# Unit 31 A Hypothesis Test about Correlation and Slope in a Simple Linear Regression

Size: px
Start display at page: ## Transcription

1 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 statistically significant slope in a linear regression and testing for a statistically significant linear relationship (i.e., correlation) are actually the same test Recall that we can use the Pearson correlation r and the least squares line to describe a linear relationship between two quantitative variables X and Y. We called a sample of observations on two such quantitative variables bivariate data, represented as (x 1, y 1 ), (x, y ),, (x n, y n ). Typically, we let Y represent the response variable and let X represent the explanatory variable. From such a data set, we have previously seen how to obtain the Pearson correlation r and the least squares line. We now want a hypothesis test to decide whether a linear relationship is statistically significant. For example, return to Table 10- where data is recorded on X = the dosage of a certain drug (in grams) and Y = the reaction time to a particular stimulus (in seconds) for each subject in a study. We have reproduced this data here as Table From the calculations in Table 10-3, you previously found the correlation between dosage and reaction time to be r = 0.968, and from these same calculations you also found the least squares line to be rct = (dsg), where rct represents reaction time in seconds, and dsg represents dosage in grams. However, these are just descriptive statistics; alone, they can not tell us whether there is sufficient evidence of a linear relationship between the two quantitative variables. When we study the prediction of a quantitative response variable Y from a quantitative explanatory variable X based on the equation for a straight line, we say we are studying the linear regression to predict Y from X or the linear regression of Y on X. Deciding whether or not a linear relationship is significant is the same as deciding whether or not the slope in the linear regression to predict Y from X is different from zero. If the slope is zero, then Y does not tend to change as X changes; if the slope is not zero, then Y will tend to change as X changes. We need a hypothesis test to decide if there is sufficient evidence that the slope in a simple linear regression is different from zero. Such a test could be one-sided or two-sided. If our focus was on finding evidence that the slope is greater than zero (i.e., finding evidence that the linear relationship was a positive one), then we would use a one-sided test. If our focus was on finding evidence that the slope is less than zero (i.e., finding evidence that the linear relationship was a negative one), then we would use a one-sided test. However, if our focus was on finding evidence that the slope is different from zero in either direction (i.e., finding evidence that the linear relationship was either positive or negative), then we would use a two-sided test. With a simple random sample of bivariate data of size n, the following t statistic with n degrees of freedom is available for the desired hypothesis test: t n = s Y X b 0 ( x x), 34

