Chapter 16: Confidence intervals
|
|
- Melvyn Victor Gordon
- 7 years ago
- Views:
Transcription
1 Chapter 16: Confidence intervals Objective (1) Learn how to estimate the error-margin in "proportion" type statistics calculated from a random sample. (2) Learn how to construct confidence intervals. Concept briefs: * Sampling distribution theorem for proportions = When the conditions are met, proportions follow normal model N( p, ), with p=true population parameter, p (1- p)/ n. * Standard Error (SE) = ^p (1- ^p ) / n ( ^p = sampled statistic) * Confidence level = % chosen for constructing confidence interval. * Critical value (z*) = z-value that corresponds to confidence level chosen. * Margin of error (ME) = z* x SE. It indicates how far ^p may be from true p. * Confidence interval (CI) = ^p ± ME. It gives the interval within which the true value of p lies. * One proportion z-interval = General term for confidence int. for proportions. * Confidence interval tradeoffs: Be aware of effect of confidence level and sample size on margin of error.
2 Confidence Intervals: Concept Summary What is a confidence interval (CI)? It is a numerical range within which the true value of population parameter lies. E.g., (1) The true proportion of students who live off-campus is in the interval [5.8%,19%]. (2) The true mean GPA of students is in the interval [2.90, 3.22]. How accurate are confidence intervals? A CI is estimated based on probability. There is no guarantee that the true value is contained within the CI. However, we can estimate a CI with a very high probability that it contains the true parameter. In fact, we can estimate it to any desired level of probabilistic certainty. E.g., (1) There is a 95% probability that the true proportion of students who live off-campus is in the interval [5.8%,19%]. (2) There is an 80% probability that the true proportion is in the interval [8.2%,16.6%]. What is the margin of error? The width of the CI tells us the margin of error (ME). Technically, the ME is half the width of the confidence interval. How does the width relate to the confidence level? To capture the true parameter with higher probability (i.e., higher confidence level), the CI must be wider. Thus, for example, a 95% CI is always wider than an 80% CI. Therefore, the margin of error always gets bigger with higher confidence level. Two key interpretations of confidence level: (1) It denotes the probability that the true parameter is contained within the CI. (2) It denotes the percentage of all random samples (within the population) that will contain the true parameter within their CI.
3 Illustration: * We survey random sample of 100 students to determine % of EC students who live off-campus. * Suppose our survey estimates this statistic to be 12.4% (so our ^p =0.124). Central Limit Theorem says: If you repeat your study several times with different random samples, and take the mean of all your estimates, you'll get the true value you're seeking (i.e., p). Reality says: What is the best I can do with the statistic calculated from my single study? The goal: Find margin of error in our calculated ^p Issues/problems (1) We don't know true p, so can't get the sampling distribution model. (2) Without p, can't tell how far, or even which direction, ^p is in. ^p p?? p ^p
4 Resolution to problem1 * We know the true sampling distribution would follow the normal model with mean p( 1 p) p and SD. n * Since p is not known, we can't find the mean or SD. * To compromise, we "fudge" the SD and calculate it using ^p instead of p. This is called Standard Error (SE) instead of Standard Deviation (SD). ^p ( 1 SE ^p ). 124( ) E.g: n 100
5 Resolution to problem2 Illustration: 2 * We don't know true p, but we know that 95% of all random samples will give an answer within of this value (provided we know ). * We have a decent approximation for = (SE [^p (1-^p ) / n]). * So, we know the true answer must lie (at worst) within ± 2 SE from the sampled result. 2SE = 6.6%. Thus, the true answer is between: % and %. * Confidence interval says: "We are 95% confident that % students who live off-campus is between 5.8% and 19%." Q: Is it possible to get really unlucky & pick a sample outside the 95% that are within 2SE of the true answer?
