Lecture 14. Analysis of Variance * Correlation and Regression. The McGraw-Hill Companies, Inc., 2000
|
|
- Wilfred Anthony
- 5 years ago
- Views:
Transcription
1 Lecture 14 Analysis of Variance * Correlation and Regression
2 Outline Analysis of Variance (ANOVA) 11-1 Introduction 11-2 Scatter Plots 11-3 Correlation 11-4 Regression
3 Outline 11-5 Coefficient of Determination and Standard Error of Estimate
4 Analysis of Variance (ANOVA) When an F test is used to test a hypothesis concerning the means of three or more populations, the technique is called analysis of variance (ANOVA).
5 Assumptions for the F Test for Comparing Three or More Means The populations from which the samples were obtained must be normally or approximately normally distributed. The samples must be independent of each other. The variances of the populations must be equal.
6 Analysis of Variance Although means are being compared in this F test, variances are used in the test instead of the means. Two different estimates of the population variance are made.
7 Analysis of Variance Between-group variance - this involves computing the variance by using the means of the groups or between the groups. Within-group variance - this involves computing the variance by using all the data and is not affected by differences in the means.
8 Analysis of Variance The following hypotheses should be used when testing for the difference between three or more means. H 0 : µ 1 = µ 2 = µ 3 = = µ k H 1 : At least one mean is different from the others.
9 Analysis of Variance d.f.n. = k 1, where k is the number of groups. d.f.d. = N k, where N is the sum of the sample sizes of the groups. Note: The formulas for this test are tedious to work through, so examples will be done in MINITAB. See text for formulas.
10 Analysis of Variance -Example A marketing specialist wishes to see whether there is a difference in the average time a customer has to wait in a checkout line in three large self-service department stores. The times (in minutes) are shown on the next slide. Is there a significant difference in the mean waiting times of customers for each store using α = 0.05?
11 Analysis of Variance -Example Store A Store B Store C
12 Analysis of Variance -Example Step 1: State the hypotheses and identify the claim. H 0 : µ 1 = µ 2 = µ 3 H 1 : At least one mean is different from the others (claim).
13 Analysis of Variance -Example Step 2: Find the critical value. Since k = 3, N = 18, and α = 0.05, d.f.n. = k 1 = 3 1= 2, d.f.d. = N k = 18 3 = 15. The critical value is Step 3: Compute the test value. From the MINITAB output, F = (See your text for computations).
14 Analysis of Variance -Example Step 4: Make a decision. Since 2.70 < 3.68, the decision is not to reject the null hypothesis. Step 5: Summarize the results. There is not enough evidence to support the claim that there is a difference among the means. The ANOVA summary table is given on the next slide.
15 Analysis of Variance -Example
16 Correlation and Regression
17 11-1 Introduction Goal: to determine whether a relationship between two or more numerical or quantitative variables exists. Example: Is there a relationship between a person s age and his or her blood pressure?
18 11-1 Introduction Correlation is a statistical method used to determine whether a relationship between variables exists. Regression is a statistical method used to describe the nature of the relationship between variables that is, positive or negative, linear or nonlinear.
19 11-2 Scatter plots The independent variable is the variable in regression that can be controlled or manipulated. The dependent variable is the variable in regression that cannot be controlled or manipulated.
20 Scatter Plots A scatter plot is a graph of the ordered pairs (x, y) of numbers consisting of the independent variable, x, and the dependent variable, y.
21 11-2 Scatter plots - example Is there a relationship between number of hours studied and test scores on an exam?
22 11-2 Scatter plots - example Independent variable: number of hours studied (x). Dependent variable: the grade the student received on the exam (y). The independent and dependent variables can be plotted on a graph called a scatter plot.
23 Scatter Plots - Example Construct a scatter plot for the data obtained in a study of age and systolic blood pressure of six randomly selected subjects. The data are shown in the following table.
24 Scatter Plots - Example Subject Age, x Pressure, y A B C D E F
25 Scatter Plots - Example Positive Relationship Pressure Pressure Age Age
26 Scatter Plots - Other Examples Construct a scatter plot for the data obtained in a study on the number of absences and the final grades of seven randomly selected students from a statistics class. The data are shown on the next slide.
27 Scatter Plots - Other Examples
28 Scatter Plots - Other Examples Negative Relationship Final Final grade grade Number Number of of absences absences 15 15
29 Scatter Plots - Other Examples Construct a scatter plot for the data obtained in a study on the number of hours nine people exercise each week and the amount of milk (in ounces) each person consumes per week. The data follow.
