# 13 Two-Sample T Tests

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 CHAPTER 13 Two-Sample T Tests Chapter Outline 13.1 TESTING A HYPOTHESIS FOR DEPENDENT AND INDEPENDENT SAMPLES 270

2 Chapter 13. Two-Sample T Tests 13.1 Testing a Hypothesis for Dependent and Independent Samples Learning Objectives Identify situations that contain dependent or independent samples. Calculate the standard deviation for two independent samples. Calculate the test statistic to test hypotheses about dependent data pairs. Calculate the test statistic to test hypotheses about independent data pairs. Introduction In the previous lessons we learned about hypothesis testing for a single sample mean (comparing a sample mean to a hypothesized population mean). In this chapter we will apply the principals of hypothesis testing to situations involving two samples. There are many situations in everyday life where we would perform statistical analysis involving two samples. For example, suppose that we wanted to test a hypothesis about the effect of two medications on curing an illness. Or, we may want to test the difference between the means of males and females on the SAT. In both of these cases, we would analyze both samples and the hypothesis would address the difference between two sample means. In this lesson, we will identify situations with different types of samples, learn to calculate the test statistic, calculate the estimate for population variance for both samples and calculate the test statistic to test hypotheses about the difference of proportions or means between samples. Independent Samples Let s recall what we assumed when we conducted a hypothesis test on a single sample. We knew that we needed to select a random sample from the population, measure that sample statistic and then make an inference about the population based on that sample. When we work with two independent samples, our null hypothesis assumes that if the samples are selected at random, the samples will vary only by chance and the difference will not be statistically significant. In short, when we have independent samples we assume that the scores of one sample do not affect the other. Dependent Samples Dependent samples are a bit different. Two samples of data are dependent when each score in one sample is paired with a specific score in the other sample. In short, these types of samples are related to each other. Dependent samples can occur in two scenarios. In one, a group may be measured twice such as in a pretest-posttest situation (scores on a test before and after the lesson). The other scenario is one in which an observation in one sample is matched with an observation in the second sample such as when researchers measure the attitudes or behaviors of twins. Testing Hypotheses with Independent Samples The set up for the independent sample t-test is very similar to the single sample t-test. Now however, instead of one sample mean, we have two. Let s look at the hypotheses for the t-test: 271

3 13.1. Testing a Hypothesis for Dependent and Independent Samples H 0 : µ 1 = µ 2 H A : µ 1 µ 2 Notice our hypothesis statements have two population means, denoting the fact that we will be testing whether the means of two separate populations are equal to one another. An equivalent way of writing the hypotheses is as follows: H 0 : µ 1 µ 2 = 0 H A : µ 1 µ 2 0 Both methods of writing the hypothesis statements are valid. Let s see how the new hypothesis statements look in our t-statistic formula: Where: t = ( x 1 x 2 ) (µ 1 µ 2 ) SE ( x1 x 2 ) x 1 x 2 is the difference between the sample means µ 1 µ 2 is the difference between the hypothesized population means SE ( x1 x 2 ) is the standard error of the difference between the sample means The standard error of the difference between the sample means is calculated by:se ( x1 x 2 ) = s 2 1 n 1 + s2 2 n 2 This standard error is called an unpooled standard error, because the standard deviation of each sample is considered. Finally, just like the single sample t-test, we ll need to know the degrees of freedom so that we can find the correct critical value. Here s the formula for the degrees of freedom when using an unpooled standard error independent samples t-test: d f = ( s 2 ) 2 1 n + s2 2 1 n2 ( ) 1 s 2 2 ( ) 1 n 1 1 n + 1 s n 2 1 n 2 Nice, right? Don t worry, we will not be using this formula. Technology automatically calculates this formula for us. When we solve the upooled independent samples t-test by hand, the conservative approach is to use the lowest n of the two groups minus one: d f = n lowest 1 The Assumptions of the Independent Samples t-test Just like the single sample t-test, the independent samples t-test has some assumptions that we must consider in order for the test to be valid: A random sample of each population is used. The random samples are each made up of independent observations Each sample is independent of one another The population distribution of each population must be nearly normal, or the size of the sample is large. Notice the new assumption now that we have two means that we are examining, we need to make sure those two means are independent of one another. What dose that independence mean? It means that if an observation is assigned to one group, it cannot also be recorded in the other group. Oftentimes, independent groups are things like: males vs. females, Astros vs. Rangers fans, tall vs. short, etc. 272