6 True ^p Confidence Interval Recipe Objective: You have a sampled proportion ^p. You want to predict the range in which the true proportion p lies. Step0: Identify the sample proportion ^p, if you haven't already. Step1: Determine confidence level you want for your prediction. (e.g., 90%). Step2: Find critical value (z*) for this confidence level. Step3: Verify conditions for theorem; find SE= ^p (1- ^p ) / n. Step4: Find margin of error: ME = z* x SE. Step5: Find the confidence interval: ^p -ME p +ME. Step6: Write a sentence (or two) that states and interprets your CI. Points to note: (1) Confidence intervals always involve a tradeoff between margin of error & level of confidence. (2) If you want higher confidence, you buy this by increasing the margin of error (think of this as "margin of safety"). E.g., for 100% confidence, the margin of safety must also be 100%. (3) The only way to get high confidence with smaller margins of error is to find a way to decrease SE. The only way to do that is by increasing sample size. (4) There is an important technical name for this confidence interval: One proportion z-interval
7 How to find critical z* values: More examples Objective: For confidence level of x %, we want to find z-value that encloses the central x % of the std. normal model. Example1: z* for 80% confidence level Confidence level=80%: Lookup z-value for area=90%. Thus, z* = % 10% 10% Example2: z* for 90% confidence level Confidence level=90%: Lookup z-value for area=95%. Thus, z* = % 5% 5% Example3: z* for 98% confidence level Confidence level=98%: Lookup z-value for area=99%. Thus, z* = % 1% 1%
8 Exercise 30, pg. 448 Strategy for (b): * For newspaper: ^p = 0.53, n = 1200; Want 95% confidence; To find z* lookup standard normal table for 97.5% area --> z*=1.96 Check conditions: random, independent, sufficiently large SE = ( )/ 1200 = X (calculate this value yourself!) ME = 1.96 X Confidence interval: X to X. * For statistics class: ^p = 0.54, n = 450; Confidence level and z* same; Check conditions; SE = ( )/ 450 = Y (calculate this value yourself!) ME = 1.96 Y Confidence interval: Y to Y. Answers: Newspaper CI: [0.5018, ] OR 50.18% to 55.82% Statistics class: [0.4939, ] OR 49.39% to 58.61% Exercise 38, pg. 448 Strategy: (a) Find z* for 98% confidence level (i.e., lookup z-score for 99% area). Question wants ME=0.05. This requires: 0.05 = z* x SE. You know z*, so you can find SE [Check your answer: SE=0.0215]. Take worst case scenario (i.e., largest SE happens for ^p =0.5). Use SE = 0. 5 ( )/ n to find n. --> n = 0.5 (1-0.5) / SE 2. [Answer: n ~ 543] (b) Very similar strategy. Get answers: SE=0.0129, n ~ 1503.
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
More information5.1 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
More informationWeek 4: Standard Error and Confidence Intervals
Health Sciences M.Sc. Programme Applied Biostatistics Week 4: Standard Error and Confidence Intervals Sampling Most research data come from subjects we think of as samples drawn from a larger population.
More informationSocial Studies 201 Notes for November 19, 2003
1 Social Studies 201 Notes for November 19, 2003 Determining sample size for estimation of a population proportion Section 8.6.2, p. 541. As indicated in the notes for November 17, when sample size is
More informationConfidence 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
More informationSimple 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
More informationLecture 19: Chapter 8, Section 1 Sampling Distributions: Proportions
Lecture 19: Chapter 8, Section 1 Sampling Distributions: Proportions Typical Inference Problem Definition of Sampling Distribution 3 Approaches to Understanding Sampling Dist. Applying 68-95-99.7 Rule
More informationPoint 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
More informationObjectives. 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
More informationTwo-sample inference: Continuous data
Two-sample inference: Continuous data Patrick Breheny April 5 Patrick Breheny STA 580: Biostatistics I 1/32 Introduction Our next two lectures will deal with two-sample inference for continuous data As
More informationMULTIPLE 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
More informationCoefficient 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
More informationUnit 31 A Hypothesis Test about Correlation and Slope in a Simple Linear Regression
Unit 31 A Hypothesis Test about Correlation and Slope in a Simple Linear Regression Objectives: To perform a hypothesis test concerning the slope of a least squares line To recognize that testing for a
More informationIntroduction to Hypothesis Testing
I. Terms, Concepts. Introduction to Hypothesis Testing A. In general, we do not know the true value of population parameters - they must be estimated. However, we do have hypotheses about what the true
More informationStat 411/511 THE RANDOMIZATION TEST. Charlotte Wickham. stat511.cwick.co.nz. Oct 16 2015
Stat 411/511 THE RANDOMIZATION TEST Oct 16 2015 Charlotte Wickham stat511.cwick.co.nz Today Review randomization model Conduct randomization test What about CIs? Using a t-distribution as an approximation
More informationConfidence 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
More informationPART 7: LESSON D: There are Five Ways to Monitor Employee Performance
PART 7: LESSON D: There are Five Ways to Monitor Employee Performance Key Points There are five ways to monitor employee performance: (#1) Watch employees work. (#2) Ask for an account. (#3) Help employees
More informationCALCULATIONS & STATISTICS
CALCULATIONS & STATISTICS CALCULATION OF SCORES Conversion of 1-5 scale to 0-100 scores When you look at your report, you will notice that the scores are reported on a 0-100 scale, even though respondents
More information**Unedited Draft** Arithmetic Revisited Lesson 4: Part 3: Multiplying Mixed Numbers
. Introduction: **Unedited Draft** Arithmetic Revisited Lesson : Part 3: Multiplying Mixed Numbers As we mentioned in a note on the section on adding mixed numbers, because the plus sign is missing, it
