p = q ˆ = 1 -ˆp = sample proportion of failures in a sample size of n x n Chapter 7 Estimates and Sample Sizes
|
|
- Shannon McKenzie
- 5 years ago
- Views:
Transcription
1 Chapter 7 Estimates and Sample Sizes 7-1 Overview 7-2 Estimating a Population Proportion 7-3 Estimating a Population Mean: σ Known 7-4 Estimating a Population Mean: σ Not Known 7-5 Estimating a Population Variance Slide 1 Overview This chapter presents the beginning of inferential statistics. The two major applications of inferential statistics involve the use of sample data to (1) estimate the value of a population parameter, and (2) test some claim (or hypothesis) about a population. We introduce methods for estimating values of these important population parameters: proportions, means, and variances. We also present methods for determining sample sizes necessary to estimate those parameters. Slide 2 Section 7-2 Estimating a Population Proportion Key Concept In this section we present important methods for using a sample proportion to estimate the value of a population proportion with a confidence interval. We also present methods for finding the size of the sample needed to estimate a population proportion. Requirements for Estimating a Population Proportion 1. The sample is a simple random sample. 2. The conditions for the binomial distribution are satisfied. (See Section 5-3.) 3. There are at least 5 successes and 5 failures. Slide 3 Slide 4 p = population proportion p = x n sample proportion (pronounced p-hat ) Notation for Proportions of x successes in a sample of size n q = 1 -p = sample proportion of failures in a sample size of n A point estimate is a single value (or point) used to approximate a population parameter. Slide 5 Slide 6
2 The sample proportion p is the best point estimate of the population proportion p. Slide 7 traffic tickets. Using these survey results, find the best point estimate of the proportion of all adult Minnesotans opposed to photo-cop use. Because the sample proportion is the best point estimate of the population proportion, we conclude that the best point estimate of p is When using the survey results to estimate the percentage of all adult Minnesotans that are opposed to photo-cop use, our best estimate is 51%. Slide 8 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 as CI. A confidence level is the probability 1- α (often expressed as the equivalent percentage value) that is the proportion of times that the confidence interval actually does contain the population parameter, assuming that the estimation process is repeated a large number of times. (The confidence level is also called degree of confidence, or the confidence coefficient.) Most common choices are 90%, 95%, or 99%. (α = 10%), (α = 5%), (α = 1%) Slide 9 Slide 10 traffic tickets. Using these survey results, find the 95% confidence interval of the proportion of all adult Minnesotans opposed to photo-cop use. We are 95% confident that the interval from to does contain the true value of p. This means if we were to select many different samples of size 829 and construct the corresponding confidence intervals, 95% of them would actually contain the value of the population proportion p. Slide 11 Critical Values 1. We know from Section 6-6 that under certain conditions, the sampling distribution of sample proportions can be approximated by a normal distribution, as in Figure 7-2, following. 2. Sample proportions have a relatively small chance (with probability denoted by α) of falling in one of the red tails of Figure 7-2, following. 3. Denoting the area of each shaded tail by α/2, we see that there is a total probability of α that a sample proportion will fall in either of the two red tails. Slide 12
3 Critical Values The Critical Value z α/2 4. By the rule of complements (from Chapter 4), there is a probability of 1-α that a sample proportion will fall within the inner region of Figure 7-2, following. 5. The z score separating the right-tail is commonly denoted by z α /2 and is referred to as a critical value because it is on the borderline separating sample proportions that are likely to occur from those that are unlikely to occur. Slide 13 Figure 7-2 Slide 14 Notation for Critical Value The critical value z α/2 is the positive z value that is at the vertical boundary separating an area of α/2 in the right tail of the standard normal distribution. (The value of z α/2 is at the vertical boundary for the area of α/2 in the left tail.) The subscript α/2 is simply a reminder that the z score separates an area of α/2 in the right tail of the standard normal distribution. A critical value is the number on the borderline separating sample statistics that are likely to occur from those that are unlikely to occur. The number z α/2 is a critical value that is a z score with the property that it separates an area of α/2 in the right tail of the standard normal distribution. (See Figure 7-2). Slide 15 Slide 16 Finding z α/2 for a 95% Confidence Level Finding z α/2 for a 95% Confidence Level - cont α = 5% α/ 2 = 2.5% =.025 α = 0.05 Use Table A-2 to find a z score of z α/2 z α/2 z α/ α/2 = Critical Values Slide 17 Slide 18