4 Chapter 13. Two-Sample T Tests Example: Independent t-test The head of the English department is interested in the difference in writing scores between freshman English students who are taught by different teachers. The incoming freshmen are randomly assigned to one of two English teachers and are given a standardized writing test after the first semester. We take a sample of eight students from one class and nine from the other. Is there a difference in achievement on the writing test between the two classes? Here s the data from the two classes: TABLE 13.1: Class 1 Class Mean Standard Deviation Hypothesis Step 1: Clearly state the Null and Alternative Hypothesis. We will be testing to see if the mean score of the two classes are equal to one another: H 0 : µ 1 = µ 2 H A : µ 1 µ 2 Hypothesis Step 2: Identify the appropriate significance level and confirm the test assumptions. We ll use the standard significance test of We were told that students were randomly assigned, and we ll assume that students did not switch classes (for independence), and we ll assume the student score are independent from one another. We ll assume the underlying population of students in each class is nearly normal in the distribution of scores. Hypothesis Step 3: Analyze the data and generate the test statistic. Now we ll use our t-test to get to the analysis. First, our standard error of the difference between the sample means: SE ( x1 x 2 ) = SE ( x1 x 2 ) = SE ( x1 x 2 ) = Now, we will use that SE in the t-test equation: t = ( x 1 x 2 ) (µ 1 µ 2 ) SE ( x1 x 2 ) = ( ) (0) =

5 13.1. Testing a Hypothesis for Dependent and Independent Samples We know that our smallest group has just eight students, so our degrees of freedom is (8-1)=7. The critical value of the t-distribution for 7 degrees of freedom is ± Hypothesis Step 4: Interpret your results. Because our calculated t-value is outside the t-critical value (our value falls in the critical region of the t- distribution), we reject our Null hypothesis. We conclude that the populations of students in the two classes significantly differ in their standardized test scores at the end of the semester. As the two classes were randomly assigned, we can plausibly conclude that the difference in the scores was due to the class assignment which class the students were in (and whatever teaching technique was used). Hypotheses with Dependent Samples As we mentioned earlier, dependent sample are a bit different from the independent samples t-test. Another name for the dependent samples t-test is the paired samples t-test. That name should give you a hint as to how the test is different. In some way, the two variables we will be testing will be paired or related to one another. Now, just because we used the word related don t think correlation. This is still a t-test, and we ll be testing questions about the means of variables. Let s see how the dependent samples t-test is different from the indepdent samples. First, the hypothesis statement. In the dependent samples t-test, our hypothesis statement looks like: H 0 : δ = 0 H A : δ 0 That symbol is the Greek letter delta which is the representation of difference. So the hypothesis of the dependent samples t-test is that the difference between two variables is zero. Now, that sure sounds a lot like the hypothesis of an independent samples t-test when we test the difference between two population means. The difference is subtle but important. In the independent samples t-test we were testing the difference between two means we calculated the mean for each group and then compared them. In the dependent samples t-test we are looking at the difference between two variables within a single observation. Let s put this in context of our previous example of standardized scores. We were told that the students were tested at the end of the semester that s just one time of testing. But what if all the students were tested when they came into the class (sort of a basic knowledge test) and then again at the end of the semester. Assuming the test was the same (or very close) both times the student saw it, then any change in the scores from beginning to end, should represent the knowledge gain over the course of the semester. Because all students have two scores one at beginning and one at the end of the semester the dependent samples t-test allows us to take the difference between the two scores for every student. This difference score is only one column of data. What we are really interested in the dependent samples t-test that difference variable Let s look at the t-test formula for a better understanding: t = d δ SE d Where: d is the average of the difference between paired variables SEd is the standard error of the difference variable Notice that d in the numerator of the formula. That s the average of new variable of differences. Again, thinking about our example: If there was no knowledge change over the semester, the average of the differences in scores for each student should be zero that s why our Null hypothesis statement states that delta (the average difference in scores for the entire population of students) is equal to zero. 274

6 Chapter 13. Two-Sample T Tests The standard error in the formula is the standard error for the variable representing the difference, which is made up of the standard deviation of the differences. This the standard deviation of the differences formula should look a lot like a simple standard deviation: ( d d ) 2 s d = n 1 And the formula for the standard error is: SEd = s d n Our degrees of freedom are similar to what they were earlier. But this time since the test is only concerned about a new variable of differences for each subject, we can use the (n-1) degrees of freedom formula, where n represents the number of pairs of data. If there were eight students in the class and each was measured twice, we would still have eight difference scores one for each student. The Assumptions of the Dependent Samples t-test The assumptions of the dependent samples t-test are very much like the single sample t-test assumptions after all, the test is only concerned with a single variable (the difference). The sample of differences (hence the sample of paired observations) is random. The paired observations are independent of one another. The distribution of population differences must be nearly normal, or the size of the paired observation sample is large. Let s look at an example of the dependent samples t-test in action to put it all together: A math teacher wants to determine the effectiveness of her statistics lesson. She gives a simple skills test to nine students before the start of class (a pre-test) and the same skills test to the same students at the end of class (a post-test). Here s the data (with some calculations): TABLE 13.2: Student Pre-test Post-test Difference Mean 3.56 s 4.19 Hypothesis Step 1: Clearly state the Null and Alternative Hypothesis. The statistics instructor is interested in the improvement over the semester, for each of her students. Since there are two measures for each student, and those measures are paired, we ll need to use the dependent 275