2 where s Y X is called the standard error of estimate. The formula for this t statistic has a format similar to other t statistics we have encountered. The numerator is the difference between a sample estimate of the slope (b) and the hypothesized value of the slope (zero), and the denominator is an appropriate standard error. The calculation of this test statistic requires obtaining the slope b of the least squares line, the sum of the squared deviations of the sample x values from their mean ( x x), and the standard error of estimate s Y X. We have previously seen how to obtain b and ( x x), but we have never previously encountered the standard error of estimate s Y X. You can think of the standard error of estimate s Y X for bivariate data similar to the way you think of the standard deviation s for one sample of observations of a quantitative variable, which can be called univariate data. Recall that the standard deviation for a sample x values is obtained by dividing ( x x) by one less than the sample size, and taking the square root of the result. The standard error of estimate s Y X is a measure of the dispersion of data points around the least squares line similar to the way that the standard deviation s is a measure of the dispersion of observations (around the sample mean) in a single sample. The standard error of estimate s Y X is obtained by dividing the sum of the squared residuals by two less than the sample size, and taking the square root of the result. If you wish, you may do this calculation for the data of Table 31-1, after which you can calculate the test statistic t n. In general, a substantial amount of calculation is needed to obtain this test statistic, and the use of appropriate statistical software or a programmable calculator is recommended. Consequently, shall not concern ourselves with these calculations in detail. To illustrate the t test about the slope in a simple linear regression, let us consider using a 0.05 significance level to perform a hypothesis test to see if the data of Table 31-1 (Table 10-) provide evidence that the linear relationship between drug dosage and reaction time is significant, or in other words, evidence that the slope in the linear regression to predict reaction time from drug dosage is different from zero. The first step is to state the null and alternative hypotheses, and choose a significance level. This test is two-sided, since we are looking for a relationship which could be positive or negative (i.e., a slope which could be different from zero in either direction). We can complete the first step of the hypothesis test as follows: H 0 : slope = 0 vs. H 1 : slope 0 (α = 0.05, two-sided test). We could alternatively choose to write the hypotheses as follows: H 0 : The linear relationship between drug dosage and reaction time is not significant vs. (α = 0.05) H 1 : The linear relationship between drug dosage and reaction time is significant The second step is to collect data and calculate the value of the test statistic. Whether you choose to do all the calculations yourself or to use the appropriate statistical software or programmable calculator, you should find the test statistic to be t 6 = The third step is to define the rejection region, decide whether or not to reject the null hypothesis, and obtain the p-value of the test. Since we have chosen α = 0.05 for a two-sided test, the rejection region is defined by the t-score with df = 6 above which 0.05 of the area lies; below the negative of this t-score will lie 0.05 of the area, making a total area of From Table A.3, we find that t 6;0.05 =.447. We can then define our rejection region algebraically as t or t (i.e., t 6.447). Since our test statistic, calculated in the second step, was found to be t 6 = 8.73, which is in the rejection region, our decision is to reject H 0 : slope = 0 ; in other words, our data provides sufficient evidence to believe that H 1 : slope 0 is true. From Table A.3, we find that t 6; = 5.595, and since our observed test statistic is t 6 = 8.73, we see that p-value < To complete the fourth step of the hypothesis test, we can summarize the results of the hypothesis test as follows: 35

3 Since t 6 = 8.73 and t 6;0.05 =.447, we have sufficient evidence to reject H 0. We conclude that the slope in the linear regression to predict reaction time from drug dosage is different from zero, or in other words, the linear relationship between dosage and reaction time is significant (p-value < ). The data suggest that the slope is less than zero, or in other words, that the linear relationship is negative. If we do not reject the null hypothesis, then no further analysis is necessary, since we are concluding that the linear relationship is not significant. On the other hand, if we do reject the null hypothesis, then we would want to describe the linear relationship, which of course is easily done by finding the least squares line. This was already done previously for the data of Table 31-1 in Unit 10 where we were working with the same data in Table 10-. There are also further analyses that are possible once the least squares line has been obtained for a significant regression. These analyses include hypothesis testing and confidence intervals concerning the slope, the intercept, and various predicted values; there are also prediction intervals which can be obtained. These analyses are beyond the scope of our discussion here but are generally available from the appropriate statistical software or programmable calculator. Finally, it is important to keep in mind that the t test concerning whether or not a linear relationship is significant (i.e., the test concerning a slope) is based on the assumptions that the relationship between two quantitative variables is indeed linear and that the variable Y has a normal distribution for each given value of X. Previously, we have discussed some descriptive methods for verifying the linearity assumption. Hypothesis tests are available concerning both the linearity assumption and the normality assumption. These tests are beyond the scope of our discussion here but are generally available from the appropriate statistical software or programmable calculator. Self-Test Problem In Self-Test Problems 10-1 and 11-1, the between age and grip strength among right-handed males is being investigated, with regard to the prediction of grip strength from age. The Age and Grip Strength Data, displayed in Table 10-4, was recorded for several right-handed males. For convenience, we have redisplayed the data here in Table 31-. (a) Explain how the data in Table 31- is appropriate for a hypothesis test concerning whether or not a linear relationship is significant. (b) A 0.05 significance level is chosen for a hypothesis test to see if there is any evidence that the linear relationship between age and grip strength among right-handed males is positive, or in other words, evidence that the slope is positive in the linear regression to predict grip strength from age. Complete the four steps of the hypothesis test by completing the table titled Hypothesis Test for Self-Test Problem You should find that t 10 = (c) Verify that the least squares line (found in Self-Test Problem 11-1) is grp = 6 + (age), and write a one sentence interpretation of the slope of the least squares line. (d) Considering the results of the hypothesis test, decide which of the Type I or Type II errors is possible, and describe this error. (e) Decide whether H 0 would have been rejected or would not have been rejected with each of the following significance levels: (i) α = 0.01, (ii) α =

