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


 Scot Newman
 1 years ago
 Views:
Transcription
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 as CI.
2 Confidence level A confidence level is the probability 1 α (often expressed as the equivalent percentage value) that the confidence interval actually does contain the population parameter, assuming that the estimation process is repeated a large number of times. (The value α is later called significance level.) Most common choices are 90%, 95%, or 99%. (α = 10%), (α = 5%), (α = 1%)
3 Student t Distribution Methods for estimating a population mean is discussed when the population standard deviation is not known. With the standard s deviation unknown, we use the Student t distribution assuming that (I) data come from a normal distribution, or (II) data size is at least 30.
4 Student t Distribution If the distribution of a population is essentially normal, then the distribution of t x s n is a Student t Distribution for all samples of size n. It is often referred to as a t distribution and is used to find critical values denoted by tα.
5 Student t Distributions for n = 3 and n = 12 Figure 75
6 Degrees of freedom The number of degrees of freedom for a collection of sample data is the number of sample values that can vary after certain restrictions have been imposed on all data values. The degree of freedom is often abbreviated df. degrees of freedom = n 1 in this section.
7 Critical Values The value separating the righttail region is commonly denoted by tα and is referred to as a critical value because it is on the borderline separating values from a specified distribution that are likely to occur from those that are unlikely to occur.
8 Important Properties of the Student t Distribution 1. The Student t distribution is different for different sample sizes (see the previous slide, for the cases n = 3 and n = 12). 2. The Student t distribution has the same general symmetric bell shape as the standard normal distribution but it reflects the greater variability (with wider distributions) that is expected with small samples. 3. The Student t distribution has a mean of t = 0 (just as the standard normal distribution has a mean of z = 0). 4. The standard deviation of the Student t distribution varies with the sample size and is greater than 1 (unlike the standard normal distribution, which has a σ = 1). 5. As the sample size n gets larger, the Student t distribution gets closer to the normal distribution.
9 (1α)% Confidence Interval for a m Population Mean (σ Not Known) x E x E where E = t a / 2 s n tα/2 can be found in tdistribution table with df = n 1
10 Margin of Error E for a population m mean (With σ Not Known) s E = t a / 2 n where tα/2 has n 1 degrees of freedom.
11 Procedure for Constructing a Confidence Interval for a Population Mean (With σ Unknown) 1. Verify that the requirements are satisfied. 2. Using n 1 degrees of freedom, find the critical value tα/2 that corresponds to the desired confidence level. m 3. Evaluate the margin of error E = t a / 2 s n 4. Substitute those values in the general format for the confidence interval: x E x E 5. Round the resulting confidence interval limits.
12 Example: A common claim is that garlic lowers cholesterol levels. In a test of the effectiveness of garlic, 49 subjects were treated with doses of raw garlic, and their cholesterol levels were measured before and after the treatment. The changes in their levels of cholesterol (in mg/dl) have a mean of 0.4 and a standard deviation of Use the sample statistics of n = 49, x = 0.4 and s = 21.0 to construct a 95% confidence interval estimate of the mean net change in LDL cholesterol after the garlic treatment. What does the confidence interval suggest about the effectiveness of garlic in reducing cholesterol?
13 Example: Requirements are satisfied: independent sample data with n = 49 (i.e., n > 30). 95% implies α = With n = 49, the df = 49 1 = 48 Closest df is 50, two tails, so tα/2 = Using tα/2 = 2.009, s = 21.0 and n = 49 the margin of error is: E s = t = = a 2 n 49
14 Example: Construct the confidence interval: x E x E Because the confidence interval limits contain the value of 0, it is very possible that the mean of the changes in cholesterol is equal to 0, suggesting that the garlic treatment did not affect the cholesterol levels. It does not appear that the garlic treatment is effective in lowering cholesterol.
15 Inference on two groups Confidence interval for the difference between two groups uses sample data from two independent samples, and tests hypotheses made about two population means μ 1 and μ 2, or simply shows confidence interval estimates of the difference μ 1 μ 2 between two population means.
16 Example: A headline in USA Today proclaimed that Men, women are equal talkers. That headline referred to a study of the numbers of words that samples of men and women spoke in a day. Construct a 95% confidence interval estimate of the difference between the mean number of words spoken by men and the mean number of words spoken by women.
