# Sampling (cont d) and Confidence Intervals Lecture 9 8 March 2006 R. Ryznar

Save this PDF as:

Size: px
Start display at page:

Download "Sampling (cont d) and Confidence Intervals Lecture 9 8 March 2006 R. Ryznar"

## Transcription

1 Sampling (cont d) and Confidence Intervals Lecture 9 8 March 2006 R. Ryznar

2 Census Surveys Decennial Census Every (over 11 million) household gets the short form and 17% or 1/6 get the long form Miss approximately.12% of population overall (including about 2.78% of black population) Why do it? Current Population Survey 60,000 households interviewed every month The American Community Survey Contacts 3 million households (including some from every county) and will replace the long form in 2010

3 Gallup Polls Many do not believe a survey of respondents can represent the views of all Americans.

4 Estimation Parameter A number that describes the population. We don t know its value. Statistic A number that describes a sample. It can change from sample to sample. If we take lots of samples the statistic follows a predictable pattern.

5 Sampling Variability

6 Law of Large Numbers As the number of trials increases the average outcome approaches the mean of the population (i.e., the expected outcome) and the standard deviation of the average outcome approaches zero.

7 To reduce bias take a SRS. To reduce variability take larger samples. The margin of error is about sampling variability. We say, The president s approval rating is 40%, plus or minus 3 percentage points. We are 95% percent confident that the true population proportion is between 37% and 43%.

8 Central Limit Theorem The distribution of an average tends to be Normal, even when the distribution from which the average is computed is decidedly non-normal.

9 Quick method for a 95% confidence interval around a sample proportion is 1/ n

10 Margin of Error and Sample Size 1/ n = 1/ 1600 = 1/ 40 = 0.025or 3.0% 1/ n = 1/ 2527 = 1/ = 0.020or 2.0% 1/ n = 1/ 100 = 1/10 = 0.1or10% The size of the population has little influence on the behavior of statistics from random samples. The population size does not matter as long as it is at least 100 times larger than the sample.

11 Gallup Polls Many do not believe a survey of respondents can represent the views of all Americans.

12 Estimating a Population Proportion We take a survey (SRS) to estimate the percentage of overweight children aged 6 11 years in the general population. count of successes in the sample 408 pˆ = = = 15.3% n 2673

13 Sampling Distribution of a Sample Proportion If the sample is large enough, the sampling distribution of is approximately normal. pˆ The mean of the sampling distribution is p. The standard deviation of the sampling distribution is p ( 1 p) n

14 The standard deviation from our sample is: p(1 n p) = pˆ(1 n pˆ).153(.847) 2673 =

15 The 95% Confidence Interval around our estimate is: pˆ ± z α / 2 pˆ(1 n pˆ).153 ± 1.96(.00696).153 ± %,16.7%

16 The 95% Confidence Interval around our estimate is: pˆ ± z α / 2 pˆ(1 n pˆ).153 ± 2(.00696).153 ± %,16.7%

17 What if you wanted a 99% Confidence Interval around our estimate? pˆ ± z α / 2 pˆ(1 n pˆ).153 ± 2.58(.00696).153 ± %,17.1%

18 Sampling distribution of a sample mean Choose an SRS of size n from a population in which individuals have mean µ and standard deviation σ. Let x be the mean of the sample. Then: The sampling distribution of x is approximately normal when the sample size n is large. The mean of the sampling distribution is equal to µ. The standard deviation (standard error of the estimate) of the sampling distribution is σ = s. e. = σ / x n

19 Confidence Interval for a Population Mean (µ) When n is large (>30) the sample standard deviation s is close to σ and can be used to estimate it. Confidence Interval for a population mean: n s z x n z x z x x 2 / 2 / 2 / α α α σ σ ± ± ±

20 Suppose a program director wants to estimate the average length of time (in months) clients remain in a rehab clinic program. She takes a random sample of 100 clients records and uses the sample s mean x, to estimate µ, the population mean. We start by calculating the mean and the sample standard deviation. Assume that: x = 465 ( x x) 2 = 2,387

21 Then, x x = n = 465 = ( x x) 2,387 s = = = and s = 4.9 n 1 99 Since we have a large sample (n=100) we can substitute s for σ. A 95% confidence interval for the mean number of months spent in the program is x s 4.9 ± 2 = 4.65 ± 2 = 4.65 ± Confidence Interval = 3.67, 5.63

22 Small sample estimates of µ x ± t α / 2 s n Where t α/2 is based on (n 1) degrees of freedom. Assumption: A random sample is selected from a population with a relative frequency distribution that is approximately normal.