4 When data from a simple random sample are used to estimate a population proportion p, the margin of error, denoted by E, is the maximum likely (with probability 1 α) difference between the observed proportion p and the true value of the population proportion p. The margin of error E is also called the maximum error of the estimate and can be found by multiplying the critical value and the standard deviation of the sample proportions, as shown in Formula 7-1, following. Margin of Error of the Estimate of p E = Formula 7-1 z α / 2 p q n Slide 19 Slide 20 Confidence Interval for Population Proportion p E < p < p + E E = where z α / 2 p q n Slide 21 Confidence Interval for Population Proportion - cont p E < p < p + E p + E (p E, p + E) Slide 22 Round-Off Rule for Confidence Interval Estimates of p Round the confidence interval limits for p to three significant digits. Slide 23 Procedure for Constructing a Confidence Interval for p 1. Verify that the required assumptions are satisfied. (The sample is a simple random sample, the conditions for the binomial distribution are satisfied, and the normal distribution can be used to approximate the distribution of sample proportions because np 5, and nq 5 are both satisfied.) 2. Refer to Table A-2 and find the critical value z α /2 that corresponds to the desired confidence level. 3. Evaluate the margin of error E = p q n Slide 24
5 Procedure for Constructing a Confidence Interval for p - cont 4. Using the value of the calculated margin of error, E and the value of the sample proportion, p, find the values of p E and p + E. Substitute those values in the general format for the confidence interval: p E < p < p + E 5. Round the resulting confidence interval limits to three significant digits. a) Find the margin of error E that corresponds to a 95% confidence level. b) Find the 95% confidence interval estimate of the population proportion p. c) Based on the results, can we safely conclude that the majority of adult Minnesotans oppose use the the photo-cop? Slide 25 Slide 26 a) Find the margin of error E that corresponds to a 95% confidence level. First, we check for assumptions. We note that np = , and nq = Next, we calculate the margin of error. We have found that p = 0.51, q = = 0.49, z α/2 = 1.96, and n = 829. b) Find the 95% confidence interval for the population proportion p. We substitute our values from Part a to obtain: < p < , < p < E = 1.96 (0.51)(0.49) 829 E = Slide 27 Slide 28 c) Based on the results, can we safely conclude that the majority of adult Minnesotans oppose use of the photo-cop? Based on the survey results, we are 95% confident that the limits of 47.6% and 54.4% contain the true percentage of adult Minnesotans opposed to the photo-cop. The percentage of opposed adult Minnesotans is likely to be any value between 47.6% and 54.4%. However, a majority requires a percentage greater than 50%, so we cannot safely conclude that the majority is opposed (because the entire confidence interval is not greater than 50%). Slide 29 Sample Size Suppose we want to collect sample data with the objective of estimating some population. The question is how many sample items must be obtained? Slide 30
6 Determining Sample Size E = z α / 2 (solve for n by algebra) ( Z α / 2) 2 p q n = p q n Slide 31 Sample Size for Estimating Proportion p When an estimate of p is known: ( zα α / 2) 2 p q n = When no estimate of p is known: ( zα α / 2 ) n = Formula 7-2 Formula 7-3 Slide 32 Example: Suppose a sociologist wants to determine the current percentage of U.S. households using . How many households must be surveyed in order to be 95% confident that the sample percentage is in error by no more than four percentage points? a) Use this result from an earlier study: In 1997, 16.9% of U.S. households used (based on data from The World Almanac and Book of Facts). b) Assume that we have no prior information suggesting a possible value of p. Slide 33 Example: Suppose a sociologist wants to determine the current percentage of U.S. households using . How many households must be surveyed in order to be 95% confident that the sample percentage is in error by no more than four percentage points? a) Use this result from an earlier study: In 1997, 16.9% of U.S. households used (based on data from The World Almanac and Book of Facts). n = [z a/2 ] 2 p q = [1.96] 2 (0.169)(0.831) = = 338 households To be 95% confident that our sample percentage is within four percentage points of the true percentage for all households, we should randomly select and survey 338 households. Slide 34 Example: Suppose a sociologist wants to determine the current percentage of U.S. households using . How many households must be surveyed in order to be 95% confident that the sample percentage is in error by no more than four percentage points? b) Assume that we have no prior information suggesting a possible value of p. n = [z a/2 ] = (1.96) 2 (0.25) = = 601 households With no prior information, we need a larger sample to achieve the same results with 95% confidence and an error of no more than 4%. Slide 35 Finding the Point Estimate and E from a Confidence Interval Point estimate of p: (upper confidence limit) + (lower confidence limit) p = Margin of Error: E = (upper confidence limit) (lower confidence limit) 2 2 Slide 36
Lecture Slides. Elementary Statistics. Tenth Edition. by Mario F. Triola. and the Triola Statistics Series
Lecture Slides Elementary Statistics Tenth Edition and the Triola Statistics Series by Mario F. Triola Slide 1 Chapter 7 Estimates and Sample Sizes 7-1 Overview 7-2 Estimating a Population Proportion 7-3
More informationSections 7.1 and 7.2. This chapter presents the beginning of inferential statistics. The two major applications of inferential statistics
Sections 7.1 and 7.2 This chapter presents the beginning of inferential statistics. The two major applications of inferential statistics Estimate the value of a population parameter Test some claim (or
More informationLecture Slides. Elementary Statistics Tenth Edition. by Mario F. Triola. and the Triola Statistics Series
Lecture Slides Elementary Statistics Tenth Edition and the Triola Statistics Series by Mario F. Triola Slide 1 Chapter 7 Estimates and Sample Sizes 7-1 Overview 7-2 Estimating a Population Proportion 7-3
More informationChapter 6 Estimation and Sample Sizes
Chapter 6 Estimation and Sample Sizes This chapter presents the beginning of inferential statistics.! The two major applications of inferential statistics! Estimate the value of a population parameter!
