# Confidence Intervals in Public Health

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Confidence Intervals in Public Health When public health practitioners use health statistics, sometimes they are interested in the actual number of health events, but more often they use the statistics to assess the true underlying risk of a health problem in the community. Observed health statistics, that is, those counts, rates or percentages that are computed or estimated from health surveys, vital statistics registries, or other health surveillance systems, are not always an accurate reflection of the true underlying risk in the population. Observed rates can vary from sample to sample or year to year, even when the true underlying risk remains the same. Statistics based on samples of a population are subject to sampling error. Sampling error refers to random variation that occurs because only a subset of the entire population is sampled and used to estimate a finding for the entire population. It is often mis-termed "margin of error" in popular use. Even health events that are based on a complete count of an entire population, such as deaths, are subject to random variation because the number of events that occurred may be considered as one of a large series of possible results that could have arisen under the same circumstances. In general, sampling error or random variation gets larger when the sample, population or number of events is small. Statistical sampling theory is used to compute a confidence interval to provide an estimate of the potential discrepancy between the true population parameters and observed rates. Understanding the potential size of that discrepancy can provide information about how to interpret the observed statistic. For instance, if the state infant death rate of 5.94 increased to 6.03 in a oneyear period, is that increase something that should cause concern? If the smoking rate among teens decreased from 13% to 8%, is that cause for celebration? Technically speaking, the 95% confidence interval indicates the range of values within which the statistic would fall 95% of the time if the researcher were to calculate the statistic (e.g., a percentage or rate) from an infinite number of samples of the same size, drawn from the same population. In less technical language, the confidence interval is a range of values within which the true value of the rate is expected to occur (with 95% probability). This document describes the most common methods for calculation of 95% confidence intervals for some rates and estimates commonly used in public health. 95% Confidence Interval for a Percentage From a Survey Sample: To calculate a confidence interval for a percentage from a survey sample, one must first calculate the standard error of the percentage. A percentage is also known as the mean of a binomial distribution. The standard error of the mean is a measure of dispersion for the hypothetical distribution of means called the sampling distribution of the mean. This is a distribution of means calculated from an infinite number of samples of the same size drawn from the same population as the original sample. Once you have calculated the standard error of the percentage, you must decide how large you want the confidence interval to be. The most common alternative is a 95% confidence interval. This is the width of the interval that includes the mean (the sampling distribution of the mean, 1

2 mentioned above) 95% of the time. In a little plainer language, a 95% confidence interval for a percentage is the range of values within which the percentage will be found at least 95% of the time if you went back and got a different sample of the same size from the same population. Transforming the standard error into a 95% confidence interval is rather simple. Fortunately, the sampling distribution of the mean has a shape that is almost identical to what is known as the normal distribution. 1 You need only multiply the standard error by the Z-score of the points in the normal distribution that exclude 2.5% of the distribution on either end (two-tailed). That Z-score is A Z-score of 1.96 defines the 95% confidence interval. A Z-score of 1.65 defines a 90% confidence interval. For a simple random sample, the standard error = pq/n where: p is the rate, q is 1 minus the rate, and n is the sample size. Example: 13% of surveyed respondents indicated that they smoked cigarettes. The sample consisted of 500 persons in a simple random sample. The standard error = (0.13*0.87) /500 = A distribution is a tool that is used in statistics to associate a statistic (e.g., a percentage, average, or other statistic) with its probability. When researchers talk about a measure being "statistically significant," they have used a distribution to evaluate the probability of the statistic, and found that it would be improbable under ordinary conditions. In most cases, we can rely on measures such as rates, averages, and proportions as having an underlying normal distribution, at least when the sample size is large enough. 2

3 Then the 95% confidence interval is: * standard error = *.015 = = So the 95% confidence interval has a lower limit of 10.1% and an upper limit of 15.9% The formula used above applies to a binomial distribution, which is the distribution of two complimentary values (e.g., heads and tails, for and against). If you are calculating a confidence interval for a different statistic, such as an average, you ll need to modify the equation. The quantity (pq) is the variance of a binomial distribution. If your measure is not a proportion, but, say, an average, you must modify the formula, substituting the pq quantity with the variance. The standard error can also be calculated as the standard deviation divided by the square root of the sample size: σ n Small Samples If the sample from which the percentage was calculated was rather small (according to central limit theorem we can define small as 29 or fewer) then the shape of the sampling distribution of the mean is not the same as the shape of the normal distribution. In this special case, we can use another distribution, known as the t distribution, that has a slightly different shape than the normal distribution. 2 The procedures in this case are analogous to those above but the t-score comes from a family of distributions that depend on the degrees of freedom. The number of degrees of freedom is defined as n-1 where n is the size of the sample. For a sample of size=30 the degrees of freedom is equal to 29. So, for a 95% confidence interval, you must use the t-score associated with 29 degrees of freedom. That particular t-score is (see Appendix 1.). So you would multiply the standard error by instead of 1.96 to generate the 95% confidence interval. If our sample were a different size, say 20, then the degrees of freedom would be 19, which is associated with a t-score of for a 95% confidence interval. As you see the interval will get wider as our sample size is reduced. This reflects the uncertainty in our estimate of the variance in the population. For a 95% confidence interval with 9 degrees of freedom the t-score is Table 1. lists the t-scores for specific degrees of freedom and sizes of confidence interval. For a 95% confidence interval, you would use the t-score that defines the points on the distribution that excludes the most extreme 5% of the distribution, which is on either end of the curve. 2 Student's t-distribution, downloaded on 2/13/09 from 3

