1 Two-sample t-tests. - Independent samples - Pooled standard devation - The equal variance assumption
2 Last time, we used the mean of one sample to test against the hypothesis that the true mean was a particular value. One-sided test: Two-sided test:
3 We also applied the idea of testing against a specific value to a proportion. After all, a proportion is just a mean of zeros (nos) and ones (yeses).
4 In every one sample test, we have a given value we re comparing the sample mean against. The question: Is this given value plausible?
5 But what if we don t have a specific value to compare against? What if, instead, we re comparing the means of two groups against each other? That s a job for two-sample testing.
6 Two independent samples. The null hypothesis is that the means are the same.
7 The alternative can be two-sided (not-equal), or one-sided on either side (less than / more than)
8 Another way to say two means at the same is The difference between these means is zero.
9 By using the difference of zero mindset, we get a hint on how to handle two sample tests. Make the null hypothesis about the difference between the means, instead of just a single mean. Then our t-score formula works just fine.
10 Let 1 and 2 be the means of each of the samples. Let 1 and 2 be the true means, (that H 0 says are the same) The t-score looks like:
11 We never deal with the formula in this form. Since the null assumption is that the true means are the same.
12 This is the formula we really use for the t-score. (But now you know where it comes from) The standard error is defined by further assumptions.
13 A hairy topic, to be sure. How about an example?
14 Example: New textbook vs. old textbook. Say we wanted to test if a new brand of textbook is a better resource as measured by provincial exam scores. (alpha =.05) We ll get two classrooms of the same grade and similar skill level. We ll then flip a coin, if heads: Class A gets the current textbook, Class B get the new one. Otherwise, Class A gets the new one.
15 At the end of the semester, these are the summary stats: Old text class New text class Mean Std. Dev s Sample Size n
16 First, identify: We are comparing two mean, a two-sample test is appropriate. We want to know if the new textbook is better so this is a onesided test. But what do we do about the standard error?
17 There are values for n, and two values for standard deviation. Old text class New text class Mean Std. Dev s Sample Size n We can get a measure that collects ( pools ) the standard deviation of the whole system. It s called the pooled standard deviation.
18 The pooled standard deviation, S P, works by way of a weighted average of the variances (standard deviation squared) that we already have. It s a weighted average because if one sample is much larger (has more degrees of freedom), it should count for more.
19 In this case, the samples are roughly the same size, so the pooled standard deviation S P should be near the middle of the two standard deviations (7.1 and 7.4). The degrees of freedom of the pooled standard deviation S P is the total of the df from each sample. (n 1 1) + (n 2 1)
20 S P =7.26 is used to find the standard error.
21 This standard error formula is used when there are two samples, hence it has two sample sizes n 1 and n 2.
22 Both n 1 and n 2 have to be large to make the standard error small. This reflects that added uncertainty of having two unknown means instead of one.
23 Now we can get the t-score. SE = 2.19 X new = 68.8 X old = 64.3
24 Sample 1 has 20 df, Sample 2 has 22 df, so the total df is 42. We compare this t-score, 2.054, to the critical value in the t- table, one-sided, 0.05 significance at 40 df (always round down if it s between values). t * = 1.684, t-score = > t * Reject H 0 The new book classroom did do better.
25 Puzzled? Let s try another.
26 Example 2: Bio-equivalence. When one drug is being tested to replace another, it s important to check that the new drug has the same effects on the body as the old drug. This is typically done with a two-sample t-test. Expenza*, a name-brand drug is being used to lower blood pressure. We ve been hired to test if Thriftubin*, a cheaper generic drug, has the same effect on blood pressure. *Made up drug names
27 The name-brand drug has been used for a while, so we have lots of data on it, so its sample size is large. Blood pressure (Systolic) Name Brand Generic Mean Std. Dev s Sample Size n
28 Comparing two means Two-sample t-test. We want to show equal vs. not equal. Two-sided test.
29 First, get the pooled standard deviation. Use S P = to get the standard error.
30 Finally, use SE = 3.34, XNB = 130.0, XG = to find the t- score and compare that to the critical value. t-score = 1.944, t* = (using df=100,.05 significance, twosided), so we fail to reject, but just barely.