23 Food prices have been going up rapidly. To periodically assess the increase in prices you purchase the same items at twenty different grocery stores. The mean and standard deviation of the costs at the twenty supermarkets are: x = \$26.84 and s = \$2.63 If we assume that the distribution of costs for the grocery basket at all supermarkets is approximately normal, we can use the t-statistic to form the confidence interval. For a confidence level of 95%, we need the tabulated value of t with df = n 1 = 19. From the t table we see that t α/2 = t.025 = x s 2.63 ± t. 025 = ± = ± 1.23 = n 20 ( 25.61, 28.07) Thus, we are reasonably confident (95%) that the interval from \$25.61 to \$28.07 contains the true mean cost µ of the grocery basket. This is because if we were to employ our interval estimator on repeated occasions, 95% of the intervals constructed would contain µ.

24 Determining Sample Size How can the appropriate sample size be determined? First determine how reliable you want the estimate to be. Example: Consider the rehab program example where we estimated the mean length of time clients stayed in the program. A sample of 100 clients records produced an estimate, x, that was within.98 month of µ with probability equal to.95. What if we wanted to estimate the true mean to within.5 month with a probability equal to.95. How large a sample would be required?

25 For the sample size n = 100, we found that an approximate 95% confidence interval to be x ± x 2 σ 4.65 ±.98 If we now want our estimator to be within.5 month of µ, we must have 2σ 2 σ =.5 =. 5 x or n S=4.9 2(4.9) =.5 n 2(4.9) n =.5 2 n = 19.6 = 19.6 =

26 You would have to sample approximately 384 clients records in order to estimate the mean length of stay in the program, µ, to within.5 month with probability equal to.95.

27 Understanding Degrees of Freedom Statisticians use the terms "degrees of freedom" to describe the number of values in the final calculation of a statistic that are free to vary. Consider, for example the statistic s 2. To calculate the s 2 of a random sample, we must first calculate the mean of that sample and then compute the sum of the several squared deviations from that mean. While there will be n such squared deviations only (n - 1) of them are, in fact, free to assume any value whatsoever. This is because the final squared deviation from the mean must include the one value of X such that the sum of all the Xs divided by n will equal the obtained mean of the sample. All of the other (n - 1) squared deviations from the mean can, theoretically, have any values whatsoever. For these reasons, the statistic s 2 is said to have only (n - 1) degrees of freedom.

### Statistical Inference

Statistical Inference Idea: Estimate parameters of the population distribution using data. How: Use the sampling distribution of sample statistics and methods based on what would happen if we used this

### Sampling Distribution of a Sample Proportion

Sampling Distribution of a Sample Proportion From earlier material remember that if X is the count of successes in a sample of n trials of a binomial random variable then the proportion of success is given

### AP * Statistics Review

AP * Statistics Review Confidence Intervals Teacher Packet AP* is a trademark of the College Entrance Examination Board. The College Entrance Examination Board was not involved in the production of this

### Chapter 14: 1-6, 9, 12; Chapter 15: 8 Solutions When is it appropriate to use the normal approximation to the binomial distribution?