17 Student t Distribution If a distribution of each group is essentially normal, then the distribution of x x t s 1 2 n 1 s 2 2 n 2 approximately follows a Student t Distribution.
18 Confidence Interval Estimate of μ 1 μ 2 : Independent Samples ( x 1 x 2 ) E < ( μ 1 μ 2 ) < ( x 1 x 2 ) + E where E 2 s s = t + a 2 n n 1 2 where df = smaller than n 1 1 and n 2 1
19 Two population standard deviations are not known and not assumed to be equal, independent samples. Find the margin of Error, E; use t /2 = Construct the confidence interval use E = and x 1 15,668.5 and x 2 16, < ( μ 1 μ 2 ) <
20 Standard Error The standard error of estimate, denoted by SE is a measure of the deviation (or standard deviation) between the parameter θ of interest and the estimate that is obtained from the observed sample.
21 Confidence Interval The confidence interval, given a confidence level (1 α), is an interval which includes the parameter θ of interest. It is constructed from the estimate θ ^ and the standard error SE. ^ ^ θ t SE < θ < θ + t SE 2 2
22 Standard Error of Estimate SE 0 = 1 n +  x 2  (x x) 2 and SE = (y ^y) 2 / (n 2) 2  (x x) 2
23 Prediction Interval for parameters ^ ^ b  t SE < b < b + t SE ^ ^ b  t SE < b < b + t SE b 0 and b 1 represent the ture values for coefficents, and t 2 freedom has n 2 degrees of
Third Midterm Exam (MATH1070 Spring 2012)
Third Midterm Exam (MATH1070 Spring 2012) Instructions: This is a one hour exam. You can use a notesheet. Calculators are allowed, but other electronics are prohibited. 1. [40pts] Multiple Choice Problems
More informationWording of Final Conclusion. Slide 1
Wording of Final Conclusion Slide 1 8.3: Assumptions for Testing Slide 2 Claims About Population Means 1) The sample is a simple random sample. 2) The value of the population standard deviation σ is known
More informationChapter 7. Estimates and Sample Size
Chapter 7. Estimates and Sample Size Chapter Problem: How do we interpret a poll about global warming? Pew Research Center Poll: From what you ve read and heard, is there a solid evidence that the average
More informationBasic 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
More informationConfidence Intervals about a Population Mean
Confidence Intervals about a Population Mean MATH 130, Elements of Statistics I J. Robert Buchanan Department of Mathematics Fall 2015 Motivation Goal: to estimate a population mean µ based on data collected
More informationSection 7.2 Confidence Intervals for Population Proportions
Section 7.2 Confidence Intervals for Population Proportions 2012 Pearson Education, Inc. All rights reserved. 1 of 83 Section 7.2 Objectives Find a point estimate for the population proportion Construct
More informationfind confidence interval for a population mean when the population standard deviation is KNOWN Understand the new distribution the tdistribution
Section 8.3 1 Estimating a Population Mean Topics find confidence interval for a population mean when the population standard deviation is KNOWN find confidence interval for a population mean when the
More informationCONFIDENCE INTERVALS ON µ WHEN σ IS UNKNOWN
CONFIDENCE INTERVALS ON µ WHEN σ IS UNKNOWN A. Introduction 1. this situation, where we do not know anything about the population but the sample characteristics, is far and away the most common circumstance
More information
An interval estimate (confidence interval) is an interval, or range of values, used to estimate a population parameter. For example 0.476
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.
More information8.2 Confidence Intervals for One Population Mean When σ is Known
8.2 Confidence Intervals for One Population Mean When σ is Known Tom Lewis Fall Term 2009 8.2 Confidence Intervals for One Population Mean When σ isfall Known Term 2009 1 / 6 Outline 1 An example 2 Finding
More informationLet m denote the margin of error. Then
S:105 Statistical Methods and Computing Sample size for confidence intervals with σ known t Intervals Lecture 13 Mar. 6, 009 Kate Cowles 374 SH, 335077 kcowles@stat.uiowa.edu 1 The margin of error The
More informationStatistics for Management IISTAT 362Final Review
Statistics for Management IISTAT 362Final Review Multiple Choice Identify the letter of the choice that best completes the statement or answers the question. 1. The ability of an interval estimate to
More information1 Confidence intervals
Math 143 Inference for Means 1 Statistical inference is inferring information about the distribution of a population from information about a sample. We re generally talking about one of two things: 1.