More informationChapter 6. Estimates and Sample Sizes
Chapter 6 Estimates and Sample Sizes Lesson 6-1/6-, Part 1 Estimating a Population Proportion This chapter begins the beginning of inferential statistics. There are two major applications of inferential
More informationTwo-Sample Inferential Statistics
The t Test for Two Independent Samples 1 Two-Sample Inferential Statistics In an experiment there are two or more conditions One condition is often called the control condition in which the treatment is
More informationChapter 9 Inferences from Two Samples
Chapter 9 Inferences from Two Samples 9-1 Review and Preview 9-2 Two Proportions 9-3 Two Means: Independent Samples 9-4 Two Dependent Samples (Matched Pairs) 9-5 Two Variances or Standard Deviations Review
More informationInferential Statistics
Inferential Statistics Part 1 Sampling Distributions, Point Estimates & Confidence Intervals Inferential statistics are used to draw inferences (make conclusions/judgements) about a population from a sample.
More informationIntroduction and Descriptive Statistics p. 1 Introduction to Statistics p. 3 Statistics, Science, and Observations p. 5 Populations and Samples p.
Preface p. xi Introduction and Descriptive Statistics p. 1 Introduction to Statistics p. 3 Statistics, Science, and Observations p. 5 Populations and Samples p. 6 The Scientific Method and the Design of
More informationCh. 7 Statistical Intervals Based on a Single Sample
Ch. 7 Statistical Intervals Based on a Single Sample Before discussing the topics in Ch. 7, we need to cover one important concept from Ch. 6. Standard error The standard error is the standard deviation
More informationA3. Statistical Inference
Appendi / A3. Statistical Inference / Mean, One Sample-1 A3. Statistical Inference Population Mean μ of a Random Variable with known standard deviation σ, and random sample of size n 1 Before selecting
More informationAMS7: WEEK 7. CLASS 1. More on Hypothesis Testing Monday May 11th, 2015
AMS7: WEEK 7. CLASS 1 More on Hypothesis Testing Monday May 11th, 2015 Testing a Claim about a Standard Deviation or a Variance We want to test claims about or 2 Example: Newborn babies from mothers taking
More informationUNIVERSITY OF TORONTO MISSISSAUGA. SOC222 Measuring Society In-Class Test. November 11, 2011 Duration 11:15a.m. 13 :00p.m.
UNIVERSITY OF TORONTO MISSISSAUGA SOC222 Measuring Society In-Class Test November 11, 2011 Duration 11:15a.m. 13 :00p.m. Location: DV2074 Aids Allowed You may be charged with an academic offence for possessing
More informationEstimating a Population Mean
Estimating a Population Mean MATH 130, Elements of Statistics I J. Robert Buchanan Department of Mathematics Fall 2017 Objectives At the end of this lesson we will be able to: obtain a point estimate for
More informationSTAT100 Elementary Statistics and Probability
STAT100 Elementary Statistics and Probability Exam, Monday, August 11, 014 Solution Show all work clearly and in order, and circle your final answers. Justify your answers algebraically whenever possible.