7 13.1. Testing a Hypothesis for Dependent and Independent Samples samples t-test. We ll assume that the normal state of affaires would be that there s no change due to chance (so our delta is equal to zero). The Null and Alternative would be: H 0 : δ = 0 H A : δ 0 Hypothesis Step 2: Identify the appropriate significance level and confirm the test assumptions. We ll assume the standard significance level of We ll assume that the students in her class are random, that they the students are independent of one another, and that the distribution of all possible difference scores in the population are nearly normally distributed. Hypothesis Step 3: Analyze the data. We have the mean and standard deviation for the data not for each time variable (pre and post), but for the difference between the two variables for each student (the column on the right). This column of differences is what we ll use in the test. First, let s calculate the Standard Error: SEd = s d n SEd = 4.19 = Now, we ll use that in our dependent samples t-test: t = d δ SEd t = = 2.54 With this test, we have nine students, so we have (9-1) = 8 degrees of freedom. The t-critical value for 8 degrees of freedom is ± Hypothesis Step 4: Interpret your results. As our calculated t-statistic is greater than our t-critical value (our value lies in the critical region), we reject our Null hypothesis and conclude that there was in fact a change in student performance from the Pre-test to the Post-test. Lesson Summary In addition to testing single samples associated with a mean, we can also perform hypothesis tests with two samples. We can test two independent samples (which are samples that do not affect one another) or dependent samples which assume that the samples are related to each other. When testing a hypothesis about two independent samples, we follow a similar process as when testing one random sample. However, when computing the test statistic, we need to calculate the estimated standard error of the difference between sample means. We carry out the test on the means of two independent samples in a similar way as the testing of one random sample. However, we use the following formula to calculate the test statistic: t = ( x 1 x 2 ) (µ 1 µ 2 ) SE.( x 1 x 2 ) with the standard error defined above. We can also test the likelihood that two dependent samples are related. dependent samples, we use the formula: To calculate the test statistic for two 276 d t = δ SEd

8 Chapter 13. Two-Sample T Tests Review Questions 1. In hypothesis testing, we have scenarios that have both dependent and independent samples. Give an example of an experiment with (1) dependent samples and (2) independent samples. 2. True or False: When we test the difference between the means of males and females on the SAT, we are using independent samples. 3. A study is conducted on the effectiveness of a drug on the hyperactivity of laboratory rats. Two random samples of rats are used for the study and one group is given Drug A and the other group is given Drug B and the number of times that they push a lever is recorded. The following results for this test were calculated: TABLE 13.3: Drug A Drug B X n s s a. Does this scenario involve dependent or independent samples? Explain. b. What would the hypotheses be for this scenario? c. Calculate the estimated standard error for this scenario. d. What is the test statistic and at an alpha level of.05 what conclusions would you make about the null hypothesis? 4. A survey is conducted on attitudes towards drinking. A random sample of eight married couples is selected, and the husbands and wives respond to an attitude-toward-drinking scale. The scores are as follows: TABLE 13.4: Husbands Wives (a) What would be the hypotheses for this scenario? (b) Calculate the estimated standard deviation for this scenario. (c) Compute the standard error of the difference for these samples. (d) What is the test statistic and at an alpha level of.05 what conclusions would you make about the null hypothesis? 277

### Chapter 6: t test for dependent samples

Chapter 6: t test for dependent samples ****This chapter corresponds to chapter 11 of your book ( t(ea) for Two (Again) ). What it is: The t test for dependent samples is used to determine whether the

### Module 5 Hypotheses Tests: Comparing Two Groups

Module 5 Hypotheses Tests: Comparing Two Groups Objective: In medical research, we often compare the outcomes between two groups of patients, namely exposed and unexposed groups. At the completion of this

### Descriptive Statistics

Descriptive Statistics Primer Descriptive statistics Central tendency Variation Relative position Relationships Calculating descriptive statistics Descriptive Statistics Purpose to describe or summarize

### Solutions to Worksheet on Hypothesis Tests

s to Worksheet on Hypothesis Tests. A production line produces rulers that are supposed to be inches long. A sample of 49 of the rulers had a mean of. and a standard deviation of.5 inches. The quality

### Inferences About Differences Between Means Edpsy 580

Inferences About Differences Between Means Edpsy 580 Carolyn J. Anderson Department of Educational Psychology University of Illinois at Urbana-Champaign Inferences About Differences Between Means Slide