4 Answers to Self-Test Problems 31-1 (a) The data consists of a random sample of observations of the two quantitative variables age and grip strength. (b) Step 1: H 0 : slope = 0 vs. H 1 : slope > 0 (α = 0.05, one-sided test) OR H 0 : The linear relationship between age and grip strength is not significant vs. (α = 0.05) H 1 : The linear relationship between age and grip strength is significant Step : t 10 = Step 3: The rejection is t H 0 is rejected; < p-value < Step 4: Since t 10 = and t 10; 0.05 = 1.81, we have sufficient evidence to reject H 0. We conclude that the slope is positive in the regression to predict grip strength from age among right-handed males, or in other words, that the linear relationship between age and grip strength is positive ( < p-value < 0.005). (c) The squares line (found in Self-Test Problem 11-1) is grp = 6 + (age). With each increase of one year in age, grip strength increases on average by about lbs. (d) Since H 0 is rejected, the Type I error is possible, which is concluding that slope > 0 (i.e., the positive linear relationship is significant) when in reality slope = 0 (i.e., the linear relationship is not significant). (e) H 0 would have been rejected with both α = 0.01 and α =

5 Summary When studying a possible linear relationship between two quantitative variables X and Y, we can obtain the Pearson correlation r and the least squares line from a sample consisting of bivariate data, represented as (x 1, y 1 ), (x, y ),, (x n, y n ). With Y representing the response variable and X representing the explanatory variable, we say that we are studying the linear regression to predict Y from X or the linear regression of Y on X. Deciding whether or not the linear relationship between X and Y is significant is the same as deciding whether or not the slope in the linear regression of Y on X is different from zero. To make this decision, the t statistic with n degrees of freedom which can be used is t n = s Y X b 0 ( x x), where s Y X is called the standard error of estimate, calculated by dividing the sum of the squared residuals by two less than the sample size, and taking the square root of this result. The numerator of this t statistic is the difference between a sample estimate of the slope (b) and the hypothesized value of the slope (zero), and the denominator is an appropriate standard error. These calculations can be greatly simplified by the use of appropriate statistical software or a programmable calculator. The null hypothesis in this hypothesis test could be stated as either H 0 : slope = 0 or H 0 : The linear relationship between X and Y is not significant, and this test and the test could be one-sided or two-sided. If our focus was on finding evidence that the slope is greater than zero (i.e., finding evidence that the linear relationship was a positive one), then we would use a one-sided test. If our focus was on finding evidence that the slope is less than zero (i.e., finding evidence that the linear relationship was a negative one), then we would use a one-sided test. However, if our focus was on finding evidence that the slope is different from zero in either direction (i.e., finding evidence that the linear relationship was either positive or negative), then we would use a two-sided test. If we do not reject the null hypothesis, then no further analysis is necessary, since we are concluding that the linear relationship is not significant. On the other hand, if we do reject the null hypothesis, then we would want to describe the linear relationship, which of course is easily done by finding the least squares line. Further analyses that are possible once the least squares line has been obtained for a significant regression include hypothesis testing and confidence intervals concerning the slope, the intercept, and various predicted values; there are also prediction intervals which can be obtained. The t test concerning whether or not a linear relationship is significant (i.e., the test concerning a slope) is based on the assumptions that the relationship between two quantitative variables is indeed linear and that the variable Y has a normal distribution for each given value of X. Hypothesis tests are available concerning both the linearity assumption and the normality assumption. 38

### Unit 26 Estimation with Confidence Intervals 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

### 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

### SPSS Guide: Regression Analysis 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

### Univariate Regression Univariate Regression Correlation and Regression The regression line summarizes the linear relationship between 2 variables Correlation coefficient, r, measures strength of relationship: the closer r is

