Inferential Statistics & Variables Paper This is the first week we’re looking at a specific inferential statistic. Please read the week’s materials and “W

Inferential Statistics & Variables Paper This is the first week we’re looking at a specific inferential statistic.

Please read the week’s materials and “Which Statistical Test Do You Use?” as linked in this week’s materials. This is a road map featuring many of the inferential statistics we’ll cover in the next few weeks.

As you’re reading this document, think about your research question and the variables in your evaluation proposal. Are you looking at relationships between variables or differences between groups? Maybe both for different variables? What test/s do you think you’ll propose using in your evaluation proposal?

In your reflection, remind me of

Your research question
Your variables
And propose the statistical test/s you think you’ll use to answer your research question/s.

It’s okay if you’re not sure. I just want you to start thinking about these things. Week 9 Materials
Contents
• Readings
• An Important Note
• t-tests
o t-test for Independent Groups
o t-test for Dependent Groups
• The t Distribution
o Degrees of Freedom
o Critical Value of t
• The 6 Steps of Hypothesis Testing for a t-test
• Important Statistical Terms
• Tools for Calculating t-tests
1 Readings
Textbook Readings
• Inferential Statistics, t-tests, Chapter 9,Assignment Preparation
• Gender Differences in Searching for Health Information on the Internet and the
Virtual Patient-Physician Relationship in Germany: Exploratory Results on
How
Men
and
Women
Differ
and
Why
(Link:
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4526954/)
• Which Statistical Test Do You Use? (PDF)
An Important Note
The remaining weeks of the semester will focus on inferential statistics. If you
recall from chapter 1 (I won’t be offended if you don’t!), inferential statistics are
statistical procedures used to test hypotheses and draw conclusions from data collected
during research studies.
From here on out, we’ll use the text more as a reference. A greater emphasis will
be on the lectures and links within them. Consequently, the lectures will be longer than
in previous weeks but the text reading will be lighter.
Also, know that in a traditional statistics course, you would learn how to calculate
these inferential statistics. You won’t be expected to calculate these statistics by hand,
but you will be directed to links to statistical software programs. Know this course
offers a broad review of inferential statistics and you are encouraged to take additional
statistical courses to master this material.
2
t-tests
This lecture will cover the t-test for independent groups & the t-test for dependent
groups. This material is covered in Chapter 9.
Let’s say a marketing firm asks you to volunteer to help them with product
evaluation. You are randomly assigned to 1 of 2 groups, Group A or Group B (you don’t
3
know there is another group besides the one to which you are assigned). Group A sees
the Big Y brand soup and Group B sees the Progresso brand soup.
You (and everyone else) are then asked to evaluate the aesthetics of the product
labels on a scale from 1 (does not look appealing) to 7 (looks very appealing).
Below are the ratings from the two focus groups.
Question: Looking at these numbers, how do we know which soup is more appealing?
Answer: We compare the group means.
• Group A: x? = 22/7 = 3.14
• Group B: x? = 36/7 = 5.14
How do we know if 3.14 is statistically different than 5.14? We need to conduct
an independent groups t-test (independent because the two groups were not related in
any way, they had different people in them).
Required Video 1: Independent t-test – Explained Simply (Link:

3.1 t-test for Independent Groups
This test is used when participants are randomly assigned to either an experimental
group or a control group. This type of experiment allows us to assume a cause and effect
relationship between the independent and dependent variable.
Assumptions
• the groups are independent
• the dependent variable is normally distributed
• the variance is homogeneous
Imagine another way of doing things…
Let’s say you’ve been hired to evaluate brownies on a scale of 1 (do not like) to 7
(like very much). You taste Brownie A, rate it, take a sip of water and then taste Brownie
B and rate it. The ratings provided by you and six others are below.
How do we know which brownie tasted better? We compare the means, as we did
before
• Brownie A: x? = 28/7 = 4.00
• Brownie B: x? = 31/7 = 4.43
But we can’t do an independent samples t-test because the Brownie A ratings are
from the same people as the Brownie B ratings (i.e., they are NOT independent).
3.2 t-test for Dependent Groups
We need to use a t-test for dependent groups for the brownie ratings. Know that
a t-test for dependent groups is also called
• a repeated measures t-test
• a paired samples t-test
• a dependent-samples t-test
• a within-subjects t-test
Required
Video
2:
Paired
t-test
Lecture
(Link:

This test is used when the participants in both groups are related to each other in
some way. The pair of scores are dependent (i.e., linked).
Examples
o Before and after some treatment
? Before: How stressed do you feel?
? Treatment: Have a massage
? After: Now how stressed do you feel?
o Participants are ‘related’ in some way
? Twins
? Couples
? Similar/matched scores
o Participants receive both levels of the variables
? For example, both brownie A and brownie B
This test can often be more powerful than a t-test for independent groups because
the variability of the same participant over time is less than the variability between 2
unrelated participants.
Assumptions of the t-test for dependent groups
• the dependent variable is normally distributed
• there must be an equal # of scores in each group
• the scores within a group are independent of one another
The t Distribution
The population variance of this distribution is estimated, so the comparison
distribution is not a normal curve
• It is a t distribution
• It is like a normal curve
o Bell-shaped
o Unimodal
o Symmetrical
• But has more scores at the extremes (i.e., heavier tails) and varies according to
the degrees of freedom
4
4.1 Degrees of Freedom
Degrees of freedom are the number of values free to vary in a calculation.
Example
_ + _ + _ + 2 = 16
•
5 + 3 + 6 + 2 = 16
o 4 + 5 + 5 + 2 = 16
o 1 + 9 + 4 + 2 = 16
Three numbers in each of these equations are free to vary- the last number can only be
one value if the first three vary- so this calculation has 3 degrees of freedom.
The equation for degrees of freedom for a t distribution is:
df = n – 1
Know that as the sample size increases, the t distribution approaches the normal
distribution.
o
4.2 Critical Value of t
The table of critical values of t can be found at the back of your book in Table 3.
You need to know 3 things in order to find the critical value of t:
• the degrees of freedom (n-1)
• the level of significance (researchers usually set this, alpha, at .05)
• whether it is a directional (one-tailed) or non-directional (two-tailed) test
Example 1
You have 21 participants in your sample. You set the probability of making a Type I
error (?) at .05. You have a non-directional hypothesis (a two-tailed test).
Question: What is the corresponding critical t?
Answer: 2.086
Example 2
You have 17 participants in your sample. You set the probability of making a Type I
error (?) at .01. You have a directional (one-tailed) hypothesis.
Question: What is the corresponding critical t?
Answer: 2.583
Example 3
You have 17 participants in your sample. You set the probability of making a Type I
error (?) at .01. You have a directional (one-tailed) hypothesis.
Question: What is the corresponding critical t?
Answer: 2.583
The 6 Steps of Hypothesis Testing for a t-test
1. State the Research Problem
2. State the Statistical Hypotheses
3. Make the Decision Rule
4. Do the Calculations
5. Make the Decision
6. The Interpretation
1. State the Research Problem
“Mean A is different from mean B.”
2. State the Statistical Hypothese
H0: ??? 1 = ??? 2
Ha: ??? 1 ? ??? 2
3. Make the Decision Rule
Determine a one-tailed or a two-tailed (non-directional alternative hypothesis) test
of significance
Determine the level of significance
p =.05 (conventional level)
4. Do the Calculations (or have a statistical software program do them for you!)
Determine the critical values of t at p < .05 with whatever your degrees of freedom are Note this is (N1+N2)-2 for the dependent groups t-test. (Table 3 provides the critical values of t) 5. Make the Decision & 6. The Interpretation Compare the calculated t-value (formula-derived) with the critical values of t. • If the calculated t-value exceeds the critical/tabled t-value, then the corresponding p is less than .05 and the null hypothesis is rejected. It is concluded that the means are significantly different from each other. • If the calculated t-value does not exceed the critical/tabled t-value, then the corresponding p is greater than .05 and the null hypothesis is retained. It is 5 concluded that the means are not significantly different from each other. Important Statistical Terms Some of these terms are described in Chapter 10. Power Power is the probability that the null hypothesis will be rejected correctly. Researchers increase the power of their experiments by • having many participants, not few • making the participants as alike as possible (e.g., restrict the ages of participants) Effect Size Effect size tells us how strongly the independent variable affects the dependent variable. Effect sizes range from small (the difference between the 2 groups is small) to large (the difference between the 2 groups is large). Two common ways to measure effect size are: • The correlation coefficient of effect size • Cohen’s d The Correlation Coefficient of Effect Size 6 Notice these r values are smaller than those for the Pearson correlation coefficient r (we’ll get to this in a bit). These are 2 separate statistical procedures, though both use the coefficient r. Cohen's d Cohen’s d values can be greater than 1 (if the effect is very large). Cohen’s d tells us how many standard deviations the two means are different from each other. Standard Error Standard error is the standard deviation of the distribution of means. The formula is: Confidence Intervals Confidence intervals estimate the amount of error involved in our data. They are based on three elements: • Value of a parameter around which the confidence interval will be built (e.g. mean, correlation coefficient) • The standard error of the measure (SE) • The desired width of the confidence interval (95% or 99% confidence interval) Confidence intervals are calculated ... 