More informationStatistics Primer. ORC Staff: Jayme Palka Peter Boedeker Marcus Fagan Trey Dejong
Statistics Primer ORC Staff: Jayme Palka Peter Boedeker Marcus Fagan Trey Dejong 1 Quick Overview of Statistics 2 Descriptive vs. Inferential Statistics Descriptive Statistics: summarize and describe data
More informationLecture #16 Thursday, October 13, 2016 Textbook: Sections 9.3, 9.4, 10.1, 10.2
STATISTICS 200 Lecture #16 Thursday, October 13, 2016 Textbook: Sections 9.3, 9.4, 10.1, 10.2 Objectives: Define standard error, relate it to both standard deviation and sampling distribution ideas. Describe
More informationBusiness Statistics: Lecture 8: Introduction to Estimation & Hypothesis Testing
Business Statistics: Lecture 8: Introduction to Estimation & Hypothesis Testing Agenda Introduction to Estimation Point estimation Interval estimation Introduction to Hypothesis Testing Concepts en terminology
More informationLecture Slides. Elementary Statistics Eleventh Edition. by Mario F. Triola. and the Triola Statistics Series 9.1-1
Lecture Slides Elementary Statistics Eleventh Edition and the Triola Statistics Series by Mario F. Triola Copyright 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 9.1-1 Chapter 9 Inferences
More informationSTA Module 10 Comparing Two Proportions
STA 2023 Module 10 Comparing Two Proportions Learning Objectives Upon completing this module, you should be able to: 1. Perform large-sample inferences (hypothesis test and confidence intervals) to compare
More informationThe Chi-Square Distributions
MATH 03 The Chi-Square Distributions Dr. Neal, Spring 009 The chi-square distributions can be used in statistics to analyze the standard deviation of a normally distributed measurement and to test the
More information- E < p. ˆ p q ˆ E = q ˆ = 1 - p ˆ = sample proportion of x failures in a sample size of n. where. x n sample proportion. population proportion
1 Chapter 7 ad 8 Review for Exam Chapter 7 Estimates ad Sample Sizes 2 Defiitio Cofidece Iterval (or Iterval Estimate) a rage (or a iterval) of values used to estimate the true value of the populatio parameter
More informationCENTRAL LIMIT THEOREM (CLT)
CENTRAL LIMIT THEOREM (CLT) A sampling distribution is the probability distribution of the sample statistic that is formed when samples of size n are repeatedly taken from a population. If the sample statistic
More informationCHAPTER 10 HYPOTHESIS TESTING WITH TWO SAMPLES
CHAPTER 10 HYPOTHESIS TESTING WITH TWO SAMPLES In this chapter our hypothesis tests allow us to compare the means (or proportions) of two different populations using a sample from each population For example,
More informationEXAM 3 Math 1342 Elementary Statistics 6-7
EXAM 3 Math 1342 Elementary Statistics 6-7 Name Date ********************************************************************************************************************************************** MULTIPLE
More informationChapter 10. Chapter 10. Multinomial Experiments and. Multinomial Experiments and Contingency Tables. Contingency Tables.
Chapter 10 Multinomial Experiments and Contingency Tables 1 Chapter 10 Multinomial Experiments and Contingency Tables 10-1 1 Overview 10-2 2 Multinomial Experiments: of-fitfit 10-3 3 Contingency Tables:
More information11-2 Multinomial Experiment
Chapter 11 Multinomial Experiments and Contingency Tables 1 Chapter 11 Multinomial Experiments and Contingency Tables 11-11 Overview 11-2 Multinomial Experiments: Goodness-of-fitfit 11-3 Contingency Tables:
More informationSection 7.2 Homework Answers
25.5 30 Sample Mean P 0.1226 sum n b. The two z-scores are z 25 20(1.7) n 1.0 20 sum n 2.012 and z 30 20(1.7) n 1.0 0.894, 20 so the probability is approximately 0.1635 (0.1645 using Table A). P14. a.
More informationStochastic calculus for summable processes 1
Stochastic calculus for summable processes 1 Lecture I Definition 1. Statistics is the science of collecting, organizing, summarizing and analyzing the information in order to draw conclusions. It is a
More informationChapter 8: Confidence Intervals
Chapter 8: Confidence Intervals Introduction Suppose you are trying to determine the mean rent of a two-bedroom apartment in your town. You might look in the classified section of the newspaper, write
More informationSingle Sample Means. SOCY601 Alan Neustadtl
Single Sample Means SOCY601 Alan Neustadtl The Central Limit Theorem If we have a population measured by a variable with a mean µ and a standard deviation σ, and if all possible random samples of size
More informationCH.8 Statistical Intervals for a Single Sample
CH.8 Statistical Intervals for a Single Sample Introduction Confidence interval on the mean of a normal distribution, variance known Confidence interval on the mean of a normal distribution, variance unknown
More informationYou are allowed 3? sheets of notes and a calculator.