### Chapter 7: Simple linear regression Learning Objectives Chapter 7: Simple linear regression Learning Objectives Reading: Section 7.1 of OpenIntro Statistics Video: Correlation vs. causation, YouTube (2:19) Video: Intro to Linear Regression, YouTube (5:18) -

### 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

### Simple Regression Theory II 2010 Samuel L. Baker SIMPLE REGRESSION THEORY II 1 Simple Regression Theory II 2010 Samuel L. Baker Assessing how good the regression equation is likely to be Assignment 1A gets into drawing inferences about how close the

### Hypothesis testing - Steps Hypothesis testing - Steps Steps to do a two-tailed test of the hypothesis that β 1 0: 1. Set up the hypotheses: H 0 : β 1 = 0 H a : β 1 0. 2. Compute the test statistic: t = b 1 0 Std. error of b 1 =

### 1. What is the critical value for this 95% confidence interval? CV = z.025 = invnorm(0.025) = 1.96 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

### II. DISTRIBUTIONS distribution normal distribution. standard scores Appendix D Basic Measurement And Statistics The following information was developed by Steven Rothke, PhD, Department of Psychology, Rehabilitation Institute of Chicago (RIC) and expanded by Mary F. Schmidt,

### Introduction to Quantitative Methods Introduction to Quantitative Methods October 15, 2009 Contents 1 Definition of Key Terms 2 2 Descriptive Statistics 3 2.1 Frequency Tables......................... 4 2.2 Measures of Central Tendencies.................

### CHAPTER 13 SIMPLE LINEAR REGRESSION. Opening Example. Simple Regression. Linear Regression Opening Example CHAPTER 13 SIMPLE LINEAR REGREION SIMPLE LINEAR REGREION! Simple Regression! Linear Regression Simple Regression Definition A regression model is a mathematical equation that descries the

### Introduction to Regression and Data Analysis Statlab Workshop Introduction to Regression and Data Analysis with Dan Campbell and Sherlock Campbell October 28, 2008 I. The basics A. Types of variables Your variables may take several forms, and it

### Regression Analysis: A Complete Example Regression Analysis: A Complete Example This section works out an example that includes all the topics we have discussed so far in this chapter. A complete example of regression analysis. PhotoDisc, Inc./Getty

### " Y. Notation and Equations for Regression Lecture 11/4. Notation: 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

### The correlation coefficient The correlation coefficient Clinical Biostatistics The correlation coefficient Martin Bland Correlation coefficients are used to measure the of the relationship or association between two quantitative

### 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

### 2. Simple Linear Regression Research methods - II 3 2. Simple Linear Regression Simple linear regression is a technique in parametric statistics that is commonly used for analyzing mean response of a variable Y which changes according

### 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

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

### Section 14 Simple Linear Regression: Introduction to Least Squares Regression Slide 1 Section 14 Simple Linear Regression: Introduction to Least Squares Regression There are several different measures of statistical association used for understanding the quantitative relationship

### Simple Linear Regression Inference 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 23. Inferences for Regression Chapter 23. Inferences for Regression Topics covered in this chapter: Simple Linear Regression Simple Linear Regression Example 23.1: Crying and IQ The Problem: Infants who cry easily may be more easily

### Chapter 13 Introduction to Linear Regression and Correlation Analysis Chapter 3 Student Lecture Notes 3- Chapter 3 Introduction to Linear Regression and Correlation Analsis Fall 2006 Fundamentals of Business Statistics Chapter Goals To understand the methods for displaing

### Fairfield Public Schools Mathematics Fairfield Public Schools AP Statistics AP Statistics BOE Approved 04/08/2014 1 AP STATISTICS Critical Areas of Focus AP Statistics is a rigorous course that offers advanced students an opportunity

### STAT 350 Practice Final Exam Solution (Spring 2015) PART 1: Multiple Choice Questions: 1) A study was conducted to compare five different training programs for improving endurance. Forty subjects were randomly divided into five groups of eight subjects