### Hypothesis 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

### Inferential Statistics

Inferential Statistics Sampling and the normal distribution Z-scores Confidence levels and intervals Hypothesis testing Commonly used statistical methods Inferential Statistics Descriptive statistics are

### Two-sample t-tests. - Independent samples - Pooled standard devation - The equal variance assumption

Two-sample t-tests. - Independent samples - Pooled standard devation - The equal variance assumption Last time, we used the mean of one sample to test against the hypothesis that the true mean was a particular

### LAB 4 INSTRUCTIONS CONFIDENCE INTERVALS AND HYPOTHESIS TESTING

LAB 4 INSTRUCTIONS CONFIDENCE INTERVALS AND HYPOTHESIS TESTING In this lab you will explore the concept of a confidence interval and hypothesis testing through a simulation problem in engineering setting.

### Lesson 1: Comparison of Population Means Part c: Comparison of Two- Means

Lesson : Comparison of Population Means Part c: Comparison of Two- Means Welcome to lesson c. This third lesson of lesson will discuss hypothesis testing for two independent means. Steps in Hypothesis

### Two-sample hypothesis testing, II 9.07 3/16/2004

Two-sample hypothesis testing, II 9.07 3/16/004 Small sample tests for the difference between two independent means For two-sample tests of the difference in mean, things get a little confusing, here,

### Inferential Statistics. Probability. From Samples to Populations. Katie Rommel-Esham Education 504

Inferential Statistics Katie Rommel-Esham Education 504 Probability Probability is the scientific way of stating the degree of confidence we have in predicting something Tossing coins and rolling dice

### Stat 411/511 THE RANDOMIZATION TEST. Charlotte Wickham. stat511.cwick.co.nz. Oct 16 2015

Stat 411/511 THE RANDOMIZATION TEST Oct 16 2015 Charlotte Wickham stat511.cwick.co.nz Today Review randomization model Conduct randomization test What about CIs? Using a t-distribution as an approximation

### Statistiek I. t-tests. John Nerbonne. CLCG, Rijksuniversiteit Groningen. John Nerbonne 1/35

Statistiek I t-tests John Nerbonne CLCG, Rijksuniversiteit Groningen http://wwwletrugnl/nerbonne/teach/statistiek-i/ John Nerbonne 1/35 t-tests To test an average or pair of averages when σ is known, we

### How to Conduct a Hypothesis Test

How to Conduct a Hypothesis Test The idea of hypothesis testing is relatively straightforward. In various studies we observe certain events. We must ask, is the event due to chance alone, or is there some

### Chapter 8: Introduction to Hypothesis Testing

Chapter 8: Introduction to Hypothesis Testing We re now at the point where we can discuss the logic of hypothesis testing. This procedure will underlie the statistical analyses that we ll use for the remainder

### Psyc 250 Statistics & Experimental Design. Single & Paired Samples t-tests

Psyc 250 Statistics & Experimental Design Single & Paired Samples t-tests Part 1 Data Entry For any statistical analysis with any computer program, it is always important that data are entered correctly

### 3.4 Statistical inference for 2 populations based on two samples

3.4 Statistical inference for 2 populations based on two samples Tests for a difference between two population means The first sample will be denoted as X 1, X 2,..., X m. The second sample will be denoted

### Chapter 9. Two-Sample Tests. Effect Sizes and Power Paired t Test Calculation

Chapter 9 Two-Sample Tests Paired t Test (Correlated Groups t Test) Effect Sizes and Power Paired t Test Calculation Summary Independent t Test Chapter 9 Homework Power and Two-Sample Tests: Paired Versus

### An Introduction to Statistics Course (ECOE 1302) Spring Semester 2011 Chapter 10- TWO-SAMPLE TESTS

The Islamic University of Gaza Faculty of Commerce Department of Economics and Political Sciences An Introduction to Statistics Course (ECOE 130) Spring Semester 011 Chapter 10- TWO-SAMPLE TESTS Practice

### Lecture Notes Module 1

Lecture Notes Module 1 Study Populations A study population is a clearly defined collection of people, animals, plants, or objects. In psychological research, a study population usually consists of a specific

### Two-sample hypothesis testing, I 9.07 3/09/2004

Two-sample hypothesis testing, I 9.07 3/09/2004 But first, from last time More on the tradeoff between Type I and Type II errors The null and the alternative: Sampling distribution of the mean, m, given

### Chapter 8 Hypothesis Testing Chapter 8 Hypothesis Testing 8-1 Overview 8-2 Basics of Hypothesis Testing

Chapter 8 Hypothesis Testing 1 Chapter 8 Hypothesis Testing 8-1 Overview 8-2 Basics of Hypothesis Testing 8-3 Testing a Claim About a Proportion 8-5 Testing a Claim About a Mean: s Not Known 8-6 Testing

### Statistical Inference and t-tests

1 Statistical Inference and t-tests Objectives Evaluate the difference between a sample mean and a target value using a one-sample t-test. Evaluate the difference between a sample mean and a target value

### Introduction to Hypothesis Testing. Point estimation and confidence intervals are useful statistical inference procedures.