Exam 1 is Wed Sept You are allowed 3? sheets of notes and a calculator The exam covers survey sampling umbers refer to types of problems on exam A population is the entire set of (potential) measurements
More informationIntroduction to Statistical Data Analysis Lecture 5: Confidence Intervals
Introduction to Statistical Data Analysis Lecture 5: Confidence Intervals James V. Lambers Department of Mathematics The University of Southern Mississippi James V. Lambers Statistical Data Analysis 1
More informationIntroduction to Statistical Data Analysis Lecture 4: Sampling
Introduction to Statistical Data Analysis Lecture 4: Sampling James V. Lambers Department of Mathematics The University of Southern Mississippi James V. Lambers Statistical Data Analysis 1 / 30 Introduction
More informationhypotheses. P-value Test for a 2 Sample z-test (Large Independent Samples) n > 30 P-value Test for a 2 Sample t-test (Small Samples) n < 30 Identify α
Chapter 8 Notes Section 8-1 Independent and Dependent Samples Independent samples have no relation to each other. An example would be comparing the costs of vacationing in Florida to the cost of vacationing
More informationLecture Slides. Elementary Statistics. by Mario F. Triola. and the Triola Statistics Series
Lecture Slides Elementary Statistics Tenth Edition and the Triola Statistics Series by Mario F. Triola Slide 1 Chapter 9 Inferences from Two Samples 9-1 Overview 9-2 Inferences About Two Proportions 9-3
More informationAn inferential procedure to use sample data to understand a population Procedures
Hypothesis Test An inferential procedure to use sample data to understand a population Procedures Hypotheses, the alpha value, the critical region (z-scores), statistics, conclusion Two types of errors
More information1/18/2011. Chapter 6: Probability. Introduction to Probability. Probability Definition
Chapter 6: Probability Introduction to Probability The role of inferential statistics is to use the sample data as the basis for answering questions about the population. To accomplish this goal, inferential
More information1) What is the probability that the random variable has a value less than 3? 1)
Ch 6 and 7 Worksheet Disclaimer; The actual exam differs NOTE: ON THIS TEST YOU WILL NEED TO USE TABLES (NOT YOUR CALCULATOR) TO FIND PROBABILITIES UNDER THE NORMAL OR CHI SQUARED OR T DISTRIBUTION! SHORT
More informationPercentage point z /2
Chapter 8: Statistical Intervals Why? point estimate is not reliable under resampling. Interval Estimates: Bounds that represent an interval of plausible values for a parameter There are three types of
More informationA proportion is the fraction of individuals having a particular attribute. Can range from 0 to 1!
Proportions A proportion is the fraction of individuals having a particular attribute. It is also the probability that an individual randomly sampled from the population will have that attribute Can range
More informationStatistics for Business and Economics
Statistics for Business and Economics Chapter 6 Sampling and Sampling Distributions Ch. 6-1 6.1 Tools of Business Statistics n Descriptive statistics n Collecting, presenting, and describing data n Inferential
More informationBinomial and Poisson Probability Distributions
Binomial and Poisson Probability Distributions Esra Akdeniz March 3, 2016 Bernoulli Random Variable Any random variable whose only possible values are 0 or 1 is called a Bernoulli random variable. What
More informationHypothesis Testing. ECE 3530 Spring Antonio Paiva
Hypothesis Testing ECE 3530 Spring 2010 Antonio Paiva What is hypothesis testing? A statistical hypothesis is an assertion or conjecture concerning one or more populations. To prove that a hypothesis is
More informationReview 6. n 1 = 85 n 2 = 75 x 1 = x 2 = s 1 = 38.7 s 2 = 39.2
Review 6 Use the traditional method to test the given hypothesis. Assume that the samples are independent and that they have been randomly selected ) A researcher finds that of,000 people who said that
More informationChapter 8 - Statistical intervals for a single sample
Chapter 8 - Statistical intervals for a single sample 8-1 Introduction In statistics, no quantity estimated from data is known for certain. All estimated quantities have probability distributions of their
More informationCh. 7: Estimates and Sample Sizes
Ch. 7: Estimates and Sample Sizes Section Title Notes Pages Introduction to the Chapter 2 2 Estimating p in the Binomial Distribution 2 5 3 Estimating a Population Mean: Sigma Known 6 9 4 Estimating a
More informationBusiness Statistics. Lecture 5: Confidence Intervals
Business Statistics Lecture 5: Confidence Intervals Goals for this Lecture Confidence intervals The t distribution 2 Welcome to Interval Estimation! Moments Mean 815.0340 Std Dev 0.8923 Std Error Mean
More informationStatistical Inference for Means
Statistical Inference for Means Jamie Monogan University of Georgia February 18, 2011 Jamie Monogan (UGA) Statistical Inference for Means February 18, 2011 1 / 19 Objectives By the end of this meeting,
More informationDeciphering Math Notation. Billy Skorupski Associate Professor, School of Education
Deciphering Math Notation Billy Skorupski Associate Professor, School of Education Agenda General overview of data, variables Greek and Roman characters in math and statistics Parameters vs. Statistics
