Two Population Means Hypothesis Testing and Confidence Intervals For Matched Pairs.

Slides:



Advertisements
Similar presentations
1 1 Slide © 2003 South-Western /Thomson Learning™ Slides Prepared by JOHN S. LOUCKS St. Edward’s University.
Advertisements

Topics Today: Case I: t-test single mean: Does a particular sample belong to a hypothesized population? Thursday: Case II: t-test independent means: Are.
Inferential Statistics
1 Chapter 9 Hypothesis Testing Developing Null and Alternative Hypotheses Type I and Type II Errors One-Tailed Tests About a Population Mean: Large-Sample.
Hypothesis Testing Developing Null and Alternative Hypotheses Developing Null and Alternative Hypotheses Type I and Type II Errors Type I and Type II Errors.
1 1 Slide STATISTICS FOR BUSINESS AND ECONOMICS Seventh Edition AndersonSweeneyWilliams Slides Prepared by John Loucks © 1999 ITP/South-Western College.
© 2010 Pearson Prentice Hall. All rights reserved Two Sample Hypothesis Testing for Means from Independent Groups.
The Multiple Regression Model Prepared by Vera Tabakova, East Carolina University.
1 Matched Samples The paired t test. 2 Sometimes in a statistical setting we will have information about the same person at different points in time.
Two Population Means Hypothesis Testing and Confidence Intervals With Known Standard Deviations.
Tests of significance Confidence intervals are used when the goal of our analysis is to estimate an unknown parameter in the population. A second goal.
ANOVA Determining Which Means Differ in Single Factor Models Determining Which Means Differ in Single Factor Models.
Hypothesis Tests About  With  Unknown. Hypothesis Testing (Revisited) Five Step Procedure 1.Define Opposing Hypotheses. (  ) 2.Choose a level of risk.
Pengujian Hipotesis Nilai Tengah Pertemuan 19 Matakuliah: I0134/Metode Statistika Tahun: 2007.
BCOR 1020 Business Statistics
Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 8-1 Business Statistics: A Decision-Making Approach 6 th Edition Chapter.
BCOR 1020 Business Statistics Lecture 21 – April 8, 2008.
Two Population Means Hypothesis Testing and Confidence Intervals With Unknown Standard Deviations.
8-4 Testing a Claim About a Mean
BCOR 1020 Business Statistics Lecture 20 – April 3, 2008.
Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 8-1 TUTORIAL 6 Chapter 10 Hypothesis Testing.
5-3 Inference on the Means of Two Populations, Variances Unknown
Statistics for Managers Using Microsoft® Excel 5th Edition
Hypothesis Testing Using The One-Sample t-Test
Two Sample Tests Ho Ho Ha Ha TEST FOR EQUAL VARIANCES
Confidence Intervals and Hypothesis Testing - II
Statistical inference: confidence intervals and hypothesis testing.
Statistics for Managers Using Microsoft® Excel 7th Edition
Fundamentals of Hypothesis Testing: One-Sample Tests
Claims about a Population Mean when σ is Known Objective: test a claim.
Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap th Lesson Introduction to Hypothesis Testing.
Copyright © Cengage Learning. All rights reserved. 13 Linear Correlation and Regression Analysis.
Section 10.1 ~ t Distribution for Inferences about a Mean Introduction to Probability and Statistics Ms. Young.
Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Chapter Inference on the Least-Squares Regression Model and Multiple Regression 14.
Chapter 9 Hypothesis Testing: Single Population
+ Chapter 9 Summary. + Section 9.1 Significance Tests: The Basics After this section, you should be able to… STATE correct hypotheses for a significance.
Chapter 9 Hypothesis Testing and Estimation for Two Population Parameters.
Chapter 9 Part C. III. One-Tailed Tests B. P-values Using p-values is another approach to conducting a hypothesis test, yielding the same result. In general:
PowerPoint presentations prepared by Lloyd Jaisingh, Morehead State University Statistical Inference: Hypotheses testing for single and two populations.
1 Introduction to Hypothesis Testing. 2 What is a Hypothesis? A hypothesis is a claim A hypothesis is a claim (assumption) about a population parameter:
Hypothesis Testing CSCE 587.
Section 9.2 Testing the Mean  9.2 / 1. Testing the Mean  When  is Known Let x be the appropriate random variable. Obtain a simple random sample (of.
Copyright © Cengage Learning. All rights reserved. 10 Inferences Involving Two Populations.
1 Section 9-4 Two Means: Matched Pairs In this section we deal with dependent samples. In other words, there is some relationship between the two samples.
Jeopardy Statistics Edition. Terms Calculator Commands Sampling Distributions Confidence Intervals Hypothesis Tests: Proportions Hypothesis Tests: Means.
Copyright © Cengage Learning. All rights reserved. 13 Linear Correlation and Regression Analysis.
1 Chapter 9 Hypothesis Testing. 2 Chapter Outline  Developing Null and Alternative Hypothesis  Type I and Type II Errors  Population Mean: Known 
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-1 Chapter 8 Fundamentals of Hypothesis Testing: One-Sample Tests Statistics.
Lecture 9 Chap 9-1 Chapter 2b Fundamentals of Hypothesis Testing: One-Sample Tests.
Interval Estimation and Hypothesis Testing Prepared by Vera Tabakova, East Carolina University.
Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 9-1 σ σ.
Example 10.2 Measuring Student Reaction to a New Textbook Hypothesis Tests for a Population Mean.
Chap 8-1 Fundamentals of Hypothesis Testing: One-Sample Tests.
Inferences from sample data Confidence Intervals Hypothesis Testing Regression Model.
2010, ECON Hypothesis Testing 1: Single Coefficient Review of hypothesis testing Testing single coefficient Interval estimation Objectives.
Hypothesis Testing Errors. Hypothesis Testing Suppose we believe the average systolic blood pressure of healthy adults is normally distributed with mean.
Paired Samples Lecture 39 Section 11.3 Tue, Nov 15, 2005.
Applied Quantitative Analysis and Practices LECTURE#14 By Dr. Osman Sadiq Paracha.
Statistical Inference Statistical inference is concerned with the use of sample data to make inferences about unknown population parameters. For example,
Understanding Basic Statistics Fourth Edition By Brase and Brase Prepared by: Lynn Smith Gloucester County College Chapter Nine Hypothesis Testing.
Lecture Slides Elementary Statistics Twelfth Edition
Lecture 22 Dustin Lueker.  Similar to testing one proportion  Hypotheses are set up like two sample mean test ◦ H 0 :p 1 -p 2 =0  Same as H 0 : p 1.
Week 111 Review - Sum of Normal Random Variables The weighted sum of two independent normally distributed random variables has a normal distribution. Example.
Lecture 22 Dustin Lueker.  Similar to testing one proportion  Hypotheses are set up like two sample mean test ◦ H 0 :p 1 -p 2 =0  Same as H 0 : p 1.
Statistical hypothesis Statistical hypothesis is a method for testing a claim or hypothesis about a parameter in a papulation The statement H 0 is called.
When the means of two groups are to be compared (where each group consists of subjects that are not related) then the excel two-sample t-test procedure.
Inference about Two Means: Dependent Samples
Hypothesis Testing and Confidence Intervals
Hypothesis Testing and Confidence Intervals
Hypothesis Testing and Confidence Intervals
Presentation transcript:

Two Population Means Hypothesis Testing and Confidence Intervals For Matched Pairs

Matched Pairs matchedSometimes experiments are conducted in such a way that samples from two populations are matched with something in common so that the i-th sample taken from the first population has something in common with the i-th sample of the second population. –It is the “common element” (same date, same weight, etc.) that is chosen at random and dictates the corresponding observations from each population. –Differences between the sample values (dictated by the “common element”) from each population are computed. –If it can be assumed that the differences have a normal distribution, t-tests can then be performed or t-intervals constructed for the average value of the differences. –Pairing, in general, reduces the variability in the problem.

Hypothesis Tests and Confidence Intervals for Matched Pairs Suppose there a random sample of n elements is taken. For each a corresponding sample from each population is observed. The difference is denoted d i. So there are n observations of differences, d i ’s. Statistics calculated:

Distribution of average difference _ d Distribution of average difference _ d Distribution: t distribution

Hypothesis Tests and Confidence Intervals for Matched Pairs Hypothesis Test: H 0 :  D = v H A :  D > v Test statistic:

Hypothesis Tests and Confidence Intervals for Matched Pairs Confidence Interval: Both the hypothesis test and the confidence interval have n-1 degrees of freedom: DF=n-1

Example Objective: Compare sales at two branch stores, one in Anaheim, the other in Irvine. –Can it be concluded that average daily sales in Anaheim is at least $200 greater than average daily sales in Irvine? –Construct a 95% confidence interval for the average difference in daily sales between the Anaheim and Irvine branches.

Approach 1 Approach 1 Records of sales on seven random dates in Anaheim are selected and seven random dates in Irvine are selected. There is nothing in common between the Anaheim and Irvine samples. Would have to use Difference in Means approach. Probably not the best approach. DateAnaheimDateIrvine 15-Dec Nov Nov85008-May Jun Mar Jul50006-Mar Aug Jun Feb Oct Mar Apr3100

Approach 2 Approach 2 Do not choose the receipts at random, but choose the dates at random and observe the sales at the Anaheim and Irvine branch stores on these dates. These data are paired by the random dates. DateAnaheimIrvine 25-Nov Feb May Aug Apr Jun Dec Difference Calculate Differences _ Calculate statistics: d =400 s D = 258.2

Hypothesis Test Hypothesis Test H 0 :  D = 200 H A :  D > 200 Select α =.05. Reject H 0 (Accept H A ) if t > t.05,6 = > 1.943; thus it can be concluded that average daily sales in Anaheim > $200 more than average daily sales in Irvine.

95% Confidence Interval 400 ±  638.8

ExcelExcel For Matched Pairs Excel Hypothesis Tests –Go to Tools/Data Analysis and select t-Test Paired Two Sample for Means. p-valueLook at p-value for the test. Confidence Intervals –Create a column of differences. Mean ± ConfidenceGo to Tools/Data Analysis and select Descriptive Statistics: Mean ± Confidence

Excel - Hypothesis Test Go Tools Select Data Analysis Select t-Test: Paired Two Sample for Means

Excel: t-Test for Matched Pairs Since H A is  D > 200, enter Column B for Range 1 Column C for Range for Hypothesized Mean Difference Check Labels Designate first cell for output.

Hypothesis Test (Cont’d) p-value for one-tail test Low p-value for 1-tail test (compared to α =.05)! Can conclude average daily sales in Anaheim exceed those in Irvine by > $200 p-value for at two-tail “  ” test

95% Confidence Interval for Matched Pairs =B2-C2 Drag to D3:D8 =I3+I16 =I3-I16 Go to Tools/Data Analysis Descriptive Statistics On Column D. Store output beginning in cell H1.

Review What constitutes “matched pairs” Matched pairs normally reduces variability from difference in means tests Create a set of differences Hypothesis Tests/Confidence Intervals for average difference –By hand –By Excel