Chapter 14: 1-6, 9, 1; Chapter 15: 8 Solutions 14-1 When is it appropriate to use the normal approximation to the binomial distribution? The usual recommendation is that the approximation is good if np

### 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

### 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.

### UCLA STAT 13 Statistical Methods - Final Exam Review Solutions Chapter 7 Sampling Distributions of Estimates

UCLA STAT 13 Statistical Methods - Final Exam Review Solutions Chapter 7 Sampling Distributions of Estimates 1. (a) (i) µ µ (ii) σ σ n is exactly Normally distributed. (c) (i) is approximately Normally

### 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

### Standard Deviation Calculator

CSS.com Chapter 35 Standard Deviation Calculator Introduction The is a tool to calculate the standard deviation from the data, the standard error, the range, percentiles, the COV, confidence limits, or

### Practice Exam. 1. What is the median of this data? A) 64 B) 63.5 C) 67.5 D) 59 E) 35

Practice Exam Use the following to answer questions 1-2: A census is done in a given region. Following are the populations of the towns in that particular region (in thousands): 35, 46, 52, 63, 64, 71,

### Multiple Hypothesis Testing: The F-test

Multiple Hypothesis Testing: The F-test Matt Blackwell December 3, 2008 1 A bit of review When moving into the matrix version of linear regression, it is easy to lose sight of the big picture and get lost

### Probability and Statistics Lecture 9: 1 and 2-Sample Estimation

Probability and Statistics Lecture 9: 1 and -Sample Estimation to accompany Probability and Statistics for Engineers and Scientists Fatih Cavdur Introduction A statistic θ is said to be an unbiased estimator

### Structure of the Data. Paired Samples. Overview. The data from a paired design can be tabulated in this form. Individual Y 1 Y 2 d i = Y 1 Y

Structure of the Data Paired Samples Bret Larget Departments of Botany and of Statistics University of Wisconsin Madison Statistics 371 11th November 2005 The data from a paired design can be tabulated

### Confidence level. Most common choices are 90%, 95%, or 99%. (α = 10%), (α = 5%), (α = 1%)

Confidence Interval A confidence interval (or interval estimate) is a range (or an interval) of values used to estimate the true value of a population parameter. A confidence interval is sometimes abbreviated

### Sampling Central Limit Theorem Proportions. Outline. 1 Sampling. 2 Central Limit Theorem. 3 Proportions

Outline 1 Sampling 2 Central Limit Theorem 3 Proportions Outline 1 Sampling 2 Central Limit Theorem 3 Proportions Populations and samples When we use statistics, we are trying to find out information about

### CHAPTER 11 CHI-SQUARE: NON-PARAMETRIC COMPARISONS OF FREQUENCY

CHAPTER 11 CHI-SQUARE: NON-PARAMETRIC COMPARISONS OF FREQUENCY The hypothesis testing statistics detailed thus far in this text have all been designed to allow comparison of the means of two or more samples

### 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

### SAMPLING DISTRIBUTIONS

0009T_c07_308-352.qd 06/03/03 20:44 Page 308 7Chapter SAMPLING DISTRIBUTIONS 7.1 Population and Sampling Distributions 7.2 Sampling and Nonsampling Errors 7.3 Mean and Standard Deviation of 7.4 Shape of

### PASS Sample Size Software

Chapter 250 Introduction The Chi-square test is often used to test whether sets of frequencies or proportions follow certain patterns. The two most common instances are tests of goodness of fit using multinomial

### How 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

### 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

### Power and Sample Size Determination

Power and Sample Size Determination Bret Hanlon and Bret Larget Department of Statistics University of Wisconsin Madison November 3 8, 2011 Power 1 / 31 Experimental Design To this point in the semester,

### 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

### Chapter 8: Introduction to Hypothesis Testing

Chapter 8: Introduction to Hypothesis Testing We re now at the point where we can discuss the logic of hypothesis testing. This procedure will underlie the statistical analyses that we ll use for the remainder

### Constructing and Interpreting Confidence Intervals

Constructing and Interpreting Confidence Intervals Confidence Intervals In this power point, you will learn: Why confidence intervals are important in evaluation research How to interpret a confidence

### LAB 4 INSTRUCTIONS CONFIDENCE INTERVALS AND HYPOTHESIS TESTING

LAB 4 INSTRUCTIONS CONFIDENCE INTERVALS AND HYPOTHESIS TESTING In this lab you will explore the concept of a confidence interval and hypothesis testing through a simulation problem in engineering setting.