### 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

### 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

### Multiple Linear Regression Multiple Linear Regression A regression with two or more explanatory variables is called a multiple regression. Rather than modeling the mean response as a straight line, as in simple regression, it is

### Good luck! BUSINESS STATISTICS FINAL EXAM INSTRUCTIONS. Name: 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

### Example: Boats and Manatees Figure 9-6 Example: Boats and Manatees Slide 1 Given the sample data in Table 9-1, find the value of the linear correlation coefficient r, then refer to Table A-6 to determine whether there is a significant

### Chapter 5 Analysis of variance SPSS Analysis of variance Chapter 5 Analysis of variance SPSS Analysis of variance Data file used: gss.sav How to get there: Analyze Compare Means One-way ANOVA To test the null hypothesis that several population means are equal,

### Factors affecting online sales Factors affecting online sales Table of contents Summary... 1 Research questions... 1 The dataset... 2 Descriptive statistics: The exploratory stage... 3 Confidence intervals... 4 Hypothesis tests... 4

### 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

### SIMPLE LINEAR CORRELATION. r can range from -1 to 1, and is independent of units of measurement. Correlation can be done on two dependent variables. SIMPLE LINEAR CORRELATION Simple linear correlation is a measure of the degree to which two variables vary together, or a measure of the intensity of the association between two variables. Correlation

### ch12 practice test SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. ch12 practice test 1) The null hypothesis that x and y are is H0: = 0. 1) 2) When a two-sided significance test about a population slope has a P-value below 0.05, the 95% confidence interval for A) does

### Week TSX Index 1 8480 2 8470 3 8475 4 8510 5 8500 6 8480 1) The S & P/TSX Composite Index is based on common stock prices of a group of Canadian stocks. The weekly close level of the TSX for 6 weeks are shown: Week TSX Index 1 8480 2 8470 3 8475 4 8510 5 8500

### 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

### Correlation and Simple Linear Regression Correlation and Simple Linear Regression We are often interested in studying the relationship among variables to determine whether they are associated with one another. When we think that changes in a

### Chapter 10. Key Ideas Correlation, Correlation Coefficient (r), Chapter 0 Key Ideas Correlation, Correlation Coefficient (r), Section 0-: Overview We have already explored the basics of describing single variable data sets. However, when two quantitative variables

### 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

### Part 2: Analysis of Relationship Between Two Variables Part 2: Analysis of Relationship Between Two Variables Linear Regression Linear correlation Significance Tests Multiple regression Linear Regression Y = a X + b Dependent Variable Independent Variable

### Simple linear regression Simple linear regression Introduction Simple linear regression is a statistical method for obtaining a formula to predict values of one variable from another where there is a causal relationship between

### 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

### 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

### Course Objective This course is designed to give you a basic understanding of how to run regressions in SPSS. SPSS Regressions Social Science Research Lab American University, Washington, D.C. Web. www.american.edu/provost/ctrl/pclabs.cfm Tel. x3862 Email. SSRL@American.edu Course Objective This course is designed

### 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

### 11. Analysis of Case-control Studies Logistic Regression Research methods II 113 11. Analysis of Case-control Studies Logistic Regression This chapter builds upon and further develops the concepts and strategies described in Ch.6 of Mother and Child Health:

### Lecture 11: Chapter 5, Section 3 Relationships between Two Quantitative Variables; Correlation Lecture 11: Chapter 5, Section 3 Relationships between Two Quantitative Variables; Correlation Display and Summarize Correlation for Direction and Strength Properties of Correlation Regression Line Cengage

### 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)

### 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.

### Pearson's Correlation Tests Chapter 800 Pearson's Correlation Tests Introduction The correlation coefficient, ρ (rho), is a popular statistic for describing the strength of the relationship between two variables. The correlation

### 1. The parameters to be estimated in the simple linear regression model Y=α+βx+ε ε~n(0,σ) are: a) α, β, σ b) α, β, ε c) a, b, s d) ε, 0, σ 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

### Section Format Day Begin End Building Rm# Instructor. 001 Lecture Tue 6:45 PM 8:40 PM Silver 401 Ballerini 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