More information16.400/453J Human Factors Engineering. Design of Experiments II
J Human Factors Engineering Design of Experiments II Review Experiment Design and Descriptive Statistics Research question, independent and dependent variables, histograms, box plots, etc. Inferential
More informationLecture 11. Data Description Estimation
Lecture 11 Data Description Estimation Measures of Central Tendency (continued, see last lecture) Sample mean, population mean Sample mean for frequency distributions The median The mode The midrange 3-22
More informationProbability and Statistics
Probability and Statistics Kristel Van Steen, PhD 2 Montefiore Institute - Systems and Modeling GIGA - Bioinformatics ULg kristel.vansteen@ulg.ac.be CHAPTER 4: IT IS ALL ABOUT DATA 4a - 1 CHAPTER 4: IT
More informationLast week: Sample, population and sampling distributions finished with estimation & confidence intervals
Past weeks: Measures of central tendency (mean, mode, median) Measures of dispersion (standard deviation, variance, range, etc). Working with the normal curve Last week: Sample, population and sampling
More informationChapter 5 Confidence Intervals
Chapter 5 Confidence Intervals Confidence Intervals about a Population Mean, σ, Known Abbas Motamedi Tennessee Tech University A point estimate: a single number, calculated from a set of data, that is
More informationIntroduction to Statistics Tutorial: Large-Sample Estimation. INCOGEN, Inc. 2008
Introduction to Statistics Tutorial: Large-Sample Estimation INCOGEN, Inc. 2008 Outline Types of estimators Evaluating goodness of an estimator Calculating confidence intervals Choosing sample size Estimating
More informationChapter 22. Comparing Two Proportions. Bin Zou STAT 141 University of Alberta Winter / 15
Chapter 22 Comparing Two Proportions Bin Zou (bzou@ualberta.ca) STAT 141 University of Alberta Winter 2015 1 / 15 Introduction In Ch.19 and Ch.20, we studied confidence interval and test for proportions,
More informationThe Chi-Square Distributions
MATH 183 The Chi-Square Distributions Dr. Neal, WKU The chi-square distributions can be used in statistics to analyze the standard deviation σ of a normally distributed measurement and to test the goodness
More informationPsychology 282 Lecture #4 Outline Inferences in SLR
Psychology 282 Lecture #4 Outline Inferences in SLR Assumptions To this point we have not had to make any distributional assumptions. Principle of least squares requires no assumptions. Can use correlations
More informationDover- Sherborn High School Mathematics Curriculum Probability and Statistics
Mathematics Curriculum A. DESCRIPTION This is a full year courses designed to introduce students to the basic elements of statistics and probability. Emphasis is placed on understanding terminology and
More informationSurvey of Smoking Behavior. Survey of Smoking Behavior. Survey of Smoking Behavior
Sample HH from Frame HH One-Stage Cluster Survey Population Frame Sample Elements N =, N =, n = population smokes Sample HH from Frame HH Elementary units are different from sampling units Sampled HH but
More informationLecture on Null Hypothesis Testing & Temporal Correlation
Lecture on Null Hypothesis Testing & Temporal Correlation CS 590.21 Analysis and Modeling of Brain Networks Department of Computer Science University of Crete Acknowledgement Resources used in the slides
More information1 MA421 Introduction. Ashis Gangopadhyay. Department of Mathematics and Statistics. Boston University. c Ashis Gangopadhyay
1 MA421 Introduction Ashis Gangopadhyay Department of Mathematics and Statistics Boston University c Ashis Gangopadhyay 1.1 Introduction 1.1.1 Some key statistical concepts 1. Statistics: Art of data analysis,
More informationAssessment Schedule 2011 Statistics and Modelling: Calculate confidence intervals for population parameters (90642)
NCEA Level 3 Statistics & Modelling (90642) 2011 page 1 of 5 Assessment Schedule 2011 Statistics and Modelling: Calculate confidence intervals for population parameters (90642) Evidence Statement Q Achievement
More informationREVIEW: Midterm Exam. Spring 2012
REVIEW: Midterm Exam Spring 2012 Introduction Important Definitions: - Data - Statistics - A Population - A census - A sample Types of Data Parameter (Describing a characteristic of the Population) Statistic
More informationChapter 7: Hypothesis Testing
Chapter 7: Hypothesis Testing *Mathematical statistics with applications; Elsevier Academic Press, 2009 The elements of a statistical hypothesis 1. The null hypothesis, denoted by H 0, is usually the nullification
More informationSection 6-1 Overview. Definition. Definition. Using Area to Find Probability. Area and Probability
Chapter focus is on: Continuous random variables Normal distributions Figure 6-1 Section 6-1 Overview ( -1 e 2 x-µ σ ) 2 f(x) = σ 2 π Formula 6-1 Slide 1 Section 6-2 The Standard Normal Distribution Key
More informationSTAT100 Elementary Statistics and Probability
STAT100 Elementary Statistics and Probability Exam, Sample Test, Summer 014 Solution Show all work clearly and in order, and circle your final answers. Justify your answers algebraically whenever possible.