30 Scatter Plots - Other Examples
31 Scatter Plots - Other Examples No Relationship Y y x X
32 Correlation Coefficient The correlation coefficient computed from the sample data measures the strength and direction of a relationship between two variables. Sample correlation coefficient, r. Population correlation coefficient, ρ.
33 Range of Values for the Correlation Coefficient Strong negative relationship No linear relationship Strong positive relationship
34 Formula for the Correlation Coefficient r r = n( xy) ( x)( y) [ ( ) ( )] n x 2 2 x n( y 2) ( y) [ ] 2 Where n is the number of data pairs
35 Correlation Coefficient - Example (Verify) Compute the correlation coefficient for the age and blood pressure data. x = 345, y = 819, xy = 47, x = 20, 399, y = 112, 443. Substituting in the formula for r gives r =
36 The Significance of the Correlation Coefficient The population corelation coefficient, ρ, is the correlation between all possible pairs of data values (x, y) taken from a population.
37 The Significance of the Correlation Coefficient H 0 : ρ = 0 H 1 : ρ 0 This tests for a significant correlation between the variables in the population.
38 Formula for the t tests for the Correlation Coefficient t=r n 2 1 r 2 with d. f. = n 2
39 11-3 Example Test the significance of the correlation coefficient for the age and blood pressure data. Use α = 0.05 and r = Step 1: State the hypotheses. H 0 : ρ = 0 H 1 : ρ 0
40 11-3 Example Step 2: Find the critical values. Since α = 0.05 and there are 6 2 = 4 degrees of freedom, the critical values are t = and t = Step 3: Compute the test value. t = (verify).
41 11-3 Example Step 4: Make the decision. Reject the null hypothesis, since the test value falls in the critical region (4.059 > 2.776). Step 5: Summarize the results. There is a significant relationship between the variables of age and blood pressure.
42 Correlation and Causation You must understand the nature of the linear relationship between the independent variable x and the dependent variable y. When a hypothesis test indicates that a significant linear relationship exists between the variables (i.e., when the null hypothesis has been rejected for a specific α value), any of the following five possibilities can exist.
43 Correlation and Causation #1 There is a direct cause-and-effect relationship between the variables. That is, x causes y. For example, water causes plants to grow, poison causes death, and heat causes ice to melt.
44 Correlation and Causation #2 There is a reverse cause-and-effect relationship between the variables. That is, y causes x. Example: a researcher believes excessive coffee consumption causes nervousness, but the researcher fails to consider that the reverse situation may occur. That is, it may be that an extremely nervous person craves coffee to calm his or her nerves.
45 Correlation and Causation #3 The relationship between the variables may be caused by a third variable. For example, if a statistician correlated the number of deaths due to drowning and the number of cans of soft drink consumed during the summer, he or she would probably find a significant relationship. However, the soft drink is not necessarily responsible for the deaths, since both variables may be related to heat and humidity.
46 Correlation and Causation #4 There may be a complexity of interrelationships among many variables. For example, a researcher may find a significant relationship between students high school grades and college grades. But there probably are many other variables involved, such as IQ, hours of study, influence of parents, motivation, age, and instructors.
47 Correlation and Causation #5 The relationship may be coincidental. For example, a researcher may be able to find a significant relationship between the increase in the number of people who are exercising and the increase in the number of people who are committing crimes. But common sense dictates that any relationship between these two values must be due to coincidence.
48 Correlation and Causation Thus, when the null hypothesis is rejected, the researcher must consider all possibilities and select the appropriate one as determined by the study. Remember, correlation does not necessarily imply causation.
49 Regression The scatter plot for the age and blood pressure data displays a linear pattern. We can model this relationship with a straight line. This regression line is called the line of best fit or the regression line. The equation of the line is y = a + bx.
50 Regression Best fit means that the sum of the squares of the vertical distances from each point to the line is at a minimum. The reason one needs a line of best fit is that the values of y will be predicted from the values of x; hence, the closer the points are to the line, the better the fit and the prediction will be. See the next Figure.
51 Regression the line of best fit
52 Correlation coefficient and the line of best fit
53 Formulas for the Regression Line y = a + bx. a ( y)( x 2 ) ( x)( xy) ( ) n x 2) ( x) 2 = b = ( ) ( )( ) n( x 2) ( x) 2 n xy x y Where a is the y intercept and b is the slope of the line.
54 11-4 Example Find the equation of the regression line for the age and the blood pressure data. Substituting into the formulas give a = and b = (verify). Hence, y = x. Note, a represents the intercept and b the slope of the line.