### Stats for Strategy Exam 1 In-Class Practice Questions DIRECTIONS

Stats for Strategy Exam 1 In-Class Practice Questions DIRECTIONS Choose the single best answer for each question. Discuss questions with classmates, TAs and Professor Whitten. Raise your hand to check

### Mind on Statistics. Chapter 10

Mind on Statistics Chapter 10 Section 10.1 Questions 1 to 4: Some statistical procedures move from population to sample; some move from sample to population. For each of the following procedures, determine

### Hypothesis Testing. Bluman Chapter 8

CHAPTER 8 Learning Objectives C H A P T E R E I G H T Hypothesis Testing 1 Outline 8-1 Steps in Traditional Method 8-2 z Test for a Mean 8-3 t Test for a Mean 8-4 z Test for a Proportion 8-5 2 Test for

### Chi Square (χ 2 ) Statistical Instructions EXP 3082L Jay Gould s Elaboration on Christensen and Evans (1980)

Chi Square (χ 2 ) Statistical Instructions EXP 3082L Jay Gould s Elaboration on Christensen and Evans (1980) For the Driver Behavior Study, the Chi Square Analysis II is the appropriate analysis below.

### 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

### Math 251, Review Questions for Test 3 Rough Answers

Math 251, Review Questions for Test 3 Rough Answers 1. (Review of some terminology from Section 7.1) In a state with 459,341 voters, a poll of 2300 voters finds that 45 percent support the Republican candidate,

### Crosstabulation & Chi Square

Crosstabulation & Chi Square Robert S Michael Chi-square as an Index of Association After examining the distribution of each of the variables, the researcher s next task is to look for relationships among

### Statistics 100 Binomial and Normal Random Variables

Statistics 100 Binomial and Normal Random Variables Three different random variables with common characteristics: 1. Flip a fair coin 10 times. Let X = number of heads out of 10 flips. 2. Poll a random

### Sample Exam #1 Elementary Statistics

Sample Exam #1 Elementary Statistics Instructions. No books, notes, or calculators are allowed. 1. Some variables that were recorded while studying diets of sharks are given below. Which of the variables

### Prob & Stats. Chapter 9 Review

Chapter 9 Review Construct the indicated confidence interval for the difference between the two population means. Assume that the two samples are independent simple random samples selected from normally

### 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

### DEPARTMENT OF ECONOMICS. Unit ECON 12122 Introduction to Econometrics. Notes 4 2. R and F tests

DEPARTMENT OF ECONOMICS Unit ECON 11 Introduction to Econometrics Notes 4 R and F tests These notes provide a summary of the lectures. They are not a complete account of the unit material. You should also

### Survey Sampling. Know How No 9 guidance for research and evaluation in Fife. What this is about? Who is it for? What do you need to know?

guidance for research and evaluation in Fife What this is about? Sampling allows you to draw conclusions about a particular population by examining a part of it. When carrying out a survey, it is not usually

### Pricing and Revenue Forecast Model. Multivariate Solutions

Pricing and Revenue Forecast Model Multivariate Solutions Applications and Goals This pricing and revenue forecast model is used primarily to determine optimal pricing of a product/service, and market

### March 3, 2016 By Namrata Uberoi, Kenneth Finegold, and Emily Gee

ASPE ISSUE BRIEF HEALTH INSURANCE COVERAGE AND THE AFFORDABLE CARE ACT, 2010 2016 March 3, 2016 By Namrata Uberoi, Kenneth Finegold, and Emily Gee This issue brief reviews the most recent survey and administrative

### Statistiek I. t-tests. John Nerbonne. CLCG, Rijksuniversiteit Groningen. John Nerbonne 1/35

Statistiek I t-tests John Nerbonne CLCG, Rijksuniversiteit Groningen http://wwwletrugnl/nerbonne/teach/statistiek-i/ John Nerbonne 1/35 t-tests To test an average or pair of averages when σ is known, we

### 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.

### Predictability of Average Inflation over Long Time Horizons

Predictability of Average Inflation over Long Time Horizons Allan Crawford, Research Department Uncertainty about the level of future inflation adversely affects the economy because it distorts savings

### NUMBERS, FACTS AND TRENDS SHAPING THE WORLD FOR RELEASE DECEMBER 30, 2013

NUMBERS, FACTS AND TRENDS SHAPING THE WORLD FOR RELEASE DECEMBER 30, 2013 Maeve Duggan, Research Assistant Aaron Smith, Senior Researcher 202.419.4500 RECOMMENDED CITATION: Maeve Duggan and Aaron Smith,