More informationProbability theory and inference statistics! Dr. Paola Grosso! SNE research group!! (preferred!)!!
Probability theory and inference statistics Dr. Paola Grosso SNE research group p.grosso@uva.nl paola.grosso@os3.nl (preferred) Roadmap Lecture 1: Monday Sep. 22nd Collecting data Presenting data Descriptive
More informationNotes 3: Statistical Inference: Sampling, Sampling Distributions Confidence Intervals, and Hypothesis Testing
Notes 3: Statistical Inference: Sampling, Sampling Distributions Confidence Intervals, and Hypothesis Testing 1. Purpose of statistical inference Statistical inference provides a means of generalizing
More informationSalt Lake Community College MATH 1040 Final Exam Fall Semester 2011 Form E
Salt Lake Community College MATH 1040 Final Exam Fall Semester 011 Form E Name Instructor Time Limit: 10 minutes Any hand-held calculator may be used. Computers, cell phones, or other communication devices
More informationPubH 5450 Biostatistics I Prof. Carlin. Lecture 13
PubH 5450 Biostatistics I Prof. Carlin Lecture 13 Outline Outline Sample Size Counts, Rates and Proportions Part I Sample Size Type I Error and Power Type I error rate: probability of rejecting the null
More information37.3. The Poisson Distribution. Introduction. Prerequisites. Learning Outcomes
The Poisson Distribution 37.3 Introduction In this Section we introduce a probability model which can be used when the outcome of an experiment is a random variable taking on positive integer values and
More informationPOLI 443 Applied Political Research
POLI 443 Applied Political Research Session 4 Tests of Hypotheses The Normal Curve Lecturer: Prof. A. Essuman-Johnson, Dept. of Political Science Contact Information: aessuman-johnson@ug.edu.gh College
More informationLast two weeks: Sample, population and sampling distributions finished with estimation & confidence intervals
Past weeks: Measures of central tendency (mean, mode, median) Measures of dispersion (standard deviation, variance, range, etc). Working with the normal curve Last two weeks: Sample, population and sampling
More informationVisual interpretation with normal approximation
Visual interpretation with normal approximation H 0 is true: H 1 is true: p =0.06 25 33 Reject H 0 α =0.05 (Type I error rate) Fail to reject H 0 β =0.6468 (Type II error rate) 30 Accept H 1 Visual interpretation
More information# of 6s # of times Test the null hypthesis that the dice are fair at α =.01 significance
Practice Final Exam Statistical Methods and Models - Math 410, Fall 2011 December 4, 2011 You may use a calculator, and you may bring in one sheet (8.5 by 11 or A4) of notes. Otherwise closed book. The
More informationDay 8: Sampling. Daniel J. Mallinson. School of Public Affairs Penn State Harrisburg PADM-HADM 503
Day 8: Sampling Daniel J. Mallinson School of Public Affairs Penn State Harrisburg mallinson@psu.edu PADM-HADM 503 Mallinson Day 8 October 12, 2017 1 / 46 Road map Why Sample? Sampling terminology Probability
More informationThe point value of each problem is in the left-hand margin. You must show your work to receive any credit, except on problems 1 & 2. Work neatly.
Introduction to Statistics Math 1040 Sample Exam III Chapters 8-10 4 Problem Pages 3 Formula/Table Pages Time Limit: 90 Minutes 1 No Scratch Paper Calculator Allowed: Scientific Name: The point value of
More informationLecture 7: Confidence interval and Normal approximation
Lecture 7: Confidence interval and Normal approximation 26th of November 2015 Confidence interval 26th of November 2015 1 / 23 Random sample and uncertainty Example: we aim at estimating the average height
More informationMath 1040 Final Exam Form A Introduction to Statistics Fall Semester 2010
Math 1040 Final Exam Form A Introduction to Statistics Fall Semester 2010 Instructor Name Time Limit: 120 minutes Any calculator is okay. Necessary tables and formulas are attached to the back of the exam.