95% CI = M ± (z * SE) Or, M – (z * SE) to M + (z * SE) Where… • M = the mean of the distribution of means • SE = the standard error of the mean o The standard deviation of the sampling distribution • z = the z score for the particular confidence interval of interest The values between -1.96 and +1.96 cover 95% of the cases if the means are normally distributed (the mean is 0 and the Standard Deviation = 1). Tools for Calculating t-tests For those of you who want to learn how to calculate t-tests using the Excel Toolpak, please check out the folder titled, "Beyond the Basics" at the top of the Moodle homepage and open the "Using the Excel Toolpak" folder. 7 I recommend you read the document, watch the linked videos and run the appropriate tests. When you're done, send me your answers and I'll let you know how you did. Toolpak Support The following resources may be helpful to you as you develop competence in the use of Excel and specifically, the ToolPak. Video: Excel - Basic Descriptive Statistics (Mean, Variance, Standard Devation, etc.) (Link: https://www.youtube.com/watch?v=-tFWH7AYLek&feature=emb_title) Use the Analysis ToolPak to perform complex data analysis (Link: https://support.microsoft.com/en-us/office/use-the-analysis-toolpak-to-performcomplex-data-analysis-6c67ccf0-f4a9-487c-8dec-bdb5a2cefab6?ui=en-us&rs=enus&ad=us) Excel Easy (Link: https://www.excel-easy.com/) Which Statistical Test Do You Use?1 Are you interested in testing the difference between group means or the association/relationship among variables? Difference between group means Are you comparing two means or more than two means? Two means Do you have the same or different participants in each group? Different Participants Independent Samples t-test Same Participants Dependent Samples t-test More than two means Do you have the same or different participants in each group? Different Participants One-Way ANOVA (if you have one factor) or Factorial ANOVA (if you have more than one factor) Association/relationship among variables What type of variables do you have and how many are there? One interval or ratio predictor variable Correlation or Simple Linear Regression Two or more interval or ratio predictor variables Two nominal variables Chi-Square Multiple Linear Regression Same Participants One-Factor Repeated Measures ANOVA This decision tree is adapted from the one presented in Aron, A., Coups, E.J., & Aron, E.N. (2011). Statistics for the Behavioral & Social Sciences: A Brief Course. New York: Prentice Hall. This chart assumes that the data are normally distributed with approximately equal variance/ homogeneity of variance with unknown population parameters. Fundamental Statistics for the Social and Behavioral Sciences 2 To my kids, Meagan and Will, and my parents, Katsumi and Grayce Tokunaga. 3 Fundamental Statistics for the Social and Behavioral Sciences Howard T. Tokunaga San Jose State University 4 FOR INFORMATION: SAGE Publications, inc. 2455 Teller Road Thousand Oaks, California 91320 E-mail: order@sagepub.com SAGE Publications Ltd. 1 Oliver's Yard 55 City Road London EC1Y 1SP United Kingdom SAGE Publications India Pvt. Ltd. B 1/I 1 Mohan Cooperative Industrial Area Mathura Road, New Delhi 110 044 India SAGE Publications Asia-Pacific Pte. Ltd. 3 Church Street #10-04 Samsung Hub Singapore 049483 Copyright © 2016 by SAGE Publications, Inc. All rights reserved. No part of this book may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system, without permission in writing from the publisher. All trademarks depicted within this book, including trademarks appearing as part of a screenshot, figure, or other image are included solely for the purpose of illustration and are the property of their respective holders. The use of the trademarks in no way indicates any relationship with, or endorsement by, the holders of said trademarks. SPSS is a registered trademark of International Business Machines Incorporated. Printed in the United States of America Cataloging-in-Publication Data is available for this title from the Library of Congress. ISBN 978-1-4833-1879-0 5 This book is printed on acid-free paper. 