4 Student s t Distribution at Varying Degrees of Freedom (k) Appendix 1 lists the t-scores for specific degrees of freedom and sizes of confidence interval. For a 95% confidence interval, you would use the t-score that defines the points on the distribution that excludes the most extreme 5% of the distribution, which is on either end of the distribution. Finite Populations If the survey sampled all or most of the members of the population, then using the finite population correction factor will improve (decrease the width of) the confidence interval. The finite population correction factor = 1-f, where f is the sampling fraction f = n/n where n is the size of the sample and N is the size of the population. The sampling fraction is simply the proportion of the population that was included in the sample. The standard error of the mean for a binomial distribution for a finite sample s.e. percentage = pq/n * (1-f) When the Percentage is Close to 0% or 100% When the percentage is close to 0% or 100%, the formulas given above can result in illogical results - confidence limits that fall below 0% or above 100%. A special formula is used to calculate asymmetric confidence limits in these cases. Because survey estimates can be small percentages, the confidence intervals for the surveys on IBIS-Q are based on logit transformations. Logit transformations yield asymmetric interval boundaries that are more balanced with respect to 4

5 the probability that the true value falls below or above the interval boundaries than is the case for standard symmetric confidence intervals for small proportions. The method used is as follows: (1) Perform a logit transformation of the original percentage estimate: f = log(p)-log(1-p) where: p = the percentage estimate f = the logit transformation of the percentage (2) Transform the standard error of the percentage to a standard error of the it s logit transformation: se(f) = se(p)/(p*(1-p)) where: se = standard error (3) Calculate the lower and upper confidence bounds of the logit transformation of the percentage: Lf = f - t(alpha/2, df)*se Uf = f + t(alpha/2, df)*se where: Lf = lower confidence bound of f Uf = upper confidence bound of f t(alpha/2, df) = the value of the t-score corresponding to the desired alpha level (0.05 for a 95% Confidence Interval) and the degrees of freedom (degrees of freedom is defined as n- 1 where n is the sample size). (4) Finally, perform inverse logit transformations to get the confidence bounds of p: Lp = exp(lf)/(1+exp(lf)) Up = exp(uf)/(1+exp(uf)) where: Lp = lower confidence bound of p Up = upper confidence bound of p Complex Sample Designs The above formulas assume that the survey sample was a simple random sample. If the survey used a complex sample design (such as clustering within households or disproportionate sampling from various geographic regions), special techniques must be used to calculate the standard error of the mean. Those techniques are accomplished using statistical software such as SAS, STATA, or SUDAAN. When the Rate is Equal to 0 When the percentage or rate is equal to zero, using the above calculation will yield a confidence interval of zero, which is incorrect. A simple method you can use to estimate the 5

6 confidence interval when the percentage or rate is zero is to assume the number of cases in the numerator of your rate is 3, then calculate the confidence interval using the population size in your original calculation. 3 95% Confidence Intervals for Rare Events: In the case of rare events distributed randomly across time, the normal distribution no longer applies. A different distribution, the Poisson distribution is used to model rare events that occur across time, such as the "100 year flood." 4 It is used to calculate confidence intervals for rare health events, such as infant mortality or cancer. This distribution is not symmetric about its mean and so the associated confidence intervals will not be symmetric (the upper limit is farther from the estimate than is the lower limit). The Poisson distribution does, however, assume the shape of a normal distribution when there are 20 or more events in the numerator. So we use a Poisson distribution for rare events (when the number of events is less than 20), but we can use the normal distribution when the number of events is 20 or more. Poisson Distribution In Appendix 2 you will find lower and upper confidence factors for use in calculating a 95% confidence interval for a rate based on a specified number of events, from 1 to 20. To calculate the confidence interval, multiply the estimated rate by the confidence factor associated with the number of events on which the rate is based. 3 Lilienfeld, DE and Stolley, PD Foundations of Epidemiology (3rd Ed.). Oxford University Press, In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a number of events occurring in a fixed period of time if these events occur with a known average rate and independently of the time since the last event. Poisson Distribution, downloaded on 2/13/09 from 6