More informationMCQ TESTING OF HYPOTHESIS
MCQ TESTING OF HYPOTHESIS MCQ 13.1 A statement about a population developed for the purpose of testing is called: (a) Hypothesis (b) Hypothesis testing (c) Level of significance (d) Teststatistic MCQ
More informationStatistical 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
More informationMAT140: Applied Statistical Methods Summary of Calculating Confidence Intervals and Sample Sizes for Estimating Parameters
MAT140: Applied Statistical Methods Summary of Calculating Confidence Intervals and Sample Sizes for Estimating Parameters Inferences about a population parameter can be made using sample statistics for
More informationThe calculations lead to the following values: d 2 = 46, n = 8, s d 2 = 4, s d = 2, SEof d = s d n s d n
EXAMPLE 1: Paired ttest and tinterval DBP Readings by Two Devices The diastolic blood pressures (DBP) of 8 patients were determined using two techniques: the standard method used by medical personnel
More informationLecture Topic 6: Chapter 9 Hypothesis Testing
Lecture Topic 6: Chapter 9 Hypothesis Testing 9.1 Developing Null and Alternative Hypotheses Hypothesis testing can be used to determine whether a statement about the value of a population parameter should
More informationHypothesis Testing with One Sample. Introduction to Hypothesis Testing 7.1. Hypothesis Tests. Chapter 7
Chapter 7 Hypothesis Testing with One Sample 71 Introduction to Hypothesis Testing Hypothesis Tests A hypothesis test is a process that uses sample statistics to test a claim about the value of a population
More informationHypothesis Testing. Concept of Hypothesis Testing
Quantitative Methods 2013 Hypothesis Testing with One Sample 1 Concept of Hypothesis Testing Testing Hypotheses is another way to deal with the problem of making a statement about an unknown population
More informationCh. 8 Hypothesis Testing
Ch. 8 Hypothesis Testing 8.1 Foundations of Hypothesis Testing Definitions In statistics, a hypothesis is a claim about a property of a population. A hypothesis test is a standard procedure for testing
More informationElements of Hypothesis Testing (Summary from lecture notes)
Statistics20090 MINITAB  Lab 1 Large Sample Tests of Hypothesis About a Population Mean We use hypothesis tests to make an inference about some population parameter of interest, for example the mean
More informationMAT X Hypothesis Testing  Part I
MAT 2379 3X Hypothesis Testing  Part I Definition : A hypothesis is a conjecture concerning a value of a population parameter (or the shape of the population). The hypothesis will be tested by evaluating
More informationPower 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,
More informationI. Basics of Hypothesis Testing
Introduction to Hypothesis Testing This deals with an issue highly similar to what we did in the previous chapter. In that chapter we used sample information to make inferences about the range of possibilities
More informationStatistical Intervals. Chapter 7 Stat 4570/5570 Material from Devore s book (Ed 8), and Cengage
7 Statistical Intervals Chapter 7 Stat 4570/5570 Material from Devore s book (Ed 8), and Cengage Confidence Intervals The CLT tells us that as the sample size n increases, the sample mean X is close to
More information0.1 Estimating and Testing Differences in the Treatment means
0. Estimating and Testing Differences in the Treatment means Is the Ftest significant, we only learn that not all population means are the same, but through this test we can not determine where the differences
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 informationSociology 6Z03 Topic 15: Statistical Inference for Means
Sociology 6Z03 Topic 15: Statistical Inference for Means John Fox McMaster University Fall 2016 John Fox (McMaster University) Soc 6Z03: Statistical Inference for Means Fall 2016 1 / 41 Outline: Statistical
More informationOLS is not only unbiased it is also the most precise (efficient) unbiased estimation technique  ie the estimator has the smallest variance
Lecture 5: Hypothesis Testing What we know now: OLS is not only unbiased it is also the most precise (efficient) unbiased estimation technique  ie the estimator has the smallest variance (if the GaussMarkov
More information1 Hypotheses test about µ if σ is not known
1 Hypotheses test about µ if σ is not known In this section we will introduce how to make decisions about a population mean, µ, when the standard deviation is not known. In order to develop a confidence
More information2. The mean birthweights from this hospital is lower than 120 oz.