55 11-4 Example Pressure Pressure y = x Age Age
56 Using the Regression Line to Predict The regression line can be used to predict a value for the dependent variable (y) for a given value of the independent variable (x). Caution: Use x values within the experimental region when predicting y values.
57 11-4 Example Use the equation of the regression line to predict the blood pressure for a person who is 50 years old. Since y = x, then y = (50) = Note that the value of 50 is within the range of x values.
58 Coefficient of Determination and Standard Error of Estimate The coefficient of determination coefficient of determination, denoted by r 2, is a measure of the variation of the dependent variable that is explained by the regression line and the independent variable.
59 Coefficient of Determination and Standard Error of Estimate r 2 is the square of the correlation coefficient. The coefficient of nondetermination is (1 r 2 ). Example: If r = 0.90, then r 2 = 0.81.
60 Coefficient of Determination and Standard Error of Estimate The standard error of estimate, denoted by s est, is the standard deviation of the observed y values about the predicted y values. The formula is given on the next slide.
61 Formula for the Standard Error of Estimate s est or = ( ) y y n 2 2 s est y a y b xy = 2 n 2
62 Standard Error of Estimate - Example From the regression equation, y = x and n = 6, find s est. Here, a = 55.57, b = 8.13, and n = 6. Substituting into the formula gives s est = 6.48 (verify).
63 Prediction Interval A prediction interval prediction interval is an interval constructed about a predicted y value, y, for a specified x value.
64 Prediction Interval For given α value, we can state with (1 α)100% confidence that the interval will contain the actual mean of the y values that correspond to the given value of x.
65 Formula for the Prediction 5 Formula for the Prediction Interval about a Value Interval about a Value y ( ) ) ( x x n X x n n est s t y + + α ( ) ) ( x x n X x n n est s t y α < < y 2.. = n f d with
66 Prediction interval - Example A researcher collects the data shown on the next slide and determines that there is a significant relationship between the age of a copy machine and its monthly maintenance cost. The regression equation is y = x. Find the 95% prediction interval for the monthly maintenance cost of a machine that is 3 years old.
67 Prediction Interval - Example Machine Age, x (Years) Monthly cost, y A 1 $62 B 2 $78 C 3 $70 D 4 $90 E 4 $93 F 6 $103
68 Prediction Interval - Example Step 1: Find Σx, Σx 2 and X. Σx = 20, Σx 2 20 = 82, X = = Step 2: Find y for x = 3. y = (3) = Step 3: Find s est s est = 6.48 as shown in previous example.
69 Prediction Interval - Example Step 4: Substitute in the formula and solve. t α/2 = 2.776, d.f. = 6 2 = 4 for 95% < y < (verify) Hence, one can be 95% confident that the interval < y < contains the actual value of y.
70 Biomathematics That Is All 2002 János FODOR
Lecture 14. Outline. Outline. Analysis of Variance * Correlation and Regression Analysis of Variance (ANOVA)
Outline Lecture 14 Analysis of Variance * Correlation and Regression Analysis of Variance (ANOVA) 11-1 Introduction 11- Scatter Plots 11-3 Correlation 11-4 Regression Outline 11-5 Coefficient of Determination
More informationChapter 10. Correlation and Regression. McGraw-Hill, Bluman, 7th ed., Chapter 10 1
Chapter 10 Correlation and Regression McGraw-Hill, Bluman, 7th ed., Chapter 10 1 Chapter 10 Overview Introduction 10-1 Scatter Plots and Correlation 10- Regression 10-3 Coefficient of Determination and
More informationChapter 10. Correlation and Regression. McGraw-Hill, Bluman, 7th ed., Chapter 10 1
Chapter 10 Correlation and Regression McGraw-Hill, Bluman, 7th ed., Chapter 10 1 Example 10-2: Absences/Final Grades Please enter the data below in L1 and L2. The data appears on page 537 of your textbook.
More informationCorrelation and Regression
Correlation and Regression Linear Correlation: Does one variable increase or decrease linearly with another? Is there a linear relationship between two or more variables? Types of linear relationships:
More informationReview 6. n 1 = 85 n 2 = 75 x 1 = x 2 = s 1 = 38.7 s 2 = 39.2
Review 6 Use the traditional method to test the given hypothesis. Assume that the samples are independent and that they have been randomly selected ) A researcher finds that of,000 people who said that
More informationThis document contains 3 sets of practice problems.