More informationChapter 8 Hypothesis Testing Chapter 8 Hypothesis Testing 8-1 Overview 8-2 Basics of Hypothesis Testing
Chapter 8 Hypothesis Testing 1 Chapter 8 Hypothesis Testing 8-1 Overview 8-2 Basics of Hypothesis Testing 8-3 Testing a Claim About a Proportion 8-5 Testing a Claim About a Mean: s Not Known 8-6 Testing
More informationChapter 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
More informationName: Date: Use the following to answer questions 3-4:
Name: Date: 1. Determine whether each of the following statements is true or false. A) The margin of error for a 95% confidence interval for the mean increases as the sample size increases. B) The margin
More informationSimple Inventory Management
Jon Bennett Consulting http://www.jondbennett.com Simple Inventory Management Free Up Cash While Satisfying Your Customers Part of the Business Philosophy White Papers Series Author: Jon Bennett September
More informationConfidence 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,
More informationMind 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
More informationChapter 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
More informationChapter 5: Normal Probability Distributions - Solutions
Chapter 5: Normal Probability Distributions - Solutions Note: All areas and z-scores are approximate. Your answers may vary slightly. 5.2 Normal Distributions: Finding Probabilities If you are given that
More informationConfidence Intervals for Cpk
Chapter 297 Confidence Intervals for Cpk Introduction This routine calculates the sample size needed to obtain a specified width of a Cpk confidence interval at a stated confidence level. Cpk is a process
More informationGood luck! BUSINESS STATISTICS FINAL EXAM INSTRUCTIONS. Name:
Glo bal Leadership M BA BUSINESS STATISTICS FINAL EXAM Name: INSTRUCTIONS 1. Do not open this exam until instructed to do so. 2. Be sure to fill in your name before starting the exam. 3. You have two hours
More informationNeed 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-
More informationLesson 7 Z-Scores and Probability
Lesson 7 Z-Scores and Probability Outline Introduction Areas Under the Normal Curve Using the Z-table Converting Z-score to area -area less than z/area greater than z/area between two z-values Converting
More informationThe 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
More informationAP STATISTICS (Warm-Up Exercises)
AP STATISTICS (Warm-Up Exercises) 1. Describe the distribution of ages in a city: 2. Graph a box plot on your calculator for the following test scores: {90, 80, 96, 54, 80, 95, 100, 75, 87, 62, 65, 85,
More informationSENSITIVITY ANALYSIS AND INFERENCE. Lecture 12
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike License. Your use of this material constitutes acceptance of that license and the conditions of use of materials on this
More informationLecture 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
More information6.4 Normal Distribution
Contents 6.4 Normal Distribution....................... 381 6.4.1 Characteristics of the Normal Distribution....... 381 6.4.2 The Standardized Normal Distribution......... 385 6.4.3 Meaning of Areas under
More informationNormal 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
More informationConfidence Intervals
Section 6.1 75 Confidence Intervals Section 6.1 C H A P T E R 6 4 Example 4 (pg. 284) Constructing a Confidence Interval Enter the data from Example 1 on pg. 280 into L1. In this example, n > 0, so the
More informationMathematical Induction
Mathematical Induction In logic, we often want to prove that every member of an infinite set has some feature. E.g., we would like to show: N 1 : is a number 1 : has the feature Φ ( x)(n 1 x! 1 x) How
More informationUnit 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
More informationMind on Statistics. Chapter 12
Mind on Statistics Chapter 12 Sections 12.1 Questions 1 to 6: For each statement, determine if the statement is a typical null hypothesis (H 0 ) or alternative hypothesis (H a ). 1. There is no difference
More informationConfidence 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
More informationFutureleaders Training: Negotiation Skills. Trainer: Sam Obeng-Dokyi
Futureleaders Training: Negotiation Skills Trainer: Sam Obeng-Dokyi 23 September 2014 Purpose & objectives 2 Aim The aim of this training session is to provide participants with an understanding of the
More informationExperimental Design. Power and Sample Size Determination. Proportions. Proportions. Confidence Interval for p. The Binomial Test
Experimental Design Power and Sample Size Determination Bret Hanlon and Bret Larget Department of Statistics University of Wisconsin Madison November 3 8, 2011 To this point in the semester, we have largely
More informationLesson 17: Margin of Error When Estimating a Population Proportion
Margin of Error When Estimating a Population Proportion Classwork In this lesson, you will find and interpret the standard deviation of a simulated distribution for a sample proportion and use this information
More information4. Continuous Random Variables, the Pareto and Normal Distributions
4. Continuous Random Variables, the Pareto and Normal Distributions A continuous random variable X can take any value in a given range (e.g. height, weight, age). The distribution of a continuous random
More informationMath 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,
More informationChapter 7 - Practice Problems 2
Chapter 7 - Practice Problems 2 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the requested value. 1) A researcher for a car insurance company
More informationComparing Means in Two Populations
Comparing Means in Two Populations Overview The previous section discussed hypothesis testing when sampling from a single population (either a single mean or two means from the same population). Now we
More informationThe sample space for a pair of die rolls is the set. The sample space for a random number between 0 and 1 is the interval [0, 1].