Probability February 13, 2013 Debdeep Pati Hypothesis testing Two sample inference Obstetrics: Mothers with low socioeconomic status (SES) deliver babies with lower than normal birthweights. The mean birthweights
More informationExample for testing one population mean:
Today: Sections 13.1 to 13.3 ANNOUNCEMENTS: We will finish hypothesis testing for the 5 situations today. See pages 586587 (end of Chapter 13) for a summary table. Quiz for week 8 starts Wed, ends Monday
More informationCONFIDENCE INTERVALS I
CONFIDENCE INTERVALS I ESTIMATION: the sample mean Gx is an estimate of the population mean µ point of sampling is to obtain estimates of population values Example: for 55 students in Section 105, 45 of
More informationEstimation; Sampling; The T distribution
Estimation; Sampling; The T distribution I. Estimation A. In most statistical studies, the population parameters are unknown and must be estimated. Therefore, developing methods for estimating as accurately
More informationCONFIDENCE INTERVALS FOR MEANS AND PROPORTIONS
LESSON SEVEN CONFIDENCE INTERVALS FOR MEANS AND PROPORTIONS An interval estimate for μ of the form a margin of error would provide the user with a measure of the uncertainty associated with the point estimate.
More informationWhen σ Is Known: Recall the Mystery Mean Activity where x bar = 240.79 and we have an SRS of size 16
8.3 ESTIMATING A POPULATION MEAN When σ Is Known: Recall the Mystery Mean Activity where x bar = 240.79 and we have an SRS of size 16 Task was to estimate the mean when we know that the situation is Normal
More informationProbability and Statistics Lecture 9: 1 and 2Sample 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
More information12.1 Inference for Linear Regression
12.1 Inference for Linear Regression Least Squares Regression Line y = a + bx You might want to refresh your memory of LSR lines by reviewing Chapter 3! 1 Sample Distribution of b p740 Shape Center Spread
More informationChapter 7. Section Introduction to Hypothesis Testing
Section 7.1  Introduction to Hypothesis Testing Chapter 7 Objectives: State a null hypothesis and an alternative hypothesis Identify type I and type II errors and interpret the level of significance Determine
More informationIn Chapter 2, we used linear regression to describe linear relationships. The setting for this is a
Math 143 Inference on Regression 1 Review of Linear Regression In Chapter 2, we used linear regression to describe linear relationships. The setting for this is a bivariate data set (i.e., a list of cases/subjects
More informationHypothesis 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
More information6.1 The Elements of a Test of Hypothesis
University of California, Davis Department of Statistics Summer Session II Statistics 13 August 22, 2012 Date of latest update: August 20 Lecture 6: Tests of Hypothesis Suppose you wanted to determine
More informationThe Basics of a Hypothesis Test
Overview The Basics of a Test Dr Tom Ilvento Department of Food and Resource Economics Alternative way to make inferences from a sample to the Population is via a Test A hypothesis test is based upon A
More informationSampling (cont d) and Confidence Intervals Lecture 9 8 March 2006 R. Ryznar
Sampling (cont d) and Confidence Intervals 11.220 Lecture 9 8 March 2006 R. Ryznar Census Surveys Decennial Census Every (over 11 million) household gets the short form and 17% or 1/6 get the long form
More informationSample Size Determination
Sample Size Determination Population A: 10,000 Population B: 5,000 Sample 10% Sample 15% Sample size 1000 Sample size 750 The process of obtaining information from a subset (sample) of a larger group (population)
More information9.1 (a) The standard deviation of the four sample differences is given as.68. The standard error is SE (ȳ1  ȳ 2 ) = SE d  = s d n d
CHAPTER 9 Comparison of Paired Samples 9.1 (a) The standard deviation of the four sample differences is given as.68. The standard error is SE (ȳ1  ȳ 2 ) = SE d  = s d n d =.68 4 =.34. (b) H 0 : The mean
More informationTesting: is my coin fair?