31 We can t detect at the 5% level that the generic drug has a different effect on blood pressure than the name-brand drug. But it s close, (t-score was almost as large as t*) so we ll want to mention that to whomever hired us. If we had a larger sample, we d be better able to detect a difference. Which sample size would we increase to do this?
32 We can t detect at the 5% level that the generic drug has a different effect on blood pressure than the name-brand drug. But it s close, (t-score was almost as large as t*) so we ll want to mention that to whomever hired us. If we had a larger sample, we d be better able to detect a difference. Which sample size would we increase to do this? The generic drug sample (it s smaller, so that s where a lot of our uncertainty is coming from)
33 I hope you re not feeling in the dark.
34 Is Syria more dangerous now than two years ago? The WHO (World Health Organization) is interested in knowing if the fighting in Syria is affecting the general health of the population. They give you the death records to two days in the capital, Damascus; one in 2010, and one in the same day in 2012 (to avoid seasonal effects).
35 *2010 Mean from world factbook. All else made up. Age at Death Syria Then Syria Now Mean Std. Dev Sample Size We take the average age of the deaths on each day to estimate life expectancy. Is life expectancy in 2012 less than in 2010? (alpha =.01) (Two sample, one side test.)
36 The pooled S P assumes that the true standard deviation of both groups is the same, and that we can use both sample standard deviations together to get a better estimate of that true standard deviation. In this case, the standard deviation (and hence the variance) is much larger in the 2012 death ages. The assumption of equal variance isn t reasonable, so we can t get a pooled estimate.
37 Instead, we use a standard error formula that includes both standard deviations, but separately.
38 Just make sure S 1 and n 1 are referring to the same sample.
39 The rest is getting the t-score and comparing to a critical. t* = for a one sided test, alpha = 0.01, and df=44. t-score = -0.81, which is between and 2.414, so we fail to reject. We cannot find evidence that life expectancy as a whole has been reduced in Damascus, Syria.
40 By using two different standard deviations, we lose some degrees of freedom. SPSS computes the degrees of freedom exactly. (formula available on request) However, whenever we re doing this by hand, we ll use the worst case scenario of the lower of (n 1 1) and (n 2 1) So that s why df=45 1 = 44, instead of (45 1) + (52 1) = 95
41 How do we know if the standard deviations are too different to use the pooled estimate S P? SPSS has a test to see if the pooled standard deviation is reasonable, and will give us both answers. (Levene s test for equal variances) To do it ourselves, we have the F-statistic and F-test, which we ll cover in the ANOVA section near the end of the semester.
42 Also, if you don t assume equal variance and use the standard error formula with separate standard deviations, you ll get an answer very close the standard error from S P. Textbook SE using S P Textbook SE using S 1, S Drugs using S P 3.34 Drugs using S 1, S
43 Next day: Paired t-tests Two sample t-tests for proportions SPSS two samples
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
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
Odds ratio, Odds ratio test for independence, chi-squared statistic. Announcements: Assignment 5 is live on webpage. Due Wed Aug 1 at 4:30pm. (9 days, 1 hour, 58.5 minutes ) Final exam is Aug 9. Review
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
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
Unit 31 A Hypothesis Test about Correlation and Slope in a Simple Linear Regression Objectives: To perform a hypothesis test concerning the slope of a least squares line To recognize that testing for a
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
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
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
Simple Linear Regression Inference 1 Inference requirements The Normality assumption of the stochastic term e is needed for inference even if it is not a OLS requirement. Therefore we have: Interpretation
Chapter 47 Confidence Intervals for the Difference Between Two Means Introduction This procedure calculates the sample size necessary to achieve a specified distance from the difference in sample means
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
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,
Glo bal Leadership M BA BUSINESS STATISTICS FINAL EXAM Name: INSTRUCTIONS 1. Do not open this exam until instructed to do so. 2. Be sure to fill in your name before starting the exam. 3. You have two hours
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
Statistics: An introduction to sample size calculations Rosie Cornish. 2006. 1 Introduction One crucial aspect of study design is deciding how big your sample should be. If you increase your sample size
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.
