal to 0.7? Last update: 5-Oct-2015CSCI3220 Algorithms for Bioinformatics | Kevin Yip-cse-cuhk | Fall 201511 **Statistical** estimation Questions we can ask (cont’d): – Maximum likelihood estimation: Given a model with unknown parameter values, what / ln a > ln b) – This value can be found by differentiating the log likelihood **and** equating it to zero: – The **formula** for estimating the prior **probabilities** Pr(Y) can be similarly derived Last update: 5-Oct-2015CSCI3220 Algorithms for Bioinformatics | Kevin /

substantially more complex. See Wong **and** Lee p. 151 compared to p. 155 Gore/Bush 2000 by State Is there evidence of clustering? Join Count **Statistic** for Gore/Bush 2000 by State See spatstat.xls (JC-%vote tab) for data (assumes free or normality sampling) – The JC-%state tab uses % of states won, calculated using the same **formulae** – **Probably** not legitimate: need to use/

are unaffected by previous outcomes, **and** independent events are easier to analyze **and** result in simpler calculations **and** **formulas**. 6.1 - 73 Copyright © 2010, 2007, 2004 Pearson Education, Inc. Caution Many methods of **statistics** require a simple random sample. /. Each trial must have all outcomes classified into two categories (commonly, success **and** failure). 4.The **probability** of success remains the same in all trials. Solve by binomial **probability** **formula**, Table A-1, or technology. 6.1 - 95 Copyright © 2010,/

**Statistics** comprises those methods used to organize **and** describe information that has been collected.Descriptive **Statistics** comprises those methods used to organize **and** describe information that has been collected. Inferential **Statistics** involves the theory of **probability** **and** comprises those methods **and** techniques/= SS/ Σf The variance for a grouped data of a sample, denoted by s 2, is defined by the following **formula**: s 2 = SS/ (Σf -1) s 2 = SS/ (Σf -1) Standard Deviation The standard deviation is /

energy, that is: 3.1.7 Kinds of **statistical** system MaxwellMaxwell-Boltzmann **statistics** usually called Boltzmann statisticsBoltzmann Bose-Einstein statisticsEinstein FermiFermi-Dirac statisticsDirac 3.2 Boltzmann **Statistics** Microcosmic state number of localized system Most **probable** distribution of localized system Degeneration Degeneration **and** Microcosmic state number Most **probable** distribution of non-localized system The other form of Boltzmann **formula** Entropy in Helmholz free energy expression 3.2.1/

that the flight time is uniformly distributed between 4 hours **and** 5 hours. a)What is the **probability** that the flight will be no more than 10 minutes late? b)What is the **probability** that the flight will be no more than 30 /100) = 1 – e - x = 0.8647 Or you can use Excel **formula**, =EXPONDIST(100,1/50,1) P (X > 100) = 1 – 0.8647 = 0.1353 Exponential **Probability** Distribution - Example 25 STAT 500 – **Statistics** for Managers Agenda for this Session# 3 Part 3 Continuous Random Variables Uniform Distribution /

“calculation **formulae**” Classic Descriptive **Statistics**: Bivariate Calculation **Formulae** for Pearson Product Moment Correlation Coefficient (r) As we explore spatial **statistics**, we will see many analogies to the mean, the variance, **and** the correlation coefficient, **and** their various **formulae** There is/the assumption made regarding the type of sampling involved: –Free (or normality) sampling assumes that the **probability** of a polygon having a particular value is not affected by the number or arrangement of the /

**STATISTICAL** TECHNIQUES 2 **STATISTICAL** TECHNIQUES 2 Correlation **and** regression Correlation **and** regression **STATISTICAL** TECHNIQUES 3 **STATISTICAL** TECHNIQUES 3 **Probabilities** **and** estimation **Probabilities** **and** estimation **AND** OR **AND** Repayments **and** credit Repayments **and**/ Foundation Savings Growth - Nuffield Foundation Savings Growth - Nuffield Foundation Savings Facts **and** **Formulae** - Nuffield Foundation Savings Facts **and** **Formulae** - Nuffield Foundation Student Loans 1 – MEI (Username: mei-imps Password: imps/

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