Testing: is my coin fair? Formally: we want to make some inference about P(head) Try it: toss coin several times (say 7 times) Assume that it is fair ( P(head)= ), and see if this assumption is compatible
More informationUnit 26: Small Sample Inference for One Mean
Unit 26: Small Sample Inference for One Mean Prerequisites Students need the background on confidence intervals and significance tests covered in Units 24 and 25. Additional Topic Coverage Additional coverage
More informationSampling Distribution of a Normal Variable
Ismor Fischer, 5/9/01 5.1 5. Formal Statement and Examples Comments: Sampling Distribution of a Normal Variable Given a random variable. Suppose that the population distribution of is known to be normal,
More informationChapter 11: Linear Regression  Inference in Regression Analysis  Part 2
Chapter 11: Linear Regression  Inference in Regression Analysis  Part 2 Note: Whether we calculate confidence intervals or perform hypothesis tests we need the distribution of the statistic we will use.
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 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 informationSampling & Confidence Intervals
Sampling & Confidence Intervals Mark Lunt Arthritis Research UK Centre for Excellence in Epidemiology University of Manchester 18/10/2016 Principles of Sampling Often, it is not practical to measure every
More informationInference in Regression Analysis. Dr. Frank Wood
Inference in Regression Analysis Dr. Frank Wood Inference in the Normal Error Regression Model Y i = β 0 + β 1 X i + ɛ i Y i value of the response variable in the i th trial β 0 and β 1 are parameters
More informationA correlation exists between two variables when one of them is related to the other in some way.
Lecture #10 Chapter 10 Correlation and Regression The main focus of this chapter is to form inferences based on sample data that come in pairs. Given such paired sample data, we want to determine whether
More informationChapter Five. Hypothesis Testing: Concepts
Chapter Five The Purpose of Hypothesis Testing... 110 An Initial Look at Hypothesis Testing... 112 Formal Hypothesis Testing... 114 Introduction... 114 Null and Alternate Hypotheses... 114 Procedure for
More informationChapter 8 Hypothesis Testing
Chapter 8 Hypothesis Testing Chapter problem: Does the MicroSort method of gender selection increase the likelihood that a baby will be girl? MicroSort: a genderselection method developed by Genetics
More informationMath 140: Introductory Statistics Instructor: Julio C. Herrera Exam 3 January 30, 2015
Name: Exam Score: Instructions: This exam covers the material from chapter 7 through 9. Please read each question carefully before you attempt to solve it. Remember that you have to show all of your work
More informationSTA102 Class Notes Chapter Hypothesis Testing
10. Hypothesis Testing In this chapter, we continue to study methods for making inference about a population using our sample. In chapter 9, we used CIs, essentially, to answer questions of the type, What
More information7: Paired Samples. Data. Paired samples vs. independent sample. Illustrative dataset oatbran
7: Paired Samples Data Paired samples vs. independent sample This chapter considers the analysis of a quantitative outcome based on paired samples. Paired samples (also called dependent samples) are samples
More informationHYPOTHESIS TESTING (TWO SAMPLE)  CHAPTER 8 1. how can a sample be used to estimate the unknown parameters of a population
HYPOTHESIS TESTING (TWO SAMPLE)  CHAPTER 8 1 PREVIOUSLY estimation how can a sample be used to estimate the unknown parameters of a population use confidence intervals around point estimates of central
More informationThe ChiSquare Distributions
MATH 183 The ChiSquare Distributions Dr. Neal, WKU The chisquare distributions can be used in statistics to analyze the standard deviation " of a normally distributed measurement and to test the goodness
More informationFor eg: The yield of a new paddy variety will be 3500 kg per hectare scientific hypothesis. In Statistical language if may be stated as the random
Lecture.9 Test of significance Basic concepts null hypothesis alternative hypothesis level of significance Standard error and its importance steps in testing Test of Significance Objective To familiarize
More informationStatistics 104: Section 7
Statistics 104: Section 7 Section Overview Reminders Comments on Midterm Common Mistakes on Problem Set 6 Statistical Week in Review Comments on Midterm Overall, the midterms were good with one notable
More information7.1 Inference for comparing means of two populations
Objectives 7.1 Inference for comparing means of two populations Matched pair t confidence interval Matched pair t hypothesis test http://onlinestatbook.com/2/tests_of_means/correlated.html Overview of