Probability Theory Probability Spaces and Events Consider a random experiment with several possible outcomes. For example, we might roll a pair of dice, flip a coin three times, or choose a random real
More information= 2.0702 N(280, 2.0702)
Name Test 10 Confidence Intervals Homework (Chpt 10.1, 11.1, 12.1) Period For 1 & 2, determine the point estimator you would use and calculate its value. 1. How many pairs of shoes, on average, do female
More informationSummary 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
More information1 The EOQ and Extensions
IEOR4000: Production Management Lecture 2 Professor Guillermo Gallego September 9, 2004 Lecture Plan 1. The EOQ and Extensions 2. Multi-Item EOQ Model 1 The EOQ and Extensions This section is devoted to
More informationz-scores AND THE NORMAL CURVE MODEL
z-scores AND THE NORMAL CURVE MODEL 1 Understanding z-scores 2 z-scores A z-score is a location on the distribution. A z- score also automatically communicates the raw score s distance from the mean A
More informationLesson 4 Measures of Central Tendency
Outline Measures of a distribution s shape -modality and skewness -the normal distribution Measures of central tendency -mean, median, and mode Skewness and Central Tendency Lesson 4 Measures of Central
More informationChapter 8 Section 1. Homework A
Chapter 8 Section 1 Homework A 8.7 Can we use the large-sample confidence interval? In each of the following circumstances state whether you would use the large-sample confidence interval. The variable
More informationChapter 3 RANDOM VARIATE GENERATION
Chapter 3 RANDOM VARIATE GENERATION In order to do a Monte Carlo simulation either by hand or by computer, techniques must be developed for generating values of random variables having known distributions.
More informationHypothesis Testing for Beginners
Hypothesis Testing for Beginners Michele Piffer LSE August, 2011 Michele Piffer (LSE) Hypothesis Testing for Beginners August, 2011 1 / 53 One year ago a friend asked me to put down some easy-to-read notes
More information22. HYPOTHESIS TESTING
22. HYPOTHESIS TESTING Often, we need to make decisions based on incomplete information. Do the data support some belief ( hypothesis ) about the value of a population parameter? Is OJ Simpson guilty?
More informationRepresentation of functions as power series
Representation of functions as power series Dr. Philippe B. Laval Kennesaw State University November 9, 008 Abstract This document is a summary of the theory and techniques used to represent functions
More informationChapter 6: Probability
Chapter 6: Probability In a more mathematically oriented statistics course, you would spend a lot of time talking about colored balls in urns. We will skip over such detailed examinations of probability,
More informationGetting Started with Statistics. Out of Control! ID: 10137
Out of Control! ID: 10137 By Michele Patrick Time required 35 minutes Activity Overview In this activity, students make XY Line Plots and scatter plots to create run charts and control charts (types of
More informationSample Size Issues for Conjoint Analysis
Chapter 7 Sample Size Issues for Conjoint Analysis I m about to conduct a conjoint analysis study. How large a sample size do I need? What will be the margin of error of my estimates if I use a sample
More informationSimple Linear Regression
STAT 101 Dr. Kari Lock Morgan Simple Linear Regression SECTIONS 9.3 Confidence and prediction intervals (9.3) Conditions for inference (9.1) Want More Stats??? If you have enjoyed learning how to analyze
More informationHYPOTHESIS TESTING (ONE SAMPLE) - CHAPTER 7 1. used confidence intervals to answer questions such as...
