Chapter 9.3 (323) A Test of the Mean of a Normal Distribution: Population Variance Unknown Given a random sample of n observations from a normal population.

Slides:



Advertisements
Similar presentations
Tests of Hypotheses Based on a Single Sample
Advertisements

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Chapter 9 Inferences Based on Two Samples.
Section 9.3 Inferences About Two Means (Independent)
9-1 Hypothesis Testing Statistical Hypotheses Statistical hypothesis testing and confidence interval estimation of parameters are the fundamental.
1/55 EF 507 QUANTITATIVE METHODS FOR ECONOMICS AND FINANCE FALL 2008 Chapter 10 Hypothesis Testing.
Chap 11-1 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chapter 11 Hypothesis Testing II Statistics for Business and Economics.
BCOR 1020 Business Statistics
1/45 Chapter 11 Hypothesis Testing II EF 507 QUANTITATIVE METHODS FOR ECONOMICS AND FINANCE FALL 2008.
Hypothesis Testing for Population Means and Proportions
T-Tests Lecture: Nov. 6, 2002.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Statistics for Business and Economics 7 th Edition Chapter 9 Hypothesis Testing: Single.
Chapter 11: Inference for Distributions
Inferences About Process Quality
Chapter 9 Hypothesis Testing.
Chapter 8 Introduction to Hypothesis Testing
Definitions In statistics, a hypothesis is a claim or statement about a property of a population. A hypothesis test is a standard procedure for testing.
Chapter 9 Hypothesis Testing II. Chapter Outline  Introduction  Hypothesis Testing with Sample Means (Large Samples)  Hypothesis Testing with Sample.
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Chapter 8 Tests of Hypotheses Based on a Single Sample.
Chapter 15 Nonparametric Statistics
Hypothesis Testing and T-Tests. Hypothesis Tests Related to Differences Copyright © 2009 Pearson Education, Inc. Chapter Tests of Differences One.
Chapter 9 Title and Outline 1 9 Tests of Hypotheses for a Single Sample 9-1 Hypothesis Testing Statistical Hypotheses Tests of Statistical.
Statistical Inference for Two Samples
AM Recitation 2/10/11.
McGraw-Hill/IrwinCopyright © 2009 by The McGraw-Hill Companies, Inc. All Rights Reserved. Chapter 9 Hypothesis Testing.
Chapter 10 Hypothesis Testing
Week 9 Chapter 9 - Hypothesis Testing II: The Two-Sample Case.
Chapter 7 Using sample statistics to Test Hypotheses about population parameters Pages
Overview Definition Hypothesis
Statistics for Managers Using Microsoft® Excel 7th Edition
Sections 8-1 and 8-2 Review and Preview and Basics of Hypothesis Testing.
Fundamentals of Hypothesis Testing: One-Sample Tests
Chapter 5 Sampling and Statistics Math 6203 Fall 2009 Instructor: Ayona Chatterjee.
Claims about a Population Mean when σ is Known Objective: test a claim.
Inference about Two Population Standard Deviations.
Copyright © Cengage Learning. All rights reserved. 10 Inferences Involving Two Populations.
McGraw-Hill/Irwin Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. Statistical Inferences Based on Two Samples Chapter 9.
Chapter 9 Hypothesis Testing: Single Population
1 Power and Sample Size in Testing One Mean. 2 Type I & Type II Error Type I Error: reject the null hypothesis when it is true. The probability of a Type.
Chapter 9 Hypothesis Testing and Estimation for Two Population Parameters.
Chapter 10 Hypothesis Testing
1 1 Slide © 2008 Thomson South-Western. All Rights Reserved Chapter 11 Inferences About Population Variances n Inference about a Population Variance n.
Mid-Term Review Final Review Statistical for Business (1)(2)
9-1 Hypothesis Testing Statistical Hypotheses Definition Statistical hypothesis testing and confidence interval estimation of parameters are.
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Chapter 9 Inferences Based on Two Samples.
Chapter 8 Introduction to Hypothesis Testing ©. Chapter 8 - Chapter Outcomes After studying the material in this chapter, you should be able to: 4 Formulate.
Chapter 7 Sampling and Sampling Distributions ©. Simple Random Sample simple random sample Suppose that we want to select a sample of n objects from a.
EMIS 7300 SYSTEMS ANALYSIS METHODS FALL 2005 Dr. John Lipp Copyright © Dr. John Lipp.
McGraw-Hill/Irwin Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. Chapter 8 Hypothesis Testing.
Chap 8-1 A Course In Business Statistics, 4th © 2006 Prentice-Hall, Inc. A Course In Business Statistics 4 th Edition Chapter 8 Introduction to Hypothesis.
Interval Estimation and Hypothesis Testing Prepared by Vera Tabakova, East Carolina University.
Two-Sample Hypothesis Testing. Suppose you want to know if two populations have the same mean or, equivalently, if the difference between the population.
1 9 Tests of Hypotheses for a Single Sample. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 9-1.
1 Objective Compare of two population variances using two samples from each population. Hypothesis Tests and Confidence Intervals of two variances use.
Chap 8-1 Fundamentals of Hypothesis Testing: One-Sample Tests.
Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 1 Understandable Statistics S eventh Edition By Brase and Brase Prepared by: Lynn Smith.
© Copyright McGraw-Hill 2004
Formulating the Hypothesis null hypothesis 4 The null hypothesis is a statement about the population value that will be tested. null hypothesis 4 The null.
Ch8.2 Ch8.2 Population Mean Test Case I: A Normal Population With Known Null hypothesis: Test statistic value: Alternative Hypothesis Rejection Region.
Statistical Inference Statistical inference is concerned with the use of sample data to make inferences about unknown population parameters. For example,
§2.The hypothesis testing of one normal population.
Chapter 8 Estimation ©. Estimator and Estimate estimator estimate An estimator of a population parameter is a random variable that depends on the sample.
Hypothesis Tests u Structure of hypothesis tests 1. choose the appropriate test »based on: data characteristics, study objectives »parametric or nonparametric.
Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Statistics for Business and Economics 8 th Edition Chapter 9 Hypothesis Testing: Single.
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Chapter 7 Inferences Concerning Means.
Statistical Decision Making. Almost all problems in statistics can be formulated as a problem of making a decision. That is given some data observed from.
Independent Samples: Comparing Means Lecture 39 Section 11.4 Fri, Apr 1, 2005.
Two-Sample Hypothesis Testing
Chapter 4. Inference about Process Quality
Chapter 9 Hypothesis Testing.
Nonparametric Statistics
Presentation transcript:

Chapter 9.3 (323) A Test of the Mean of a Normal Distribution: Population Variance Unknown Given a random sample of n observations from a normal population with mean . Using the sample mean and standard deviation X and s we can use the following test with significance level , (i) To test either null hypothesis against the alternative the decision rule is Or equivalently

A Test of the Mean of a Normal Distribution: Population Variance Unknown (continued) (ii) To test either null hypothesis against the alternative the decision rule is Or equivalently

A Test of the Mean of a Normal Distribution: Population Variance Unknown (continued) (iii) To test the null hypothesis against the alternative the decision rule is equivalently where tn-1,/2 is the student t-value for n – 1 degrees of freedom* and upper tail probability** /2. The p-values for these tests are computed in the same way as we did for tests with known variance except that the student t value is substituted*** for the normal Z value. *frígráður **líkur í efri hala ***sett í stað

Chapter 9.4 (327) Tests of the Population Proportion (Large Sample Size) We begin by assuming a random sample of n observations from a population that has a proportion  whose members possess a particular attribute (tiltekið viðhorf) . If (1 - ) > 9 and the sample proportion is p the following tests have significance level : (i) To test either null hypothesis against the alternative (valtilgátunni) the decision rule is (ákvörðunarreglunni)

Tests of the Population Proportion (Large Sample Size) (Continued) (ii) To test either null hypothesis against the alternative the decision rule is

Tests of the Population Proportion (Large Sample Size) (Continued) (iii) To test the null hypothesis against the two-sided alternative the decision rule is For all of these tests the p-value is the smallest significance level at which the null hypothesis can be rejected.