More informationA Trial Analogy for Statistical. Hypothesis Testing. Legal Trial Begin with claim: Statistical Significance Test Hypotheses (statements)
A Trial Analogy for Statistical Slide 1 Hypothesis Testing Legal Trial Begin with claim: Smith is not guilty If this is rejected, we accept Smith is guilty reasonable doubt Present evidence (facts) Evaluate
More informationChapter 9. Section Correlation
Chapter 9 Section 9.1  Correlation Objectives: Introduce linear correlation, independent and dependent variables, and the types of correlation Find a correlation coefficient Test a population correlation
More informationChapter 7 Section 7.1: Inference for the Mean of a Population
Chapter 7 Section 7.1: Inference for the Mean of a Population Now let s look at a similar situation Take an SRS of size n Normal Population : N(, ). Both and are unknown parameters. Unlike what we used
More informationSampling 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
More informationStats for Strategy Exam 1 InClass Practice Questions DIRECTIONS
Stats for Strategy Exam 1 InClass Practice Questions DIRECTIONS Choose the single best answer for each question. Discuss questions with classmates, TAs and Professor Whitten. Raise your hand to check
More informationChapter 2 Drawing Statistical Conclusions
Chapter 2 Drawing Statistical Conclusions STAT 3022 School of Statistic, University of Minnesota 2013 spring 1 / 50 Population and Parameters Suppose there s a population of interest that we wish to study:
More informationUnit 21 Student s t Distribution in Hypotheses Testing
Unit 21 Student s t Distribution in Hypotheses Testing Objectives: To understand the difference between the standard normal distribution and the Student's t distributions To understand the difference between
More information4) The role of the sample mean in a confidence interval estimate for the population mean is to: 4)
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) Assume that the change in daily closing prices for stocks on the New York Stock Exchange is a random
More informationHomework Zero. Max H. Farrell Chicago Booth BUS41100 Applied Regression Analysis. Complete before the first class, do not turn in
Homework Zero Max H. Farrell Chicago Booth BUS41100 Applied Regression Analysis Complete before the first class, do not turn in This homework is intended as a self test of your knowledge of the basic statistical
More informationThe t Test. April 38, exp (2πσ 2 ) 2σ 2. µ ln L(µ, σ2 x) = 1 σ 2. i=1. 2σ (σ 2 ) 2. ˆσ 2 0 = 1 n. (x i µ 0 ) 2 = i=1
The t Test April 38, 008 Again, we begin with independent normal observations X, X n with unknown mean µ and unknown variance σ The likelihood function Lµ, σ x) exp πσ ) n/ σ x i µ) Thus, ˆµ x ln Lµ,
More informationChiSquare Distribution. is distributed according to the chisquare distribution. This is usually written
ChiSquare Distribution If X i are k independent, normally distributed random variables with mean 0 and variance 1, then the random variable is distributed according to the chisquare distribution. This
More informationSTA218 Introduction to Hypothesis Testing
STA218 Introduction to Hypothesis Testing Al Nosedal. University of Toronto. Fall 2015 October 29, 2015 Who wants to be a millionaire? Let s say that one of you is invited to this popular show. As you
More informationSimple Linear Regression Chapter 11
Simple Linear Regression Chapter 11 Rationale Frequently decisionmaking situations require modeling of relationships among business variables. For instance, the amount of sale of a product may be related
More informationNull Hypothesis H 0. The null hypothesis (denoted by H 0
Hypothesis test In statistics, a hypothesis is a claim or statement about a property of a population. A hypothesis test (or test of significance) is a standard procedure for testing a claim about a property
More informationPractice 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 12: 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,
More information11. CONFIDENCE INTERVALS FOR THE MEAN; KNOWN VARIANCE
11. CONFIDENCE INTERVALS FOR THE MEAN; KNOWN VARIANCE We assume here that the population variance σ 2 is known. This is an unrealistic assumption, but it allows us to give a simplified presentation which
More informationHypothesis Testing. Bluman Chapter 8
CHAPTER 8 Learning Objectives C H A P T E R E I G H T Hypothesis Testing 1 Outline 81 Steps in Traditional Method 82 z Test for a Mean 83 t Test for a Mean 84 z Test for a Proportion 85 2 Test for
More informationData Analysis and Uncertainty Part 3: Hypothesis Testing/Sampling
Data Analysis and Uncertainty Part 3: Hypothesis Testing/Sampling Instructor: Sargur N. University at Buffalo The State University of New York srihari@cedar.buffalo.edu Topics 1. Hypothesis Testing 1.