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
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
INTERPRETING THE ONE-WAY ANALYSIS OF VARIANCE (ANOVA) As with other parametric statistics, we begin the one-way ANOVA with a test of the underlying assumptions. Our first assumption is the assumption of
Notation: Notation and Equations for Regression Lecture 11/4 m: The number of predictor variables in a regression Xi: One of multiple predictor variables. The subscript i represents any number from 1 through
Chicago Booth BUSINESS STATISTICS 41000 Final Exam Fall 2011 Name: Section: I pledge my honor that I have not violated the Honor Code Signature: This exam has 34 pages. You have 3 hours to complete this
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
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 565 Randomized Block Analysis of Variance Introduction This module analyzes a randomized block analysis of variance with up to two treatment factors and their interaction. It provides tables of
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
1 Final Review 2 Review 2.1 CI 1-propZint Scenario 1 A TV manufacturer claims in its warranty brochure that in the past not more than 10 percent of its TV sets needed any repair during the first two years
t-tests in Excel By Mark Harmon Copyright 2011 Mark Harmon No part of this publication may be reproduced or distributed without the express permission of the author. email@example.com www.excelmasterseries.com
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
Chapter 10 Hypothesis Testing: Two Means, Paired Data, Two Proportions 10.1 Hypothesis Testing: Two Population Means and Two Population Proportions 1 10.1.1 Student Learning Objectives By the end of this
Chapter 2 Probability Topics SPSS T tests Data file used: gss.sav In the lecture about chapter 2, only the One-Sample T test has been explained. In this handout, we also give the SPSS methods to perform
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
STATISTICS PROJECT: Hypothesis Testing See my comments in red. Scoring last page. INTRODUCTION My topic is the average tuition cost of a 4-yr. public college. Since I will soon be transferring to a 4-yr.
Variations of the t-test: Sample tail Sample t-test (unequal sample sizes and unequal variances) Like the last example, below we have ceramic sherd thickness measurements (in cm) of two samples representing
Comparing Means in Two Populations Overview The previous section discussed hypothesis testing when sampling from a single population (either a single mean or two means from the same population). Now we
Parkland College A with Honors Projects Honors Program 2014 Calculating P-Values Isela Guerra Parkland College Recommended Citation Guerra, Isela, "Calculating P-Values" (2014). A with Honors Projects.
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
Descriptive Statistics Primer Descriptive statistics Central tendency Variation Relative position Relationships Calculating descriptive statistics Descriptive Statistics Purpose to describe or summarize
Two Related Samples t Test In this example 1 students saw five pictures of attractive people and five pictures of unattractive people. For each picture, the students rated the friendliness of the person
Consider a study in which How many subjects? The importance of sample size calculations Office of Research Protections Brown Bag Series KB Boomer, Ph.D. Director, firstname.lastname@example.org A researcher conducts
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
Comparing Two Groups Chapter 7 describes two ways to compare two populations on the basis of independent samples: a confidence interval for the difference in population means and a hypothesis test. The
1: Analysis of Variance Introduction EDA Hypothesis Test Introduction In Chapter 8 and again in Chapter 11 we compared means from two independent groups. In this chapter we extend the procedure to consider
HYPOTHESIS TESTING (ONE SAMPLE) - CHAPTER 7 1 PREVIOUSLY used confidence intervals to answer questions such as... You know that 0.25% of women have red/green color blindness. You conduct a study of men
BA 275 Review Problems - Week 5 (10/23/06-10/27/06) CD Lessons: 48, 49, 50, 51, 52 Textbook: pp. 380-394 1. Does vigorous exercise affect concentration? In general, the time needed for people to complete
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
Difference of Means and Problems Dr. Tom Ilvento FREC 408 Accounting Firm Study An accounting firm specializes in auditing the financial records of large firm It is interested in evaluating its fee structure,particularly
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
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
Permutation Tests for Comparing Two Populations Ferry Butar Butar, Ph.D. Jae-Wan Park Abstract Permutation tests for comparing two populations could be widely used in practice because of flexibility of