HYPOTHESIS TESTING (ONE SAMPLE) - CHAPTER 7 1 PREVIOUSLY used confidence intervals to answer questions such as... You know that 0.25% of women have red/green color blindness. You conduct a study of men
More informationProbability Distributions
CHAPTER 5 Probability Distributions CHAPTER OUTLINE 5.1 Probability Distribution of a Discrete Random Variable 5.2 Mean and Standard Deviation of a Probability Distribution 5.3 The Binomial Distribution
More informationLecture 2: Discrete Distributions, Normal Distributions. Chapter 1
Lecture 2: Discrete Distributions, Normal Distributions Chapter 1 Reminders Course website: www. stat.purdue.edu/~xuanyaoh/stat350 Office Hour: Mon 3:30-4:30, Wed 4-5 Bring a calculator, and copy Tables
More informationCOMP6053 lecture: Relationship between two variables: correlation, covariance and r-squared. jn2@ecs.soton.ac.uk
COMP6053 lecture: Relationship between two variables: correlation, covariance and r-squared jn2@ecs.soton.ac.uk Relationships between variables So far we have looked at ways of characterizing the distribution
More informationAugust 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,
More informationConfidence Intervals for Spearman s Rank Correlation
Chapter 808 Confidence Intervals for Spearman s Rank Correlation Introduction This routine calculates the sample size needed to obtain a specified width of Spearman s rank correlation coefficient confidence
More informationMath 58. Rumbos Fall 2008 1. Solutions to Review Problems for Exam 2
Math 58. Rumbos Fall 2008 1 Solutions to Review Problems for Exam 2 1. For each of the following scenarios, determine whether the binomial distribution is the appropriate distribution for the random variable
More informationA PRACTICAL GUIDE TO db CALCULATIONS
A PRACTICAL GUIDE TO db CALCULATIONS This is a practical guide to doing db (decibel) calculations, covering most common audio situations. You see db numbers all the time in audio. You may understand that
More information1. The Fly In The Ointment
Arithmetic Revisited Lesson 5: Decimal Fractions or Place Value Extended Part 5: Dividing Decimal Fractions, Part 2. The Fly In The Ointment The meaning of, say, ƒ 2 doesn't depend on whether we represent
More informationPrime Factorization 0.1. Overcoming Math Anxiety
0.1 Prime Factorization 0.1 OBJECTIVES 1. Find the factors of a natural number 2. Determine whether a number is prime, composite, or neither 3. Find the prime factorization for a number 4. Find the GCF
More informationLesson 9 Hypothesis Testing
Lesson 9 Hypothesis Testing Outline Logic for Hypothesis Testing Critical Value Alpha (α) -level.05 -level.01 One-Tail versus Two-Tail Tests -critical values for both alpha levels Logic for Hypothesis
More informationPractice problems for Homework 12 - confidence intervals and hypothesis testing. Open the Homework Assignment 12 and solve the problems.
Practice problems for Homework 1 - confidence intervals and hypothesis testing. Read sections 10..3 and 10.3 of the text. Solve the practice problems below. Open the Homework Assignment 1 and solve the
More informationProbability. Distribution. Outline
7 The Normal Probability Distribution Outline 7.1 Properties of the Normal Distribution 7.2 The Standard Normal Distribution 7.3 Applications of the Normal Distribution 7.4 Assessing Normality 7.5 The
More informationAirplane Buying Cheat Sheet. How To Get A Discount On Your Next Airplane:
Airplane Buying Cheat Sheet Disclaimer: Make sure you always get professional legal advice & used licenced technicians when you are doing your inspections. This information is used at your own risk...
More informationReview. 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
More information3 Some Integer Functions
3 Some Integer Functions A Pair of Fundamental Integer Functions The integer function that is the heart of this section is the modulo function. However, before getting to it, let us look at some very simple
More informationC. The null hypothesis is not rejected when the alternative hypothesis is true. A. population parameters.