Chapter 9.5 (330) Tests of Variance of a Normal Population Given a random sample of n observations from a normally distributed population with variance 2. If we observe the sample variance sx2, then the following tests have significance level : (i) To test either the null hypothesis against the alternative the decision rule is

Tests of Variance of a Normal Population (continued) (ii) To test either null hypothesis against the alternative the decision rule is

Tests of Variance of a Normal Population (continued) (iii) To test the null hypothesis against the alternative the decision rule is Where 2n-1 is a chi-square random variable and P(2n-1 > 2n-1,) = . The p-value for these tests is the smallest significance level at which the null hypothesis can be rejected given the sample variance.

Some Probabilities for the Chi-Square Distribution (Figure 9 Some Probabilities for the Chi-Square Distribution (Figure 9.5) Section 9.5 page 331 f(2v) /2 1 -  /2 2v,1-/2 2v,/2

Chapter 9.6 (334) Tests of the Difference Between Population Means: Matched Pairs Suppose that we have a random sample of n matched (samstæð) pairs of observations (mælinga/athugana) from distributions with means X and Y . Let D and sd denote the observed sample mean and standard deviation for the n differences Di = (xi – yi) . If the population distribution of the differences is a normal distribution, then the following tests have significance level . (i) To test either null hypothesis against the alternative the decision rule is

Tests of the Difference Between Population Means: Matched Pairs (continued) (ii) To test either null hypothesis against the alternative the decision rule is

Tests of the Difference Between Population Means: Matched Pairs (continued) (iii) To test the null hypothesis against the two-sided alternative the decision rule is Here tn-1, is the number for which P(tn-1 > tn-1, ) =  where the random variable tn-1 follows a Student’s t distribution with (n – 1) degrees of freedom. When we want to test the null hypothesis that the two population means are equal, we set D0 = 0 in the formulas. P-values for all of these tests are interpreted as the smallest significance level at which the null hypothesis can be rejected (hægt er að hafna) given the test statistic.

Tests of the Difference Between Population Means: Independent Samples (Known Variances) Suppose that we have two independent random samples of nx and ny observations from normal distributions with means X and Y and variances 2x and 2y . If the observed sample means are X and Y, then the following tests have significance level . (i) To test either null hypothesis against the alternative the decision rule is

Tests of the Difference Between Population Means: Independent Samples (Known Variances) (continued) (ii) To test either null hypothesis against the alternative the decision rule is

Tests of the Difference Between Population Means: Independent Samples (Known Variances) (continued) (iii) To test the null hypothesis against the alternative the decision rule is If the sample sizes are large (n > 100) then a good approximation at significance level  can be made if the population variances are replaced by the sample variances. In addition the central limit leads to good approximations even if the populations are not normally distributed. P-values for all these tests are interpreted as the smallest significance level at which the null hypothesis can be rejected given the test statistic.

Tests of the Difference Between Population Means: Population Variances Unknown and Equal These tests assume that we have two independent random samples of nx and ny observations from normally distributed populations with means X and Y and a common variance. The sample variances sx2 and sy2 are used to compute a pooled variance estimator Then using the observed sample means are X and Y, the following tests have significance level : (i) To test either null hypothesis against the alternative the decision rule is

Tests of the Difference Between Population Means: Population Variances Unknown and Equal (continued) (ii) To test either null hypothesis against the alternative the decision rule is

Tests of the Difference Between Population Means: Population Variances Unknown and Equal (continued) (iii) To test the null hypothesis against the alternative the decision rule is Here tnx+ny-2, is the number for which P(tnx+ny-2, > tnx+ny-2, ) = . P-values for all these tests are interpreted as the smallest significance level at which the null hypothesis can be rejected given the test statistic.