Introduction to Hypothesis Testing Point estimation and confidence intervals are useful statistical inference procedures. Another type of inference is used frequently used concerns tests of hypotheses.

### Section 13, Part 1 ANOVA. Analysis Of Variance

Section 13, Part 1 ANOVA Analysis Of Variance Course Overview So far in this course we ve covered: Descriptive statistics Summary statistics Tables and Graphs Probability Probability Rules Probability

### 12 Hypothesis Testing

CHAPTER 12 Hypothesis Testing Chapter Outline 12.1 HYPOTHESIS TESTING 12.2 CRITICAL VALUES 12.3 ONE-SAMPLE T TEST 247 12.1. Hypothesis Testing www.ck12.org 12.1 Hypothesis Testing Learning Objectives Develop

### Power 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,

### SCHOOL OF HEALTH AND HUMAN SCIENCES DON T FORGET TO RECODE YOUR MISSING VALUES

SCHOOL OF HEALTH AND HUMAN SCIENCES Using SPSS Topics addressed today: 1. Differences between groups 2. Graphing Use the s4data.sav file for the first part of this session. DON T FORGET TO RECODE YOUR

### Hypothesis Testing. Bluman Chapter 8

CHAPTER 8 Learning Objectives C H A P T E R E I G H T Hypothesis Testing 1 Outline 8-1 Steps in Traditional Method 8-2 z Test for a Mean 8-3 t Test for a Mean 8-4 z Test for a Proportion 8-5 2 Test for

### C. The null hypothesis is not rejected when the alternative hypothesis is true. A. population parameters.

Sample Multiple Choice Questions for the material since Midterm 2. Sample questions from Midterms and 2 are also representative of questions that may appear on the final exam.. A randomly selected sample

### Mind on Statistics. Chapter 13

Mind on Statistics Chapter 13 Sections 13.1-13.2 1. Which statement is not true about hypothesis tests? A. Hypothesis tests are only valid when the sample is representative of the population for the question

### Nonparametric Statistics

1 14.1 Using the Binomial Table Nonparametric Statistics In this chapter, we will survey several methods of inference from Nonparametric Statistics. These methods will introduce us to several new tables

### Introduction. Hypothesis Testing. Hypothesis Testing. Significance Testing

Introduction Hypothesis Testing Mark Lunt Arthritis Research UK Centre for Ecellence in Epidemiology University of Manchester 13/10/2015 We saw last week that we can never know the population parameters

### MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Sample Practice problems - chapter 12-1 and 2 proportions for inference - Z Distributions Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide

### Association Between Variables

Contents 11 Association Between Variables 767 11.1 Introduction............................ 767 11.1.1 Measure of Association................. 768 11.1.2 Chapter Summary.................... 769 11.2 Chi

### Psychology 60 Fall 2013 Practice Exam Actual Exam: Next Monday. Good luck!

Psychology 60 Fall 2013 Practice Exam Actual Exam: Next Monday. Good luck! Name: 1. The basic idea behind hypothesis testing: A. is important only if you want to compare two populations. B. depends on

### Sections 4.5-4.7: Two-Sample Problems. Paired t-test (Section 4.6)

Sections 4.5-4.7: Two-Sample Problems Paired t-test (Section 4.6) Examples of Paired Differences studies: Similar subjects are paired off and one of two treatments is given to each subject in the pair.

### HypoTesting. Name: Class: Date: Multiple Choice Identify the choice that best completes the statement or answers the question.

Name: Class: Date: HypoTesting Multiple Choice Identify the choice that best completes the statement or answers the question. 1. A Type II error is committed if we make: a. a correct decision when the

### EXCEL Analysis TookPak [Statistical Analysis] 1. First of all, check to make sure that the Analysis ToolPak is installed. Here is how you do it:

EXCEL Analysis TookPak [Statistical Analysis] 1 First of all, check to make sure that the Analysis ToolPak is installed. Here is how you do it: a. From the Tools menu, choose Add-Ins b. Make sure Analysis

### General Procedure for Hypothesis Test. Five types of statistical analysis. 1. Formulate H 1 and H 0. General Procedure for Hypothesis Test

Five types of statistical analysis General Procedure for Hypothesis Test Descriptive Inferential Differences Associative Predictive What are the characteristics of the respondents? What are the characteristics