15 16 17 18 19 10 9 8 7 6 5 4 3 2 1 Acquisitions Editor: Vicki Knight Editorial Assistant: Yvonne McDuffee Associate Editor: Katie Bierach Production Editor: Kelly DeRosa Copy Editor: Gillian Dickens Typesetter: C&M Digitals (P) Ltd. Proofreader: Jennifer Grubba Indexer: Marilyn Augst Cover Designer: Candice Harman Marketing Manager: Nicole Elliott 6 Detailed Contents Preface Acknowledgments About the Author Chapter 1. Introduction to Statistics Chapter 2. Examining Data: Tables and Figures Chapter 3. Measures of Central Tendency Chapter 4. Measures of Variability Chapter 5. Normal Distributions Chapter 6. Probability and Introduction to Hypothesis Testing Chapter 7. Testing One Sample Mean Chapter 8. Estimating the Mean of a Population Chapter 9. Testing the Difference between Two Means Chapter 10. Errors in Hypothesis Testing, Statistical Power, and Effect Size Chapter 11. One-Way Analysis of Variance (ANOVA) Chapter 12. Two-Way Analysis of Variance (ANOVA) Chapter 13. Correlation and Linear Regression Chapter 14. Chi-Square Tables Appendix: Review of Basic Mathematics Glossary References Index 7 Preface It may surprise students to learn they have something in common with writers of books such as this one: When you get close to finishing a writing assignment, you get a bit tired and a bit lazy. The first attempt at this preface was written shortly after final drafts of chapters were sent to my editor at SAGE, Vicki Knight. After reading it, she said, “It's not bad, but it reads like the ‘typical’ Preface. I think it would be useful for the reader to have a sense of why you wrote this book and why you wrote it the way you did.” In responding to my editor's plea for self-analysis, I found that this book's journey began in college. When I entered college, I thought my path would take me to law school; however, taking an Intro to Psych class my freshman year made me realize I enjoy the challenge of trying to understand the human mind. The school I attended, UC Santa Cruz, was a fairly unconventional university, but somehow in the midst of a sea of humanistic psychologists, I became attracted to the empirical and methodological aspects of psychology. This was a result of taking classes with instructors such as David Harrington and Dane Archer, who showed me that statistics could appeal to students if taught using a gentle, guiding approach that addresses questions relevant to students' lives. After graduating from college, I was able to get a job as a research assistant for a human resource consulting firm. Despite my lack of work experience, I was hired primarily as a result of having taken statistics and research methods courses, which taught me that learning statistics has benefits both inside and outside of the classroom. Several years later, I started grad school at UC Berkeley, where two events critical to this book took place. First, serving as a teaching assistant, I found I really enjoyed helping students, particularly in statistics and research methods classes that were often viewed with fear and suspicion. Second, I took graduate classes from Geoff Keppel, who had developed his own unique method and system for analyzing experimental research designs. His lectures and books were instrumental in showing me that statistics can be taught in a systematic way that highlights similarities rather than differences between different research situations. Geoff managed to transform something as daunting sounding as a “3 × 2 × 4 research design” into the mathematical equivalent of playing with wooden toy alphabet blocks labeled “A,” “B,” and “C.” For a long time, I thought my gratitude to Geoff was an isolated occurrence. However, the appreciation others have for his approach to teaching statistics was made apparent to me several years later when I watched him receive an American Psychological Association (APA) Lifetime Achievement award. After leaving Cal, I took on a teaching position at San Jose State, where my teaching responsibilities included an introductory statistics course aimed at students with a wide range of background, ability, and motivation. As I needed to select a textbook to use in this course, for the first time I looked carefully at the wide range of offerings. What I found 8 striking (and still find striking) was that the majority of books focused on providing formulas and very small sets of data designed to demonstrate how to correctly calculate the correct numbers from these formulas. Little emphasis, however, was given to what these numbers meant or implied. Given my own experiences learning statistics, I thought a book was needed that discusses statistics in a thematic manner, focusing on how they are used to answer questions and test ideas within the larger research process. The primary purpose of this book is to not just teach students how to calculate statistics but how to interpret the results of statistical analyses in light of a study's research hypothesis and to communicate one's results and interpretations to a broader audience. Hopefully, this book will not only help students understand the purpose and use of statistics but also give them a greater understanding of how research studies are conceived, conducted, and communicated. The 14 chapters of this book may be placed into three general categories. The first four chapters are designed to introduce students to the research process and how data that have been collected may be organized, presented, and summarized. Chapters 5 through 10 discuss the process of conducting statistical analyses to test research questions and hy... Purchase answer to see full attachment