More informationHypothesis Tests for a Population Proportion
Hypothesis Tests for a Population Proportion MATH 130, Elements of Statistics I J. Robert Buchanan Department of Mathematics Fall 2015 Review: Steps of Hypothesis Testing 1. A statement is made regarding
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 informationInferential Statistics
Inferential Statistics Sampling and the normal distribution Zscores Confidence levels and intervals Hypothesis testing Commonly used statistical methods Inferential Statistics Descriptive statistics are
More information6. Duality between confidence intervals and statistical tests
6. Duality between confidence intervals and statistical tests Suppose we carry out the following test at a significance level of 100α%. H 0 :µ = µ 0 H A :µ µ 0 Then we reject H 0 if and only if µ 0 does
More information62 The Standard Normal Distribution. Uniform Distribution. Density Curve. Area and Probability. Using Area to Find Probability
62 The Standard Normal Distribution This section presents the standard normal distribution which has three properties: 1. Its graph is bellshaped. 2. Its mean is equal to 0 (μ = 0). 3. Its standard deviation
More informationCHAPTERS 46: Hypothesis Tests Read sections 4.3, 4.5, 5.1.5, Confidence Interval vs. Hypothesis Test (4.3):
CHAPTERS 46: Hypothesis Tests Read sections 4.3, 4.5, 5.1.5, 6.1.3 Confidence Interval vs. Hypothesis Test (4.3): The purpose of a confidence interval is to estimate the value of a parameter. The purpose
More informationSection I: Multiple Choice Select the best answer for each question.
Chapter 15 (Regression Inference) AP Statistics Practice Test (TPS 4 p796) Section I: Multiple Choice Select the best answer for each question. 1. Which of the following is not one of the conditions that
More informationDetermining the Sample Size: One Sample
Sample Sie Laboratory Determining the Sample Sie: One Sample OVERVIEW In this lab, you will be determining the sample sie for population means and population proportions. This lab is intended to help you
More informationHypothesis Tests: Two Related Samples
Hypothesis Tests: Two Related Samples AKA Dependent Samples Tests AKA Pairs Tests Cal State Northridge Ψ320 Andrew Ainsworth PhD Major Points Related samples? Samples? Difference scores? An example t
More informationA POPULATION MEAN, CONFIDENCE INTERVALS AND HYPOTHESIS TESTING
CHAPTER 5. A POPULATION MEAN, CONFIDENCE INTERVALS AND HYPOTHESIS TESTING 5.1 Concepts When a number of animals or plots are exposed to a certain treatment, we usually estimate the effect of the treatment
More informationCHAPTER 9 HYPOTHESIS TESTING
CHAPTER 9 HYPOTHESIS TESTING A hypothesis test is a formal way to make a decision based on statistical analysis. A hypothesis test has the following general steps: Set up two contradictory hypotheses.
More informationAn Introduction to Bayesian Statistics
An Introduction to Bayesian Statistics Robert Weiss Department of Biostatistics UCLA School of Public Health robweiss@ucla.edu April 2011 Robert Weiss (UCLA) An Introduction to Bayesian Statistics UCLA
More informationOneSample ttest. Example 1: Mortgage Process Time. Problem. Data set. Data collection. Tools
OneSample ttest Example 1: Mortgage Process Time Problem A faster loan processing time produces higher productivity and greater customer satisfaction. A financial services institution wants to establish
More informationAP * 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
More informationMath 251: Practice Questions for Test III Hints and Answers.. Then. P (67 x 69) = P (.33 z.33) = =.2586.
Math 251: Practice Questions for Test III Hints and Answers III.A: The Central Limit Theorem and Sampling Distributions. III.A.1 The heights of 18yearold men are approximately normally distributed with
More information