7 For example, in a given geographic area, there were 722 births in a single year, and seven infant deaths. The infant mortality rate in was 9.7 per 1,000 live births, calculated as [(7/722)*1,000]. The lower and upper confidence limits are calculated using the confidence factors found on Appendix 2. The factors for seven events are.4021 and for the lower and upper limits of the confidence interval, respectively. The lower limit of the confidence interval = 9.7*.4021 = 3.90, and the upper limit = 9.7* = 19.99, for a rate of 9.7 and a 95% confidence interval from 3.90 to If this same rate had been based on 100 deaths then the confidence factors would be.8136 and The lower limit would be 9.7*.8136, and the upper limit 9.7* for an estimate of 9.7 with a confidence interval from 7.89 to This interval is much smaller due to the greater number of deaths on which the rate is based. In the Utah IBIS-PH query system, starting in March 2010, the confidence factors are obtained by using SAS software that requires specification of the percentage of the inverse gamma distribution to be excluded on either end of the distribution (2.5% for a 95% confidence interval), and the two parameters associated with the distribution function: the mean and the variance. In the case of the crude rate, where the variance and mean are equal, this is the special case of the gamma family of distributions known as the Poisson distribution. Directly Age-Adjusted Rates When comparing across geographic areas, some method of age adjusting is typically used to control for area-to-area differences in health events that can be explained by differing ages of the area populations. For example, an area that has an older population will have higher crude (not ageadjusted) rates for cancer, even though its exposure levels and cancer rates for specific age groups are the same as those of other areas. One might incorrectly attribute the high cancer rates to some characteristic of the area other than age. Age-adjusted rates control for age effects, allowing better comparability of rates across areas. Direct standardization adjusts the age-specific rates observed in the small area to the age distribution of a standard population (Lilienfeld & Stolley, 1994) 5. The directly age-adjusted death rate is a weighted average of the age-specific death rates where the age-specific weights represent the relative age distribution of the standard population. Directly age-adjusted death rate (DAADR) = W si * D i / P i = W si * R i Where.. W si = the weight for the i th age group in the standard population (the proportion of the standard population in the i th age group) = P si / P si P si = the population in age group i in the standard population D i = number of deaths (or other event) in age group i of the study population P i = the population in age group i in the study population R i = the age-specific rate in the i th age group 5 Lilienfeld, DE and Stolley, PD (1994) 7

9 there are fewer than 20 (some say 25) cases in the index population, indirect standardization of rates should be used. Indirectly standardized rates are based on the Standardized Mortality Ratio (SMR) and the crude rate for a standard population. Indirect standardization adjusts the overall standard population rate to the age distribution of the small area (Lilienfeld & Stolley, 1994) 8. Strictly speaking, it is valid to compare indirectly standardized rates only with the rate in the standard population, not with each other. 9 An indirectly standardized death or disease rate (ISR) can be computed as: Where... ISR = SMR*Rs SMR = observed deaths/disease in the small area = D = D expected deaths/disease in the small area e (Rsi * ni) SMR = observed deaths in the small area/expected deaths in the small area D = observed number of deaths in the small area e = (Rsi * n i ) = expected number of deaths in small area R s = the crude death rate in the standard population R si = the age-specific death rate in age group i of the standard population ( # deaths / population count, before applying the constant) n i = the population count in age group i of the small area For indirectly standardized rates based on events that follow a Poisson distribution and for which the ratio of events to total population is small (<.3) and the sample size is large, the following two methods can be used to calculate confidence interval (Kahn & Sempos, 1989) 10. (1) When the number of events >20: CI ISR = (SMR/e) * R s * K 8 Lilienfeld & Stolley (1994) 9 Rothman, Kenneth J. and Greenland, Sander (1998) Modern Epidemiology (2nd Ed.). Philadelphia, PA: Lippincott. 10 Harold A. Kahn and Christopher T. Sempos (1989) Statistical Methods in Epidemiology. New York: Oxford University Press. 9