### Module 5: Multiple Regression Analysis Using Statistical Data Using to Make Statistical Decisions: Data Multiple to Make Regression Decisions Analysis Page 1 Module 5: Multiple Regression Analysis Tom Ilvento, University of Delaware, College

### BA 275 Review Problems - Week 5 (10/23/06-10/27/06) CD Lessons: 48, 49, 50, 51, 52 Textbook: pp. 380-394 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

### Outline. Topic 4 - Analysis of Variance Approach to Regression. Partitioning Sums of Squares. Total Sum of Squares. Partitioning sums of squares Topic 4 - Analysis of Variance Approach to Regression Outline Partitioning sums of squares Degrees of freedom Expected mean squares General linear test - Fall 2013 R 2 and the coefficient of correlation

### NCSS Statistical Software Principal Components Regression. In ordinary least squares, the regression coefficients are estimated using the formula ( ) Chapter 340 Principal Components Regression Introduction is a technique for analyzing multiple regression data that suffer from multicollinearity. When multicollinearity occurs, least squares estimates

### 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

### HYPOTHESIS TESTING: CONFIDENCE INTERVALS, T-TESTS, ANOVAS, AND REGRESSION HYPOTHESIS TESTING: CONFIDENCE INTERVALS, T-TESTS, ANOVAS, AND REGRESSION HOD 2990 10 November 2010 Lecture Background This is a lightning speed summary of introductory statistical methods for senior undergraduate

### August 2012 EXAMINATIONS Solution Part I August 01 EXAMINATIONS Solution Part I (1) In a random sample of 600 eligible voters, the probability that less than 38% will be in favour of this policy is closest to (B) () In a large random sample,

### This chapter will demonstrate how to perform multiple linear regression with IBM SPSS CHAPTER 7B Multiple Regression: Statistical Methods Using IBM SPSS This chapter will demonstrate how to perform multiple linear regression with IBM SPSS first using the standard method and then using the

### Final Exam Practice Problem Answers Final Exam Practice Problem Answers The following data set consists of data gathered from 77 popular breakfast cereals. The variables in the data set are as follows: Brand: The brand name of the cereal

### Calculating P-Values. Parkland College. Isela Guerra Parkland College. Recommended Citation 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.

### Pearson s Correlation Pearson s Correlation Correlation the degree to which two variables are associated (co-vary). Covariance may be either positive or negative. Its magnitude depends on the units of measurement. Assumes the

### 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

### 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

### 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

### 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

### 1.5 Oneway Analysis of Variance 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

### Odds ratio, Odds ratio test for independence, chi-squared statistic. 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

### ANALYSIS OF TREND CHAPTER 5 ANALYSIS OF TREND CHAPTER 5 ERSH 8310 Lecture 7 September 13, 2007 Today s Class Analysis of trends Using contrasts to do something a bit more practical. Linear trends. Quadratic trends. Trends in SPSS.

### Elementary Statistics Sample Exam #3 Elementary Statistics Sample Exam #3 Instructions. No books or telephones. Only the supplied calculators are allowed. The exam is worth 100 points. 1. A chi square goodness of fit test is considered to

### INTERPRETING THE ONE-WAY ANALYSIS OF VARIANCE (ANOVA) 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

### Definition 8.1 Two inequalities are equivalent if they have the same solution set. Add or Subtract the same value on both sides of the inequality. 8 Inequalities Concepts: Equivalent Inequalities Linear and Nonlinear Inequalities Absolute Value Inequalities (Sections 4.6 and 1.1) 8.1 Equivalent Inequalities Definition 8.1 Two inequalities are equivalent

### 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

### Comparing Nested Models Comparing Nested Models ST 430/514 Two models are nested if one model contains all the terms of the other, and at least one additional term. The larger model is the complete (or full) model, and the smaller

### 17. SIMPLE LINEAR REGRESSION II 17. SIMPLE LINEAR REGRESSION II The Model In linear regression analysis, we assume that the relationship between X and Y is linear. This does not mean, however, that Y can be perfectly predicted from X.