### Chapter 8. Hypothesis Testing

Chapter 8 Hypothesis Testing Hypothesis 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

### Notes 8: Hypothesis Testing

Notes 8: Hypothesis Testing Julio Garín Department of Economics Statistics for Economics Spring 2012 (Stats for Econ) Hypothesis Testing Spring 2012 1 / 44 Introduction Why we conduct surveys? We want

### Statistical 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

### AP Statistics 2002 Scoring Guidelines

AP Statistics 2002 Scoring Guidelines The materials included in these files are intended for use by AP teachers for course and exam preparation in the classroom; permission for any other use must be sought

### Unit 29 Chi-Square Goodness-of-Fit Test

Unit 29 Chi-Square Goodness-of-Fit Test Objectives: To perform the chi-square hypothesis test concerning proportions corresponding to more than two categories of a qualitative variable To perform the Bonferroni

### research/scientific includes the following: statistical hypotheses: you have a null and alternative you accept one and reject the other

1 Hypothesis Testing Richard S. Balkin, Ph.D., LPC-S, NCC 2 Overview When we have questions about the effect of a treatment or intervention or wish to compare groups, we use hypothesis testing Parametric

### Experimental Design. Power and Sample Size Determination. Proportions. Proportions. Confidence Interval for p. The Binomial Test

Experimental Design Power and Sample Size Determination Bret Hanlon and Bret Larget Department of Statistics University of Wisconsin Madison November 3 8, 2011 To this point in the semester, we have largely

### Two-Sample T-Tests Assuming Equal Variance (Enter Means)

Chapter 4 Two-Sample T-Tests Assuming Equal Variance (Enter Means) Introduction This procedure provides sample size and power calculations for one- or two-sided two-sample t-tests when the variances of

### Linear Models in STATA and ANOVA

Session 4 Linear Models in STATA and ANOVA Page Strengths of Linear Relationships 4-2 A Note on Non-Linear Relationships 4-4 Multiple Linear Regression 4-5 Removal of Variables 4-8 Independent Samples

### Name: Date: Use the following to answer questions 3-4:

Name: Date: 1. Determine whether each of the following statements is true or false. A) The margin of error for a 95% confidence interval for the mean increases as the sample size increases. B) The margin

### Study Guide for the Final Exam

Study Guide for the Final Exam When studying, remember that the computational portion of the exam will only involve new material (covered after the second midterm), that material from Exam 1 will make

### SPSS on two independent samples. Two sample test with proportions. Paired t-test (with more SPSS)

SPSS on two independent samples. Two sample test with proportions. Paired t-test (with more SPSS) State of the course address: The Final exam is Aug 9, 3:30pm 6:30pm in B9201 in the Burnaby Campus. (One

### Construct a scatterplot for the given data. 2) x Answer:

Review for Test 5 STA 2023 spr 2014 Name Given the linear correlation coefficient r and the sample size n, determine the critical values of r and use your finding to state whether or not the given r represents

### One-Way Analysis of Variance

One-Way Analysis of Variance Note: Much of the math here is tedious but straightforward. We ll skim over it in class but you should be sure to ask questions if you don t understand it. I. Overview A. We

### AP Statistics 2012 Scoring Guidelines

AP Statistics 2012 Scoring Guidelines The College Board The College Board is a mission-driven not-for-profit organization that connects students to college success and opportunity. Founded in 1900, the

### Independent samples t-test. Dr. Tom Pierce Radford University

Independent samples t-test Dr. Tom Pierce Radford University The logic behind drawing causal conclusions from experiments The sampling distribution of the difference between means The standard error of

### Chapter 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

### Outline. Definitions Descriptive vs. Inferential Statistics The t-test - One-sample t-test

The t-test Outline Definitions Descriptive vs. Inferential Statistics The t-test - One-sample t-test - Dependent (related) groups t-test - Independent (unrelated) groups t-test Comparing means Correlation

### General Method: Difference of Means. 3. Calculate df: either Welch-Satterthwaite formula or simpler df = min(n 1, n 2 ) 1.

General Method: Difference of Means 1. Calculate x 1, x 2, SE 1, SE 2. 2. Combined SE = SE1 2 + SE2 2. ASSUMES INDEPENDENT SAMPLES. 3. Calculate df: either Welch-Satterthwaite formula or simpler df = min(n

### Hypothesis Testing or How to Decide to Decide Edpsy 580

Hypothesis Testing or How to Decide to Decide Edpsy 580 Carolyn J. Anderson Department of Educational Psychology University of Illinois at Urbana-Champaign Hypothesis Testing or How to Decide to Decide

### Sampling 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

### Recall this chart that showed how most of our course would be organized:

Chapter 4 One-Way ANOVA Recall this chart that showed how most of our course would be organized: Explanatory Variable(s) Response Variable Methods Categorical Categorical Contingency Tables Categorical

### Hypothesis Testing & Data Analysis. Statistics. Descriptive Statistics. What is the difference between descriptive and inferential statistics?