10 Where... SMR = observed deaths in the small area/expected deaths in the small area e = expected deaths in the small area = (R si * n i ) R s = the crude death rate in the standard population R si = the age-specific death rate in age group i of the standard population ( # deaths / population count) n i = the population count in age group i of the small area K = a constant (e.g., 100,000) that is being used to communicate the rate (2) When the number of events <=20: LL ISR = (Lower limit for parameter estimate from Poisson table/e)) * R s * K UL ISR = (Upper limit for parameter estimate from Poisson table/e)) * R s * K Where LL is the lower confidence interval limit, and UL is the upper confidence interval limit. Revision History 1/15/2002 Document created Brian Paoli, Lois M. Haggard, Gulzar Shah, Office of Public Health Assessment, Utah Department of Health 8/11/2009 Images of sampling distributions, error correction, when rate=zero. 12/02/2009 Asymmetric confidence intervals for percentages close to 0% or 100%. 4/23/2010 Paragraph on CI for nonbinomial distributions 4/30/2010 Added text in 95% Confidence Intervals for Rare Events and Directly Age-adjusted Rates re: using SAS procedure to calculate confidence intervals for rare events. Lois M. Haggard, Community Health Assessment Program, New Mexico Department of Health Michael Friedrichs, Chronic Disease Bureau, Utah Department of Health, Kathryn Marti, Brian Paoli, Office of Public Health Assessment, Utah Department of Health Lois M. Haggard, Community Health Assessment Program, New Mexico Department of Health Kathryn Marti, Brian Paoli, Office of Public Health Assessment, Utah Department of Health 10

11 Appendix 1. Upper critical values of Student's t distribution with K degrees of freedom Probability of exceeding the critical value K

12 Appendix 2. 95% Confidence Interval Factors for Poisson-Distributed Events number of events 95% Confidence Interval, Lower Limit Factor 95% Confidence Interval, Upper Limit Factor

### Sampling 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

### HYPOTHESIS 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

### Inferential Statistics

Inferential Statistics Sampling and the normal distribution Z-scores Confidence levels and intervals Hypothesis testing Commonly used statistical methods Inferential Statistics Descriptive statistics are

### Life Table Analysis using Weighted Survey Data

Life Table Analysis using Weighted Survey Data James G. Booth and Thomas A. Hirschl June 2005 Abstract Formulas for constructing valid pointwise confidence bands for survival distributions, estimated using

### 4. 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

### 7 Hypothesis testing - one sample tests

7 Hypothesis testing - one sample tests 7.1 Introduction Definition 7.1 A hypothesis is a statement about a population parameter. Example A hypothesis might be that the mean age of students taking MAS113X

### Advanced Statistical Analysis of Mortality. Rhodes, Thomas E. and Freitas, Stephen A. MIB, Inc. 160 University Avenue. Westwood, MA 02090

Advanced Statistical Analysis of Mortality Rhodes, Thomas E. and Freitas, Stephen A. MIB, Inc 160 University Avenue Westwood, MA 02090 001-(781)-751-6356 fax 001-(781)-329-3379 trhodes@mib.com Abstract

### Simple Random Sampling

Source: Frerichs, R.R. Rapid Surveys (unpublished), 2008. NOT FOR COMMERCIAL DISTRIBUTION 3 Simple Random Sampling 3.1 INTRODUCTION Everyone mentions simple random sampling, but few use this method for

### 5.1 Identifying the Target Parameter

University of California, Davis Department of Statistics Summer Session II Statistics 13 August 20, 2012 Date of latest update: August 20 Lecture 5: Estimation with Confidence intervals 5.1 Identifying

### MINITAB ASSISTANT WHITE PAPER

MINITAB ASSISTANT WHITE PAPER This paper explains the research conducted by Minitab statisticians to develop the methods and data checks used in the Assistant in Minitab 17 Statistical Software. One-Way

### Technical Briefing 3. Commonly used public health statistics and their confidence intervals. Purpose. Contents

Technical Briefing 3 March 2008 Commonly used public health statistics and their confidence intervals Purpose This is the third in a series of technical briefings produced by the Association of Public

### Basic 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

### Errata and updates for ASM Exam C/Exam 4 Manual (Sixteenth Edition) sorted by page

Errata for ASM Exam C/4 Study Manual (Sixteenth Edition) Sorted by Page 1 Errata and updates for ASM Exam C/Exam 4 Manual (Sixteenth Edition) sorted by page Practice exam 1:9, 1:22, 1:29, 9:5, and 10:8

### Confidence Intervals for the Difference Between Two Means

Chapter 47 Confidence Intervals for the Difference Between Two Means Introduction This procedure calculates the sample size necessary to achieve a specified distance from the difference in sample means

### 4. Introduction to Statistics

Statistics for Engineers 4-1 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

### Chi 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

### Week 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.

### Statistical Rules of Thumb

Statistical Rules of Thumb Second Edition Gerald van Belle University of Washington Department of Biostatistics and Department of Environmental and Occupational Health Sciences Seattle, WA WILEY AJOHN