### 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

### Statistical Models in R Statistical Models in R Some Examples Steven Buechler Department of Mathematics 276B Hurley Hall; 1-6233 Fall, 2007 Outline Statistical Models Linear Models in R Regression Regression analysis is the appropriate

### Part 3. Comparing Groups. Chapter 7 Comparing Paired Groups 189. Chapter 8 Comparing Two Independent Groups 217 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

### X X X a) perfect linear correlation b) no correlation c) positive correlation (r = 1) (r = 0) (0 < r < 1) CORRELATION AND REGRESSION / 47 CHAPTER EIGHT CORRELATION AND REGRESSION Correlation and regression are statistical methods that are commonly used in the medical literature to compare two or more variables.

### KSTAT MINI-MANUAL. Decision Sciences 434 Kellogg Graduate School of Management 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

### 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

### Using R for Linear Regression Using R for Linear Regression In the following handout words and symbols in bold are R functions and words and symbols in italics are entries supplied by the user; underlined words and symbols are optional

### Having a coin come up heads or tails is a variable on a nominal scale. Heads is a different category from tails. Chi-square Goodness of Fit Test The chi-square test is designed to test differences whether one frequency is different from another frequency. The chi-square test is designed for use with data on a nominal

### COMPARISONS OF CUSTOMER LOYALTY: PUBLIC & PRIVATE INSURANCE COMPANIES. 277 CHAPTER VI COMPARISONS OF CUSTOMER LOYALTY: PUBLIC & PRIVATE INSURANCE COMPANIES. This chapter contains a full discussion of customer loyalty comparisons between private and public insurance companies

### MULTIPLE REGRESSION AND ISSUES IN REGRESSION ANALYSIS MULTIPLE REGRESSION AND ISSUES IN REGRESSION ANALYSIS MSR = Mean Regression Sum of Squares MSE = Mean Squared Error RSS = Regression Sum of Squares SSE = Sum of Squared Errors/Residuals α = Level of Significance

### CURVE FITTING LEAST SQUARES APPROXIMATION CURVE FITTING LEAST SQUARES APPROXIMATION Data analysis and curve fitting: Imagine that we are studying a physical system involving two quantities: x and y Also suppose that we expect a linear relationship

### Course Text. Required Computing Software. Course Description. Course Objectives. StraighterLine. Business Statistics Course Text Business Statistics Lind, Douglas A., Marchal, William A. and Samuel A. Wathen. Basic Statistics for Business and Economics, 7th edition, McGraw-Hill/Irwin, 2010, ISBN: 9780077384470 [This

### Business Statistics. Successful completion of Introductory and/or Intermediate Algebra courses is recommended before taking Business Statistics. Business Course Text Bowerman, Bruce L., Richard T. O'Connell, J. B. Orris, and Dawn C. Porter. Essentials of Business, 2nd edition, McGraw-Hill/Irwin, 2008, ISBN: 978-0-07-331988-9. Required Computing

### An analysis appropriate for a quantitative outcome and a single quantitative explanatory. 9.1 The model behind linear regression Chapter 9 Simple Linear Regression An analysis appropriate for a quantitative outcome and a single quantitative explanatory variable. 9.1 The model behind linear regression When we are examining the relationship

### HYPOTHESIS TESTING (ONE SAMPLE) - CHAPTER 7 1. used confidence intervals to answer questions such as... 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

### AP STATISTICS (Warm-Up Exercises) AP STATISTICS (Warm-Up Exercises) 1. Describe the distribution of ages in a city: 2. Graph a box plot on your calculator for the following test scores: {90, 80, 96, 54, 80, 95, 100, 75, 87, 62, 65, 85,

### In the general population of 0 to 4-year-olds, the annual incidence of asthma is 1.4% 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

### 2 Sample t-test (unequal sample sizes and unequal variances) 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 Principles of Hypothesis Testing for Public Health Laura Lee Johnson, Ph.D. Statistician National Center for Complementary and Alternative Medicine johnslau@mail.nih.gov Fall 2011 Answers to Questions 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