2 Hypothesis Testing & Data Analysis 5 What is the difference between descriptive and inferential statistics? Statistics 8 Tools to help us understand our data. Makes a complicated mess simple to understand.

### Hypothesis Testing COMP 245 STATISTICS. Dr N A Heard. 1 Hypothesis Testing 2 1.1 Introduction... 2 1.2 Error Rates and Power of a Test...

Hypothesis Testing COMP 45 STATISTICS Dr N A Heard Contents 1 Hypothesis Testing 1.1 Introduction........................................ 1. Error Rates and Power of a Test.............................

### Chapter 11-12 1 Review

Chapter 11-12 Review Name 1. In formulating hypotheses for a statistical test of significance, the null hypothesis is often a statement of no effect or no difference. the probability of observing the data

### Hypothesis Testing - Relationships

- Relationships Session 3 AHX43 (28) 1 Lecture Outline Correlational Research. The Correlation Coefficient. An example. Considerations. One and Two-tailed Tests. Errors. Power. for Relationships AHX43

### Step 6: Writing Your Hypotheses Written and Compiled by Amanda J. Rockinson-Szapkiw

Step 6: Writing Your Hypotheses Written and Compiled by Amanda J. Rockinson-Szapkiw Introduction To determine if a theory has the ability to explain, predict, or describe, you conduct experimentation and

### Tests for Two Proportions

Chapter 200 Tests for Two Proportions Introduction This module computes power and sample size for hypothesis tests of the difference, ratio, or odds ratio of two independent proportions. The test statistics

### UNDERSTANDING THE DEPENDENT-SAMPLES t TEST

UNDERSTANDING THE DEPENDENT-SAMPLES t TEST A dependent-samples t test (a.k.a. matched or paired-samples, matched-pairs, samples, or subjects, simple repeated-measures or within-groups, or correlated groups)

### Two-Sample T-Tests Allowing Unequal Variance (Enter Difference)

Chapter 45 Two-Sample T-Tests Allowing Unequal Variance (Enter Difference) Introduction This procedure provides sample size and power calculations for one- or two-sided two-sample t-tests when no assumption

### Introduction to Hypothesis Testing. Hypothesis Testing. Step 1: State the Hypotheses

Introduction to Hypothesis Testing 1 Hypothesis Testing A hypothesis test is a statistical procedure that uses sample data to evaluate a hypothesis about a population Hypothesis is stated in terms of the

### THE FIRST SET OF EXAMPLES USE SUMMARY DATA... EXAMPLE 7.2, PAGE 227 DESCRIBES A PROBLEM AND A HYPOTHESIS TEST IS PERFORMED IN EXAMPLE 7.

THERE ARE TWO WAYS TO DO HYPOTHESIS TESTING WITH STATCRUNCH: WITH SUMMARY DATA (AS IN EXAMPLE 7.17, PAGE 236, IN ROSNER); WITH THE ORIGINAL DATA (AS IN EXAMPLE 8.5, PAGE 301 IN ROSNER THAT USES DATA FROM

### Unit 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

### Analysis of Variance ANOVA

Analysis of Variance ANOVA Overview We ve used the t -test to compare the means from two independent groups. Now we ve come to the final topic of the course: how to compare means from more than two populations.

### Multiple Hypothesis Testing: The F-test

Multiple Hypothesis Testing: The F-test Matt Blackwell December 3, 2008 1 A bit of review When moving into the matrix version of linear regression, it is easy to lose sight of the big picture and get lost

### Homework 6 Solutions

Math 17, Section 2 Spring 2011 Assignment Chapter 20: 12, 14, 20, 24, 34 Chapter 21: 2, 8, 14, 16, 18 Chapter 20 20.12] Got Milk? The student made a number of mistakes here: Homework 6 Solutions 1. Null

### Chapter 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

### Introduction to Hypothesis Testing

I. Terms, Concepts. Introduction to Hypothesis Testing A. In general, we do not know the true value of population parameters - they must be estimated. However, we do have hypotheses about what the true

### Answer: C. The strength of a correlation does not change if units change by a linear transformation such as: Fahrenheit = 32 + (5/9) * Centigrade

Statistics Quiz Correlation and Regression -- ANSWERS 1. Temperature and air pollution are known to be correlated. We collect data from two laboratories, in Boston and Montreal. Boston makes their measurements

### AP Statistics 2011 Scoring Guidelines

AP Statistics 2011 Scoring Guidelines The College Board The College Board is a not-for-profit membership association whose mission is to connect students to college success and opportunity. Founded in