### Summary 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

### SAS Software to Fit the Generalized Linear Model

SAS Software to Fit the Generalized Linear Model Gordon Johnston, SAS Institute Inc., Cary, NC Abstract In recent years, the class of generalized linear models has gained popularity as a statistical modeling

### Chapter 2: Systems of Linear Equations and Matrices:

At the end of the lesson, you should be able to: Chapter 2: Systems of Linear Equations and Matrices: 2.1: Solutions of Linear Systems by the Echelon Method Define linear systems, unique solution, inconsistent,

### The SURVEYFREQ Procedure in SAS 9.2: Avoiding FREQuent Mistakes When Analyzing Survey Data ABSTRACT INTRODUCTION SURVEY DESIGN 101 WHY STRATIFY?

The SURVEYFREQ Procedure in SAS 9.2: Avoiding FREQuent Mistakes When Analyzing Survey Data Kathryn Martin, Maternal, Child and Adolescent Health Program, California Department of Public Health, ABSTRACT

### Business 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, McGraw-Hill/Irwin, 2008, ISBN: 978-0-07-331988-9. Required Computing

### 13. Poisson Regression Analysis

136 Poisson Regression Analysis 13. Poisson Regression Analysis We have so far considered situations where the outcome variable is numeric and Normally distributed, or binary. In clinical work one often

### 1) 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

### Need for Sampling. Very large populations Destructive testing Continuous production process

Chapter 4 Sampling and Estimation Need for Sampling Very large populations Destructive testing Continuous production process The objective of sampling is to draw a valid inference about a population. 4-

### HYPOTHESIS 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

### Module 5 Hypotheses Tests: Comparing Two Groups

Module 5 Hypotheses Tests: Comparing Two Groups Objective: In medical research, we often compare the outcomes between two groups of patients, namely exposed and unexposed groups. At the completion of this

### Confidence Intervals for Cp

Chapter 296 Confidence Intervals for Cp Introduction This routine calculates the sample size needed to obtain a specified width of a Cp confidence interval at a stated confidence level. Cp is a process

### MCQ S OF MEASURES OF CENTRAL TENDENCY

MCQ S OF MEASURES OF CENTRAL TENDENCY MCQ No 3.1 Any measure indicating the centre of a set of data, arranged in an increasing or decreasing order of magnitude, is called a measure of: (a) Skewness (b)

### Biostatistics: DESCRIPTIVE STATISTICS: 2, VARIABILITY

Biostatistics: DESCRIPTIVE STATISTICS: 2, VARIABILITY 1. Introduction Besides arriving at an appropriate expression of an average or consensus value for observations of a population, it is important to

: Table of Contents... 1 Overview of Model... 1 Dispersion... 2 Parameterization... 3 Sigma-Restricted Model... 3 Overparameterized Model... 4 Reference Coding... 4 Model Summary (Summary Tab)... 5 Summary

### A POPULATION MEAN, CONFIDENCE INTERVALS AND HYPOTHESIS TESTING

CHAPTER 5. A POPULATION MEAN, CONFIDENCE INTERVALS AND HYPOTHESIS TESTING 5.1 Concepts When a number of animals or plots are exposed to a certain treatment, we usually estimate the effect of the treatment

### Statistics 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 1.1-1.6) Objectives

### Regression Analysis: A Complete Example

Regression Analysis: A Complete Example This section works out an example that includes all the topics we have discussed so far in this chapter. A complete example of regression analysis. PhotoDisc, Inc./Getty

### Northumberland Knowledge

Northumberland Knowledge Know Guide How to Analyse Data - November 2012 - This page has been left blank 2 About this guide The Know Guides are a suite of documents that provide useful information about

### Statistical Functions in Excel

Statistical Functions in Excel There are many statistical functions in Excel. Moreover, there are other functions that are not specified as statistical functions that are helpful in some statistical analyses.

### Exercise Answers. Exercise 3.1 1. B 2. C 3. A 4. B 5. A

Exercise Answers Exercise 3.1 1. B 2. C 3. A 4. B 5. A Exercise 3.2 1. A; denominator is size of population at start of study, numerator is number of deaths among that population. 2. B; denominator is

### MBA 611 STATISTICS AND QUANTITATIVE METHODS

MBA 611 STATISTICS AND QUANTITATIVE METHODS Part I. Review of Basic Statistics (Chapters 1-11) A. Introduction (Chapter 1) Uncertainty: Decisions are often based on incomplete information from uncertain

### Study Guide for the Final Exam

Study Guide for the Final Exam When studying, remember that the computational portion of the exam will only involve new material (covered after the second midterm), that material from Exam 1 will make