Sample Multiple Choice Questions for the material since Midterm 2. Sample questions from Midterms and 2 are also representative of questions that may appear on the final exam.. A randomly selected sample
More informationFactors affecting online sales
Factors affecting online sales Table of contents Summary... 1 Research questions... 1 The dataset... 2 Descriptive statistics: The exploratory stage... 3 Confidence intervals... 4 Hypothesis tests... 4
More informationSolutions to Math 51 First Exam January 29, 2015
Solutions to Math 5 First Exam January 29, 25. ( points) (a) Complete the following sentence: A set of vectors {v,..., v k } is defined to be linearly dependent if (2 points) there exist c,... c k R, not
More informationX = rnorm(5e4,mean=1,sd=2) # we need a total of 5 x 10,000 = 5e4 samples X = matrix(data=x,nrow=1e4,ncol=5)
ECL 290 Statistical Models in Ecology using R Problem set for Week 6 Monte Carlo simulation, power analysis, bootstrapping 1. Monte Carlo simulations - Central Limit Theorem example To start getting a
More informationUniversity of Chicago Graduate School of Business. Business 41000: Business Statistics
Name: University of Chicago Graduate School of Business Business 41000: Business Statistics Special Notes: 1. This is a closed-book exam. You may use an 8 11 piece of paper for the formulas. 2. Throughout
More informationHow To Calculate Confidence Intervals In A Population Mean
Chapter 8 Confidence Intervals 8.1 Confidence Intervals 1 8.1.1 Student Learning Objectives By the end of this chapter, the student should be able to: Calculate and interpret confidence intervals for one
More information8. THE NORMAL DISTRIBUTION
8. THE NORMAL DISTRIBUTION The normal distribution with mean μ and variance σ 2 has the following density function: The normal distribution is sometimes called a Gaussian Distribution, after its inventor,
More informationMcKinsey Problem Solving Test Top Tips
McKinsey Problem Solving Test Top Tips 1 McKinsey Problem Solving Test You re probably reading this because you ve been invited to take the McKinsey Problem Solving Test. Don t stress out as part of the
More informationIntroduction. Hypothesis Testing. Hypothesis Testing. Significance Testing
Introduction Hypothesis Testing Mark Lunt Arthritis Research UK Centre for Ecellence in Epidemiology University of Manchester 13/10/2015 We saw last week that we can never know the population parameters
More informationHypothesis Testing: Two Means, Paired Data, Two Proportions
Chapter 10 Hypothesis Testing: Two Means, Paired Data, Two Proportions 10.1 Hypothesis Testing: Two Population Means and Two Population Proportions 1 10.1.1 Student Learning Objectives By the end of this
More informationReading and Taking Notes on Scholarly Journal Articles
Reading and Taking Notes on Scholarly Journal Articles Set aside enough time in your schedule to read material thoroughly and repeatedly, until you understand what the author is studying, arguing, or discussing.
More information1. 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
More informationTI 83/84 Calculator The Basics of Statistical Functions
What you want to do How to start What to do next Put Data in Lists STAT EDIT 1: EDIT ENTER Clear numbers already in a list: Arrow up to L1, then hit CLEAR, ENTER. Then just type the numbers into the appropriate
More informationA) 0.1554 B) 0.0557 C) 0.0750 D) 0.0777
Math 210 - Exam 4 - Sample Exam 1) What is the p-value for testing H1: µ < 90 if the test statistic is t=-1.592 and n=8? A) 0.1554 B) 0.0557 C) 0.0750 D) 0.0777 2) The owner of a football team claims that
More informationFixed-Effect Versus Random-Effects Models
CHAPTER 13 Fixed-Effect Versus Random-Effects Models Introduction Definition of a summary effect Estimating the summary effect Extreme effect size in a large study or a small study Confidence interval
More informationThe 2014 Ultimate Career Guide
The 2014 Ultimate Career Guide Contents: 1. Explore Your Ideal Career Options 2. Prepare For Your Ideal Career 3. Find a Job in Your Ideal Career 4. Succeed in Your Ideal Career 5. Four of the Fastest
More informationCA200 Quantitative Analysis for Business Decisions. File name: CA200_Section_04A_StatisticsIntroduction
CA200 Quantitative Analysis for Business Decisions File name: CA200_Section_04A_StatisticsIntroduction Table of Contents 4. Introduction to Statistics... 1 4.1 Overview... 3 4.2 Discrete or continuous
More informationMy Secure Backup: How to reduce your backup size
My Secure Backup: How to reduce your backup size As time passes, we find our backups getting bigger and bigger, causing increased space charges. This paper takes a few Newsletter and other articles I've
More informationSample Size and Power in Clinical Trials
Sample Size and Power in Clinical Trials Version 1.0 May 011 1. Power of a Test. Factors affecting Power 3. Required Sample Size RELATED ISSUES 1. Effect Size. Test Statistics 3. Variation 4. Significance
More information