### Slide 1 Math 1520, Lecture 23. This lecture covers mean, median, mode, odds, and expected value.

Slide 1 Math 1520, Lecture 23 This lecture covers mean, median, mode, odds, and expected value. Slide 2 Mean, Median and Mode Mean, Median and mode are 3 concepts used to get a sense of the central tendencies

### 1.5 Oneway Analysis of Variance

Statistics: Rosie Cornish. 200. 1.5 Oneway Analysis of Variance 1 Introduction Oneway analysis of variance (ANOVA) is used to compare several means. This method is often used in scientific or medical experiments

### Prediction and Confidence Intervals in Regression

Fall Semester, 2001 Statistics 621 Lecture 3 Robert Stine 1 Prediction and Confidence Intervals in Regression Preliminaries Teaching assistants See them in Room 3009 SH-DH. Hours are detailed in the syllabus.

### Standard Deviation Estimator

CSS.com Chapter 905 Standard Deviation Estimator Introduction Even though it is not of primary interest, an estimate of the standard deviation (SD) is needed when calculating the power or sample size of

### F. 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

### Statistical Inference and t-tests

1 Statistical Inference and t-tests Objectives Evaluate the difference between a sample mean and a target value using a one-sample t-test. Evaluate the difference between a sample mean and a target value

### Association Between Variables

Contents 11 Association Between Variables 767 11.1 Introduction............................ 767 11.1.1 Measure of Association................. 768 11.1.2 Chapter Summary.................... 769 11.2 Chi

### Objectives. 6.1, 7.1 Estimating with confidence (CIS: Chapter 10) CI)

Objectives 6.1, 7.1 Estimating with confidence (CIS: Chapter 10) Statistical confidence (CIS gives a good explanation of a 95% CI) Confidence intervals. Further reading http://onlinestatbook.com/2/estimation/confidence.html

### Hypothesis Testing I

ypothesis Testing I The testing process:. Assumption about population(s) parameter(s) is made, called null hypothesis, denoted. 2. Then the alternative is chosen (often just a negation of the null hypothesis),

### BY Aaron Smith NUMBERS, FACTS AND TRENDS SHAPING THE WORLD FOR RELEASE MARCH 10, 2016 FOR MEDIA OR OTHER INQUIRIES:

NUMBERS, FACTS AND TRENDS SHAPING THE WORLD FOR RELEASE MARCH 10, 2016 BY Aaron Smith FOR MEDIA OR OTHER INQUIRIES: Aaron Smith, Associate Director, Research Dana Page, Senior Communications Manager 202.419.4372

### Nonparametric Test Procedures

Nonparametric Test Procedures 1 Introduction to Nonparametrics Nonparametric tests do not require that samples come from populations with normal distributions or any other specific distribution. Hence

### BY Maeve Duggan NUMBERS, FACTS AND TRENDS SHAPING THE WORLD FOR RELEASE AUGUST 19, 2015 FOR FURTHER INFORMATION ON THIS REPORT:

NUMBERS, FACTS AND TRENDS SHAPING THE WORLD FOR RELEASE AUGUST 19, 2015 BY Maeve Duggan FOR FURTHER INFORMATION ON THIS REPORT: Maeve Duggan, Research Associate Dana Page, Senior Communications Manager

### MEASURES OF VARIATION

NORMAL DISTRIBTIONS MEASURES OF VARIATION In statistics, it is important to measure the spread of data. A simple way to measure spread is to find the range. But statisticians want to know if the data are

### Data Analysis. Lecture Empirical Model Building and Methods (Empirische Modellbildung und Methoden) SS Analysis of Experiments - Introduction

Data Analysis Lecture Empirical Model Building and Methods (Empirische Modellbildung und Methoden) Prof. Dr. Dr. h.c. Dieter Rombach Dr. Andreas Jedlitschka SS 2014 Analysis of Experiments - Introduction

### BY Aaron Smith NUMBERS, FACTS AND TRENDS SHAPING THE WORLD. FOR RELEASE: February 11, 2016 FOR MEDIA OR OTHER INQUIRIES:

NUMBERS, FACTS AND TRENDS SHAPING THE WORLD FOR RELEASE: February 11, 2016 BY Aaron Smith FOR MEDIA OR OTHER INQUIRIES: Aaron Smith, Associate Director, Research Dana Page, Senior Communications Manager

### L10: Probability, statistics, and estimation theory

L10: Probability, statistics, and estimation theory Review of probability theory Bayes theorem Statistics and the Normal distribution Least Squares Error estimation Maximum Likelihood estimation Bayesian

### Week 3&4: Z tables and the Sampling Distribution of X

Week 3&4: Z tables and the Sampling Distribution of X 2 / 36 The Standard Normal Distribution, or Z Distribution, is the distribution of a random variable, Z N(0, 1 2 ). The distribution of any other normal

### NUMBERS, FACTS AND TRENDS SHAPING THE WORLD FOR RELEASE NOVEMBER 3, 2014 FOR FURTHER INFORMATION ON THIS REPORT: Aaron Smith, Senior Researcher

NUMBERS, FACTS AND TRENDS SHAPING THE WORLD FOR RELEASE NOVEMBER 3, 2014 FOR FURTHER INFORMATION ON THIS REPORT: Aaron Smith, Senior Researcher 202.419.4372 RECOMMENDED CITATION: Pew Research Center, November,

### WHERE DOES THE 10% CONDITION COME FROM?

1 WHERE DOES THE 10% CONDITION COME FROM? The text has mentioned The 10% Condition (at least) twice so far: p. 407 Bernoulli trials must be independent. If that assumption is violated, it is still okay

### 4: 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

### FINAL EXAM REVIEW - Fa 13

FINAL EXAM REVIEW - Fa 13 Determine which of the four levels of measurement (nominal, ordinal, interval, ratio) is most appropriate. 1) The temperatures of eight different plastic spheres. 2) The sample

### IPSOS / Enterprise Community Partners POLL DATA Prepared by Ipsos Public Affairs

Ipsos Poll Conducted for Enterprise Community Partners Make Room Topline 5.17.2016 These are findings from an Ipsos poll conducted February 29-March 4, 2016 on behalf of Enterprise Community Partners.

### MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Sample Practice problems - chapter 12-1 and 2 proportions for inference - Z Distributions Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide

### Chapter 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

### August 2012 EXAMINATIONS Solution Part I

August 01 EXAMINATIONS Solution Part I (1) In a random sample of 600 eligible voters, the probability that less than 38% will be in favour of this policy is closest to (B) () In a large random sample,

### Math 2015 Lesson 21. We discuss the mean and the median, two important statistics about a distribution. p(x)dx = 0.5

ean and edian We discuss the mean and the median, two important statistics about a distribution. The edian The median is the halfway point of a distribution. It is the point where half the population has

### Fixed vs. Random Effects

Statistics 203: Introduction to Regression and Analysis of Variance Fixed vs. Random Effects Jonathan Taylor - p. 1/19 Today s class Implications for Random effects. One-way random effects ANOVA. Two-way

### The Standard Normal distribution

The Standard Normal distribution 21.2 Introduction Mass-produced 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

### Introductory Statistics Notes

Introductory Statistics Notes Jamie DeCoster Department of Psychology University of Alabama 348 Gordon Palmer Hall Box 870348 Tuscaloosa, AL 35487-0348 Phone: (205) 348-4431 Fax: (205) 348-8648 August

### Statistics - Written Examination MEC Students - BOVISA

Statistics - Written Examination MEC Students - BOVISA Prof.ssa A. Guglielmi 26.0.2 All rights reserved. Legal action will be taken against infringement. Reproduction is prohibited without prior consent.

### MT426 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 2004-2012 by Jenny A. Baglivo. All Rights Reserved. Contents 3 MT426 Notebook 3 3 3.1 Definitions............................................