### Statistical Confidence Calculations

Statistical Confidence Calculations Statistical Methodology Omniture Test&Target utilizes standard statistics to calculate confidence, confidence intervals, and lift for each campaign. The student s T

### Chapter 5 Discrete Probability Distribution. Learning objectives

Chapter 5 Discrete Probability Distribution Slide 1 Learning objectives 1. Understand random variables and probability distributions. 1.1. Distinguish discrete and continuous random variables. 2. Able

### Normality Testing in Excel

Normality Testing in Excel By Mark Harmon Copyright 2011 Mark Harmon No part of this publication may be reproduced or distributed without the express permission of the author. mark@excelmasterseries.com

### Normal distribution Approximating binomial distribution by normal 2.10 Central Limit Theorem

1.1.2 Normal distribution 1.1.3 Approimating binomial distribution by normal 2.1 Central Limit Theorem Prof. Tesler Math 283 October 22, 214 Prof. Tesler 1.1.2-3, 2.1 Normal distribution Math 283 / October

### ESTIMATION - CHAPTER 6 1

ESTIMATION - CHAPTER 6 1 PREVIOUSLY use known probability distributions (binomial, Poisson, normal) and know population parameters (mean, variance) to answer questions such as......given 20 births and

Lecture 4: Transformations Regression III: Advanced Methods William G. Jacoby Michigan State University Goals of the lecture The Ladder of Roots and Powers Changing the shape of distributions Transforming

### Chapter 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

### Statistics Review PSY379

Statistics Review PSY379 Basic concepts Measurement scales Populations vs. samples Continuous vs. discrete variable Independent vs. dependent variable Descriptive vs. inferential stats Common analyses

### Probability 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 Distributions-VI Today, I am going to introduce

### Review. March 21, 2011. 155S7.1 2_3 Estimating a Population Proportion. Chapter 7 Estimates and Sample Sizes. Test 2 (Chapters 4, 5, & 6) Results

MAT 155 Statistical Analysis Dr. Claude Moore Cape Fear Community College Chapter 7 Estimates and Sample Sizes 7 1 Review and Preview 7 2 Estimating a Population Proportion 7 3 Estimating a Population

### Chapter 8. Hypothesis Testing

Chapter 8 Hypothesis Testing Hypothesis In statistics, a hypothesis is a claim or statement about a property of a population. A hypothesis test (or test of significance) is a standard procedure for testing

### Chapter 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:

### Introduction to. Hypothesis Testing CHAPTER LEARNING OBJECTIVES. 1 Identify the four steps of hypothesis testing.

Introduction to Hypothesis Testing CHAPTER 8 LEARNING OBJECTIVES After reading this chapter, you should be able to: 1 Identify the four steps of hypothesis testing. 2 Define null hypothesis, alternative

### Course Text. Required Computing Software. Course Description. Course Objectives. StraighterLine. Business Statistics

Course Text Business Statistics Lind, Douglas A., Marchal, William A. and Samuel A. Wathen. Basic Statistics for Business and Economics, 7th edition, McGraw-Hill/Irwin, 2010, ISBN: 9780077384470 [This

### 1. What is the critical value for this 95% confidence interval? CV = z.025 = invnorm(0.025) = 1.96

1 Final Review 2 Review 2.1 CI 1-propZint Scenario 1 A TV manufacturer claims in its warranty brochure that in the past not more than 10 percent of its TV sets needed any repair during the first two years

### Calculating P-Values. Parkland College. Isela Guerra Parkland College. Recommended Citation

Parkland College A with Honors Projects Honors Program 2014 Calculating P-Values Isela Guerra Parkland College Recommended Citation Guerra, Isela, "Calculating P-Values" (2014). A with Honors Projects.

### Models for Count Data With Overdispersion

Models for Count Data With Overdispersion Germán Rodríguez November 6, 2013 Abstract This addendum to the WWS 509 notes covers extra-poisson variation and the negative binomial model, with brief appearances

### TImath.com. F Distributions. Statistics

F Distributions ID: 9780 Time required 30 minutes Activity Overview In this activity, students study the characteristics of the F distribution and discuss why the distribution is not symmetric (skewed

### Simple Regression Theory II 2010 Samuel L. Baker

SIMPLE REGRESSION THEORY II 1 Simple Regression Theory II 2010 Samuel L. Baker Assessing how good the regression equation is likely to be Assignment 1A gets into drawing inferences about how close the

### Lesson 1: Comparison of Population Means Part c: Comparison of Two- Means

Lesson : Comparison of Population Means Part c: Comparison of Two- Means Welcome to lesson c. This third lesson of lesson will discuss hypothesis testing for two independent means. Steps in Hypothesis