Tests of the Difference Between Population Means: Population Variances Unknown and Not Equal These tests assume that we have two independent random samples of nx and ny observations from normal populations with means X and Y and a common variance. The sample variances sx2 and sy2 are used. The degrees of freedom, v, for the student t statistic is given by Then using the observed sample means are X and Y, the following tests have significance level : (i) To test either null hypothesis against the alternative the decision rule is

Tests of the Difference Between Population Means: Population Variances Unknown and Not Equal (continued) (ii) To test either null hypothesis against the alternative the decision rule is

Tests of the Difference Between Population Means: Population Variances Unknown and Not Equal (continued) (iii) To test the null hypothesis against the alternative the decision rule is Here tnx+ny-2, is the number for which P(tnx+ny-2, > tnx+ny-2, ) = . P-values for all these tests are interpreted as the smallest significance level at which the null hypothesis can be rejected given the test statistic.

Chapter 9.7 (346) Testing the Equality of Population Proportions (Large Samples) Given independent random samples of nx and ny with proportion successes px and py. When we assume that the population proportions are equal, an estimate of the common proportion is For large sample sizes - - n(1 - ) > 9 - - the following tests have significance level : (i) To test either null hypothesis against the alternative the decision rule is

Testing the Equality of Population Proportions - Large Samples - (continued) (ii) To test either null hypothesis against the alternative the decision rule is

Testing the Equality of Population Proportions - Large Samples - (continued) (iii) To test the null hypothesis against the alternative the decision rule is It is also possible to compute and interpret the p-values for these tests by calculating the minimum significance level at which the null hypothesis can be rejected.

Chapter 9.8 (350) The F Distribution Given that we have two independent random samples of nx and ny observations from two normal populations with variances 2x and 2y . If the sample variances are sx2 and sy2 then the random variable Has an F distribution with numerator degrees of freedom (nx – 1) and denominator degrees of freedom (ny – 1). An F distribution with numerator degrees of freedom v1 and denominator degrees of freedom v2 will be denoted Fv1, v2 . We denote Fv1, v2,  the number for which We need to emphasize that this test is quite sensitive to the assumption of normality.

Tests for Equality of Variances from Two Normal Populations Let sx2 and sy2 be observed sample variances from independent random samples of size nx and ny from normally distributed populations with variances 2x and 2y . Use s2x to denote the larger variance. Then the following tests have significance level : (i) To test either null hypothesis against the alternative the decision rule is

Tests for Equality of Variances from Two Normal Populations (continued) (ii) To test the null hypothesis against the alternative the decision rule is Where s2x is the larger of the two sample variances. Since either sample variance could be larger this rule is actually based on a two-tailed test and hence we use /2 as the upper tail probability. Here Fnx-1,ny-1 is the number for which Where Fnx-1,ny-1 has an F distribution with (nx – 1) numerator degrees of freedom and (ny – 1) denominator degrees of freedom.

Chapter 9.9 (354) Determining the Probability of a Type II Error Consider the test against the alternative Using a decision rule Using the decision rule determine the values of the sample mean that result in accepting the null hypothesis. Now for any value of the population mean defined by the alternative hypothesis H1 find the probability that the sample mean will be in the acceptance region for the null hypothesis. This is the probability of a Type II error. Thus we consider  = * such that * > 0. Then for * the probability of a Type II error is and Power = 1 - 

Power Function for Test H0:  = 5 against H1:  > 5 ( = 0.05,  =0.1, n = 16) (Figure 9.13) .5 .05 5.00 5.05 5.10 

Key Words Alternative Hypothesis Determining the Probability of Type II Error Equality of Population Proportions F Distribution Hypothesis Testing Methodology Interpretation of the Probability value or p-value Null Hypothesis Power Function States of Nature and Decisions on Null Hypothesis Test of Mean of a Normal Distribution (Variance Known) Composite Null and Alternative Composite or Simple Null and Alternative Hypothesis

Key Words (continued) Testing the Equality of Two Population Proportions (Large Samples) Tests for Difference Between Population Means: Independent Samples Tests for Equality of Variances from Two Normal Populations Tests for the Difference Between Sample Means: Population Variances Unknown and Equal Tests for Differences Between Population Means: Matched Pairs Tests of the Mean of a Normal Distribution: Population Variance Unknown Tests of the Population Proportion (Large Sample Sizes) Tests of Variance of a Normal Population Type I Error Type II Error