### Section 13, Part 1 ANOVA. Analysis Of Variance

Section 13, Part 1 ANOVA Analysis Of Variance Course Overview So far in this course we ve covered: Descriptive statistics Summary statistics Tables and Graphs Probability Probability Rules Probability

### The normal approximation to the binomial

The normal approximation to the binomial In order for a continuous distribution (like the normal) to be used to approximate a discrete one (like the binomial), a continuity correction should be used. There

### Chapter 7 Review. Confidence Intervals. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Chapter 7 Review Confidence Intervals MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) Suppose that you wish to obtain a confidence interval for

### Chapter 11. Chapter 11 Overview. Chapter 11 Objectives 11/24/2015. Other Chi-Square Tests

11/4/015 Chapter 11 Overview Chapter 11 Introduction 11-1 Test for Goodness of Fit 11- Tests Using Contingency Tables Other Chi-Square Tests McGraw-Hill, Bluman, 7th ed., Chapter 11 1 Bluman, Chapter 11

### FE670 Algorithmic Trading Strategies. Stevens Institute of Technology

FE670 Algorithmic Trading Strategies Lecture 6. Portfolio Optimization: Basic Theory and Practice Steve Yang Stevens Institute of Technology 10/03/2013 Outline 1 Mean-Variance Analysis: Overview 2 Classical

### Confidence intervals

Confidence intervals Today, we re going to start talking about confidence intervals. We use confidence intervals as a tool in inferential statistics. What this means is that given some sample statistics,

### The Normal distribution

The Normal distribution The normal probability distribution is the most common model for relative frequencies of a quantitative variable. Bell-shaped and described by the function f(y) = 1 2σ π e{ 1 2σ

### 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

### Key Concept. Properties

MAT 155 Statistical Analysis Dr. Claude Moore Cape Fear Community College Chapter 6 Normal Probability Distributions 6 1 Review and Preview 6 2 The Standard Normal Distribution 6 3 Applications of Normal

### Characteristics of Binomial Distributions

Lesson2 Characteristics of Binomial Distributions In the last lesson, you constructed several binomial distributions, observed their shapes, and estimated their means and standard deviations. In Investigation

### Statistical tests for SPSS

Statistical tests for SPSS Paolo Coletti A.Y. 2010/11 Free University of Bolzano Bozen Premise This book is a very quick, rough and fast description of statistical tests and their usage. It is explicitly

### Margin of Error When Estimating a Population Proportion

Margin of Error When Estimating a Population Proportion Student Outcomes Students use data from a random sample to estimate a population proportion. Students calculate and interpret margin of error in

### Estimation and Confidence Intervals

Estimation and Confidence Intervals Fall 2001 Professor Paul Glasserman B6014: Managerial Statistics 403 Uris Hall Properties of Point Estimates 1 We have already encountered two point estimators: th e

### 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

### REGRESSION LINES IN STATA

REGRESSION LINES IN STATA THOMAS ELLIOTT 1. Introduction to Regression Regression analysis is about eploring linear relationships between a dependent variable and one or more independent variables. Regression

### GrowingKnowing.com 2011

GrowingKnowing.com 2011 GrowingKnowing.com 2011 1 Estimates We are often asked to predict the future! When will you complete your team project? When will you make your first million dollars? When will

### 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-

### UNDERSTANDING THE ONE-WAY ANOVA

UNDERSTANDING The One-way Analysis of Variance (ANOVA) is a procedure for testing the hypothesis that K population means are equal, where K >. The One-way ANOVA compares the means of the samples or groups

### Unit 26 Estimation with Confidence Intervals

Unit 26 Estimation with Confidence Intervals Objectives: To see how confidence intervals are used to estimate a population proportion, a population mean, a difference in population proportions, or a difference

### 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

### Survey Process White Paper Series The Six Steps in Conducting Quantitative Marketing Research

Survey Process White Paper Series The Six Steps in Conducting Quantitative Marketing Research POLARIS MARKETING RESEARCH, INC. 1455 LINCOLN PARKWAY, SUITE 320 ATLANTA, GEORGIA 30346 404.816.0353 www.polarismr.com

### Coefficient of Determination

Coefficient of Determination The coefficient of determination R 2 (or sometimes r 2 ) is another measure of how well the least squares equation ŷ = b 0 + b 1 x performs as a predictor of y. R 2 is computed