### Confidence Intervals for One Standard Deviation Using Standard Deviation

Chapter 640 Confidence Intervals for One Standard Deviation Using Standard Deviation Introduction This routine calculates the sample size necessary to achieve a specified interval width or distance from

### " Y. Notation and Equations for Regression Lecture 11/4. Notation:

Notation: Notation and Equations for Regression Lecture 11/4 m: The number of predictor variables in a regression Xi: One of multiple predictor variables. The subscript i represents any number from 1 through

### Lecture Notes Module 1

Lecture Notes Module 1 Study Populations A study population is a clearly defined collection of people, animals, plants, or objects. In psychological research, a study population usually consists of a specific

### Review of basic statistics and the simplest forecasting model: the sample mean

Review of basic statistics and the simplest forecasting model: the sample mean Robert Nau Fuqua School of Business, Duke University August 2014 Most of what you need to remember about basic statistics

### A Coefficient of Variation for Skewed and Heavy-Tailed Insurance Losses. Michael R. Powers[ 1 ] Temple University and Tsinghua University

A Coefficient of Variation for Skewed and Heavy-Tailed Insurance Losses Michael R. Powers[ ] Temple University and Tsinghua University Thomas Y. Powers Yale University [June 2009] Abstract We propose a

### Statistics 104: Section 6!

Page 1 Statistics 104: Section 6! TF: Deirdre (say: Dear-dra) Bloome Email: dbloome@fas.harvard.edu Section Times Thursday 2pm-3pm in SC 109, Thursday 5pm-6pm in SC 705 Office Hours: Thursday 6pm-7pm SC

### Technology Step-by-Step Using StatCrunch

Technology Step-by-Step 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

### Chapter 4 Lecture Notes

Chapter 4 Lecture Notes Random Variables October 27, 2015 1 Section 4.1 Random Variables A random variable is typically a real-valued function defined on the sample space of some experiment. For instance,

### Summary of Formulas and Concepts. Descriptive Statistics (Ch. 1-4)

Summary of Formulas and Concepts Descriptive Statistics (Ch. 1-4) Definitions Population: The complete set of numerical information on a particular quantity in which an investigator is interested. We assume

### THE 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

### Lecture 14. Chapter 7: Probability. Rule 1: Rule 2: Rule 3: Nancy Pfenning Stats 1000

Lecture 4 Nancy Pfenning Stats 000 Chapter 7: Probability Last time we established some basic definitions and rules of probability: Rule : P (A C ) = P (A). Rule 2: In general, the probability of one event

### CHAPTER 13. Experimental Design and Analysis of Variance

CHAPTER 13 Experimental Design and Analysis of Variance CONTENTS STATISTICS IN PRACTICE: BURKE MARKETING SERVICES, INC. 13.1 AN INTRODUCTION TO EXPERIMENTAL DESIGN AND ANALYSIS OF VARIANCE Data Collection

### ( ) ( ) Central Tendency. Central Tendency

1 Central Tendency CENTRAL TENDENCY: A statistical measure that identifies a single score that is most typical or representative of the entire group Usually, a value that reflects the middle of the distribution

### MAT 155. Key Concept. September 27, 2010. 155S5.5_3 Poisson Probability Distributions. Chapter 5 Probability Distributions

MAT 155 Dr. Claude Moore Cape Fear Community College Chapter 5 Probability Distributions 5 1 Review and Preview 5 2 Random Variables 5 3 Binomial Probability Distributions 5 4 Mean, Variance and Standard

### MATHEMATICAL METHODS OF STATISTICS

MATHEMATICAL METHODS OF STATISTICS By HARALD CRAMER TROFESSOK IN THE UNIVERSITY OF STOCKHOLM Princeton PRINCETON UNIVERSITY PRESS 1946 TABLE OF CONTENTS. First Part. MATHEMATICAL INTRODUCTION. CHAPTERS

### The American Cancer Society Cancer Prevention Study I: 12-Year Followup

Chapter 3 The American Cancer Society Cancer Prevention Study I: 12-Year Followup of 1 Million Men and Women David M. Burns, Thomas G. Shanks, Won Choi, Michael J. Thun, Clark W. Heath, Jr., and Lawrence

### Regression Analysis: Basic Concepts

The simple linear model Regression Analysis: Basic Concepts Allin Cottrell Represents the dependent variable, y i, as a linear function of one independent variable, x i, subject to a random disturbance

### Point and Interval Estimates

Point and Interval Estimates Suppose we want to estimate a parameter, such as p or µ, based on a finite sample of data. There are two main methods: 1. Point estimate: Summarize the sample by a single number