### Unit 27: Comparing Two Means

Unit 27: Comparing Two Means Prerequisites Students should have experience with one-sample t-procedures before they begin this unit. That material is covered in Unit 26, Small Sample Inference for One

### Statistics 2014 Scoring Guidelines

AP Statistics 2014 Scoring Guidelines College Board, Advanced Placement Program, AP, AP Central, and the acorn logo are registered trademarks of the College Board. AP Central is the official online home

### PASS Sample Size Software

Chapter 250 Introduction The Chi-square test is often used to test whether sets of frequencies or proportions follow certain patterns. The two most common instances are tests of goodness of fit using multinomial

### HYPOTHESIS TESTING: POWER OF THE TEST

HYPOTHESIS TESTING: POWER OF THE TEST The first 6 steps of the 9-step test of hypothesis are called "the test". These steps are not dependent on the observed data values. When planning a research project,

### Class 19: Two Way Tables, Conditional Distributions, Chi-Square (Text: Sections 2.5; 9.1)

Spring 204 Class 9: Two Way Tables, Conditional Distributions, Chi-Square (Text: Sections 2.5; 9.) Big Picture: More than Two Samples In Chapter 7: We looked at quantitative variables and compared the

### UNDERSTANDING THE TWO-WAY ANOVA

UNDERSTANDING THE e have seen how the one-way ANOVA can be used to compare two or more sample means in studies involving a single independent variable. This can be extended to two independent variables

### Two-sample inference: Continuous data

Two-sample inference: Continuous data Patrick Breheny April 5 Patrick Breheny STA 580: Biostatistics I 1/32 Introduction Our next two lectures will deal with two-sample inference for continuous data As

### Statistiek I. Proportions aka Sign Tests. John Nerbonne. CLCG, Rijksuniversiteit Groningen. http://www.let.rug.nl/nerbonne/teach/statistiek-i/

Statistiek I Proportions aka Sign Tests John Nerbonne CLCG, Rijksuniversiteit Groningen http://www.let.rug.nl/nerbonne/teach/statistiek-i/ John Nerbonne 1/34 Proportions aka Sign Test The relative frequency

### Math 251, Review Questions for Test 3 Rough Answers

Math 251, Review Questions for Test 3 Rough Answers 1. (Review of some terminology from Section 7.1) In a state with 459,341 voters, a poll of 2300 voters finds that 45 percent support the Republican candidate,

### Chapter 23 Inferences About Means

Chapter 23 Inferences About Means Chapter 23 - Inferences About Means 391 Chapter 23 Solutions to Class Examples 1. See Class Example 1. 2. We want to know if the mean battery lifespan exceeds the 300-minute

### WISE Power Tutorial All Exercises

ame Date Class WISE Power Tutorial All Exercises Power: The B.E.A.. Mnemonic Four interrelated features of power can be summarized using BEA B Beta Error (Power = 1 Beta Error): Beta error (or Type II

### BA 275 Review Problems - Week 6 (10/30/06-11/3/06) CD Lessons: 53, 54, 55, 56 Textbook: pp. 394-398, 404-408, 410-420

BA 275 Review Problems - Week 6 (10/30/06-11/3/06) CD Lessons: 53, 54, 55, 56 Textbook: pp. 394-398, 404-408, 410-420 1. Which of the following will increase the value of the power in a statistical test

### 1. Comparing Two Means: Dependent Samples

1. Comparing Two Means: ependent Samples In the preceding lectures we've considered how to test a difference of two means for independent samples. Now we look at how to do the same thing with dependent

### Independent t- Test (Comparing Two Means)

Independent t- Test (Comparing Two Means) The objectives of this lesson are to learn: the definition/purpose of independent t-test when to use the independent t-test the use of SPSS to complete an independent

### About Hypothesis Testing

About Hypothesis Testing TABLE OF CONTENTS About Hypothesis Testing... 1 What is a HYPOTHESIS TEST?... 1 Hypothesis Testing... 1 Hypothesis Testing... 1 Steps in Hypothesis Testing... 2 Steps in Hypothesis

### WHERE DOES THE 10% CONDITION COME FROM?

1 WHERE DOES THE 10% CONDITION COME FROM? The text has mentioned The 10% Condition (at least) twice so far: p. 407 Bernoulli trials must be independent. If that assumption is violated, it is still okay

### QUANTITATIVE METHODS BIOLOGY FINAL HONOUR SCHOOL NON-PARAMETRIC TESTS

QUANTITATIVE METHODS BIOLOGY FINAL HONOUR SCHOOL NON-PARAMETRIC TESTS This booklet contains lecture notes for the nonparametric work in the QM course. This booklet may be online at http://users.ox.ac.uk/~grafen/qmnotes/index.html.