Unit 26 Estimation with Confidence Intervals Objectives: To see how confidence intervals are used to estimate a population proportion, a population mean, a difference in population proportions, or a difference
Chapter 7 One-way ANOVA One-way ANOVA examines equality of population means for a quantitative outcome and a single categorical explanatory variable with any number of levels. The t-test of Chapter 6 looks
SPSS Guide: Regression Analysis I put this together to give you a step-by-step guide for replicating what we did in the computer lab. It should help you run the tests we covered. The best way to get familiar
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)
1 One-Way ANOVA using SPSS 11.0 This section covers steps for testing the difference between three or more group means using the SPSS ANOVA procedures found in the Compare Means analyses. Specifically,
Unit 31: One-Way ANOVA Summary of Video A vase filled with coins takes center stage as the video begins. Students will be taking part in an experiment organized by psychology professor John Kelly in which
Testing Group Differences using T-tests, ANOVA, and Nonparametric Measures Jamie DeCoster Department of Psychology University of Alabama 348 Gordon Palmer Hall Box 870348 Tuscaloosa, AL 35487-0348 Phone:
Statistics: Rosie Cornish. 200. 1.5 Oneway Analysis of Variance 1 Introduction Oneway analysis of variance (ANOVA) is used to compare several means. This method is often used in scientific or medical experiments
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
KSTAT MINI-MANUAL Decision Sciences 434 Kellogg Graduate School of Management Kstat is a set of macros added to Excel and it will enable you to do the statistics required for this course very easily. To
1 Simple ANalysis Of Variance (ANOVA) Oftentimes we have more than two groups that we want to compare. The purpose of ANOVA is to allow us to compare group means from several independent samples. In general,
Hypothesis Testing for a Proportion Example: We are interested in the probability of developing asthma over a given one-year period for children 0 to 4 years of age whose mothers smoke in the home In the
1 Chapter 7 Comparing Means in SPSS (t-tests) This section covers procedures for testing the differences between two means using the SPSS Compare Means analyses. Specifically, we demonstrate procedures
TIPS FOR DOING STATISTICS IN EXCEL Before you begin, make sure that you have the DATA ANALYSIS pack running on your machine. It comes with Excel. Here s how to check if you have it, and what to do if you
Part 3 Comparing Groups Chapter 7 Comparing Paired Groups 189 Chapter 8 Comparing Two Independent Groups 217 Chapter 9 Comparing More Than Two Groups 257 188 Elementary Statistics Using SAS Chapter 7 Comparing
ANOVA ANOVA Analysis of Variance Chapter 6 A procedure for comparing more than two groups independent variable: smoking status non-smoking one pack a day > two packs a day dependent variable: number of
STAT 145 (Notes) Al Nosedal email@example.com Department of Mathematics and Statistics University of New Mexico Fall 2013 CHAPTER 18 INFERENCE ABOUT A POPULATION MEAN. Conditions for Inference about mean
1 2 CONTENTS OF DAY 2 I. More Precise Definition of Simple Random Sample 3 Connection with independent random variables 3 Problems with small populations 8 II. Why Random Sampling is Important 9 A myth,
Chapter 6 Testing for Lack of Fit How can we tell if a model fits the data? If the model is correct then ˆσ 2 should be an unbiased estimate of σ 2. If we have a model which is not complex enough to fit
Data Analysis Tools This section of the notes is meant to introduce you to many of the tools that are provided by Excel under the Tools/Data Analysis menu item. If your computer does not have that tool
STA 3024 Practice Problems Exam 2 NOTE: These are just Practice Problems. This is NOT meant to look just like the test, and it is NOT the only thing that you should study. Make sure you know all the material
STATISTICS 8, FINAL EXAM NAME: KEY Seat Number: Last six digits of Student ID#: Circle your Discussion Section: 1 2 3 4 Make sure you have 8 pages. You will be provided with a table as well, as a separate
General Regression Formulae Single Predictor Standardized Parameter Model: Z Yi = β Z Xi + ε i Single Predictor Standardized Statistical Model: Z Yi = β Z Xi Estimate of Beta (Beta-hat: β = r YX (1 Standard
NEW YORK UNIVERSITY ROBERT F. WAGNER GRADUATE SCHOOL OF PUBLIC SERVICE Course Syllabus Spring 2016 Statistical Methods for Public, Nonprofit, and Health Management Section Format Day Begin End Building
Math 108 Exam 3 Solutions Spring 00 1. An ecologist studying acid rain takes measurements of the ph in 12 randomly selected Adirondack lakes. The results are as follows: 3.0 6.5 5.0 4.2 5.5 4.7 3.4 6.8
14.2 Do Two Distributions Have the Same Means or Variances? 615 that this is wasteful, since it yields much more information than just the median (e.g., the upper and lower quartile points, the deciles,