### Hypothesis Testing Level I Quantitative Methods. IFT Notes for the CFA exam

Hypothesis Testing 2014 Level I Quantitative Methods IFT Notes for the CFA exam Contents 1. Introduction... 3 2. Hypothesis Testing... 3 3. Hypothesis Tests Concerning the Mean... 10 4. Hypothesis Tests

### 2 ESTIMATION. Objectives. 2.0 Introduction

2 ESTIMATION Chapter 2 Estimation Objectives After studying this chapter you should be able to calculate confidence intervals for the mean of a normal distribution with unknown variance; be able to calculate

### DATA 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

### Appendices. 2006 Bexar County Community Health Assessment Appendices Appendix A 125

Appendices Appendix A Recent reports suggest that the number of mothers seeking dropped precipitously between 2004 and 2005. Tables 1A and 1B, below, shows information since 1990. The trend has been that

### Accurately and Efficiently Measuring Individual Account Credit Risk On Existing Portfolios

Accurately and Efficiently Measuring Individual Account Credit Risk On Existing Portfolios By: Michael Banasiak & By: Daniel Tantum, Ph.D. What Are Statistical Based Behavior Scoring Models And How Are

### Chi-square test Fisher s Exact test

Lesson 1 Chi-square test Fisher s Exact test McNemar s Test Lesson 1 Overview Lesson 11 covered two inference methods for categorical data from groups Confidence Intervals for the difference of two proportions

### Annex 6 BEST PRACTICE EXAMPLES FOCUSING ON SAMPLE SIZE AND RELIABILITY CALCULATIONS AND SAMPLING FOR VALIDATION/VERIFICATION. (Version 01.

Page 1 BEST PRACTICE EXAMPLES FOCUSING ON SAMPLE SIZE AND RELIABILITY CALCULATIONS AND SAMPLING FOR VALIDATION/VERIFICATION (Version 01.1) I. Introduction 1. The clean development mechanism (CDM) Executive

### Fairfield 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

### Algebra 1 Course Title

Algebra 1 Course Title Course- wide 1. What patterns and methods are being used? Course- wide 1. Students will be adept at solving and graphing linear and quadratic equations 2. Students will be adept

### Chapter Study Guide. Chapter 11 Confidence Intervals and Hypothesis Testing for Means

OPRE504 Chapter Study Guide Chapter 11 Confidence Intervals and Hypothesis Testing for Means I. Calculate Probability for A Sample Mean When Population σ Is Known 1. First of all, we need to find out the

### MULTIPLE LINEAR REGRESSION ANALYSIS USING MICROSOFT EXCEL. by Michael L. Orlov Chemistry Department, Oregon State University (1996)

MULTIPLE LINEAR REGRESSION ANALYSIS USING MICROSOFT EXCEL by Michael L. Orlov Chemistry Department, Oregon State University (1996) INTRODUCTION In modern science, regression analysis is a necessary part

### ANALYTIC AND REPORTING GUIDELINES

ANALYTIC AND REPORTING GUIDELINES The National Health and Nutrition Examination Survey (NHANES) Last Update: December, 2005 Last Correction, September, 2006 National Center for Health Statistics Centers

### Poisson Models for Count Data

Chapter 4 Poisson Models for Count Data In this chapter we study log-linear models for count data under the assumption of a Poisson error structure. These models have many applications, not only to the

### EMPIRICAL FREQUENCY DISTRIBUTION

INTRODUCTION TO MEDICAL STATISTICS: Mirjana Kujundžić Tiljak EMPIRICAL FREQUENCY DISTRIBUTION observed data DISTRIBUTION - described by mathematical models 2 1 when some empirical distribution approximates

### Z-table p-values: use choice 2: normalcdf(

P-values with the Ti83/Ti84 Note: The majority of the commands used in this handout can be found under the DISTR menu which you can access by pressing [ nd ] [VARS]. You should see the following: NOTE:

### An interval estimate (confidence interval) is an interval, or range of values, used to estimate a population parameter. For example 0.476<p<0.

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.

### Part 2: Analysis of Relationship Between Two Variables

Part 2: Analysis of Relationship Between Two Variables Linear Regression Linear correlation Significance Tests Multiple regression Linear Regression Y = a X + b Dependent Variable Independent Variable

### Chi Squared and Fisher's Exact Tests. Observed vs Expected Distributions

BMS 617 Statistical Techniques for the Biomedical Sciences Lecture 11: Chi-Squared and Fisher's Exact Tests Chi Squared and Fisher's Exact Tests This lecture presents two similarly structured tests, Chi-squared

### Normal distribution. ) 2 /2σ. 2π σ

Normal distribution The normal distribution is the most widely known and used of all distributions. Because the normal distribution approximates many natural phenomena so well, it has developed into a