More informationReview. A Bernoulli Trial is a very simple experiment:
Review A Bernoulli Trial is a very simple experiment: Review A Bernoulli Trial is a very simple experiment: two possible outcomes (success or failure) probability of success is always the same (p) the
More informationCh. 1: Data and Distributions
Ch. 1: Data and Distributions Populations vs. Samples How to graphically display data Histograms, dot plots, stem plots, etc Helps to show how samples are distributed Distributions of both continuous and
More informationgreen green green/green green green yellow green/yellow green yellow green yellow/green green yellow yellow yellow/yellow yellow
CHAPTER PROBLEM Did Mendel s results from plant hybridization experiments contradict his theory? Gregor Mendel conducted original experiments to study the genetic traits of pea plants. In 1865 he wrote
More informationQUIZ 4 (CHAPTER 7) - SOLUTIONS MATH 119 SPRING 2013 KUNIYUKI 105 POINTS TOTAL, BUT 100 POINTS = 100%
QUIZ 4 (CHAPTER 7) - SOLUTIONS MATH 119 SPRING 013 KUNIYUKI 105 POINTS TOTAL, BUT 100 POINTS = 100% 1) We want to conduct a study to estimate the mean I.Q. of a pop singer s fans. We want to have 96% confidence
More informationPoint Estimation and Confidence Interval
Chapter 8 Point Estimation and Confidence Interval 8.1 Point estimator The purpose of point estimation is to use a function of the sample data to estimate the unknown parameter. Definition 8.1 A parameter
More informationTest 3 SOLUTIONS. x P(x) xp(x)
16 1. A couple of weeks ago in class, each of you took three quizzes where you randomly guessed the answers to each question. There were eight questions on each quiz, and four possible answers to each
More informationChapter 23. Inferences About Means. Monday, May 6, 13. Copyright 2009 Pearson Education, Inc.
Chapter 23 Inferences About Means Sampling Distributions of Means Now that we know how to create confidence intervals and test hypotheses about proportions, we do the same for means. Just as we did before,
More informationInferences About Two Population Proportions
Inferences About Two Population Proportions MATH 130, Elements of Statistics I J. Robert Buchanan Department of Mathematics Fall 2018 Background Recall: for a single population the sampling proportion
More informationSpecial distributions
Special distributions August 22, 2017 STAT 101 Class 4 Slide 1 Outline of Topics 1 Motivation 2 Bernoulli and binomial 3 Poisson 4 Uniform 5 Exponential 6 Normal STAT 101 Class 4 Slide 2 What distributions
More informationLecture Slides. Elementary Statistics Tenth Edition. by Mario F. Triola. and the Triola Statistics Series. Slide 1
Lecture Slides Elementary Statistics Tenth Edition and the Triola Statistics Series by Mario F. Triola Slide 1 Chapter 10 Correlation and Regression 10-1 Overview 10-2 Correlation 10-3 Regression 10-4
More informationThe t-statistic. Student s t Test
The t-statistic 1 Student s t Test When the population standard deviation is not known, you cannot use a z score hypothesis test Use Student s t test instead Student s t, or t test is, conceptually, very
More informationOne-sample categorical data: approximate inference
One-sample categorical data: approximate inference Patrick Breheny October 6 Patrick Breheny Biostatistical Methods I (BIOS 5710) 1/25 Introduction It is relatively easy to think about the distribution
More informationInferential statistics
Inferential statistics Inference involves making a Generalization about a larger group of individuals on the basis of a subset or sample. Ahmed-Refat-ZU Null and alternative hypotheses In hypotheses testing,
More information23.3. Sampling Distributions. Engage Sampling Distributions. Learning Objective. Math Processes and Practices. Language Objective
23.3 Sampling Distributions Essential Question: How is the mean of a sampling distribution related to the corresponding population mean or population proportion? Explore 1 Developing a Distribution of
More informationExam III #1 Solutions
Department of Mathematics University of Notre Dame Math 10120 Finite Math Fall 2017 Name: Instructors: Basit & Migliore Exam III #1 Solutions November 14, 2017 This exam is in two parts on 11 pages and
More informationExample: Four levels of herbicide strength in an experiment on dry weight of treated plants.
The idea of ANOVA Reminders: A factor is a variable that can take one of several levels used to differentiate one group from another. An experiment has a one-way, or completely randomized, design if several
More informationLecture Slides. Elementary Statistics Tenth Edition. by Mario F. Triola. and the Triola Statistics Series. Slide 1
Lecture Slides Elementary Statistics Tenth Edition and the Triola Statistics Series by Mario F. Triola Slide 1 4-1 Overview 4-2 Fundamentals 4-3 Addition Rule Chapter 4 Probability 4-4 Multiplication Rule:
More informationCIVL /8904 T R A F F I C F L O W T H E O R Y L E C T U R E - 8
CIVL - 7904/8904 T R A F F I C F L O W T H E O R Y L E C T U R E - 8 Chi-square Test How to determine the interval from a continuous distribution I = Range 1 + 3.322(logN) I-> Range of the class interval
More information