P RACTICE PROBLEMS This document contains 3 sets of practice problems. Correlation: 3 problems Regression: 4 problems ANOVA: 8 problems You should print a copy of these practice problems and bring them
More informationSix Sigma Black Belt Study Guides
Six Sigma Black Belt Study Guides 1 www.pmtutor.org Powered by POeT Solvers Limited. Analyze Correlation and Regression Analysis 2 www.pmtutor.org Powered by POeT Solvers Limited. Variables and relationships
More information[ z = 1.48 ; accept H 0 ]
CH 13 TESTING OF HYPOTHESIS EXAMPLES Example 13.1 Indicate the type of errors committed in the following cases: (i) H 0 : µ = 500; H 1 : µ 500. H 0 is rejected while H 0 is true (ii) H 0 : µ = 500; H 1
More information11 Correlation and Regression
Chapter 11 Correlation and Regression August 21, 2017 1 11 Correlation and Regression When comparing two variables, sometimes one variable (the explanatory variable) can be used to help predict the value
More informationIntroduction to Business Statistics QM 220 Chapter 12
Department of Quantitative Methods & Information Systems Introduction to Business Statistics QM 220 Chapter 12 Dr. Mohammad Zainal 12.1 The F distribution We already covered this topic in Ch. 10 QM-220,
More informationCh 13 & 14 - Regression Analysis
Ch 3 & 4 - Regression Analysis Simple Regression Model I. Multiple Choice:. A simple regression is a regression model that contains a. only one independent variable b. only one dependent variable c. more
More informationEXAM 3 Math 1342 Elementary Statistics 6-7
EXAM 3 Math 1342 Elementary Statistics 6-7 Name Date ********************************************************************************************************************************************** MULTIPLE
More informationSimple Linear Regression
9-1 l Chapter 9 l Simple Linear Regression 9.1 Simple Linear Regression 9.2 Scatter Diagram 9.3 Graphical Method for Determining Regression 9.4 Least Square Method 9.5 Correlation Coefficient and Coefficient
More informationCorrelation and Regression
Elementary Statistics A Step by Step Approach Sixth Edition by Allan G. Bluman http://www.mhhe.com/math/stat/blumanbrief SLIDES PREPARED BY LLOYD R. JAISINGH MOREHEAD STATE UNIVERSITY MOREHEAD KY Updated
More informationWhile you wait: Enter the following in your calculator. Find the mean and sample variation of each group. Bluman, Chapter 12 1
While you wait: Enter the following in your calculator. Find the mean and sample variation of each group. Bluman, Chapter 12 1 Chapter 12 Analysis of Variance McGraw-Hill, Bluman, 7th ed., Chapter 12 2
More informationChapter 9. Correlation and Regression
Chapter 9 Correlation and Regression Lesson 9-1/9-2, Part 1 Correlation Registered Florida Pleasure Crafts and Watercraft Related Manatee Deaths 100 80 60 40 20 0 1991 1993 1995 1997 1999 Year Boats in
More informationREVIEW 8/2/2017 陈芳华东师大英语系
REVIEW Hypothesis testing starts with a null hypothesis and a null distribution. We compare what we have to the null distribution, if the result is too extreme to belong to the null distribution (p
More informationChapter 12 - Lecture 2 Inferences about regression coefficient
Chapter 12 - Lecture 2 Inferences about regression coefficient April 19th, 2010 Facts about slope Test Statistic Confidence interval Hypothesis testing Test using ANOVA Table Facts about slope In previous
More informationChapter 12 - Part I: Correlation Analysis
ST coursework due Friday, April - Chapter - Part I: Correlation Analysis Textbook Assignment Page - # Page - #, Page - # Lab Assignment # (available on ST webpage) GOALS When you have completed this lecture,
More informationDo not copy, post, or distribute
14 CORRELATION ANALYSIS AND LINEAR REGRESSION Assessing the Covariability of Two Quantitative Properties 14.0 LEARNING OBJECTIVES In this chapter, we discuss two related techniques for assessing a possible
More informationCorrelation and Regression
Correlation and Regression Dr. Bob Gee Dean Scott Bonney Professor William G. Journigan American Meridian University 1 Learning Objectives Upon successful completion of this module, the student should
More informationLinear Correlation and Regression Analysis
Linear Correlation and Regression Analysis Set Up the Calculator 2 nd CATALOG D arrow down DiagnosticOn ENTER ENTER SCATTER DIAGRAM Positive Linear Correlation Positive Correlation Variables will tend
More informationConditions for Regression Inference:
AP Statistics Chapter Notes. Inference for Linear Regression We can fit a least-squares line to any data relating two quantitative variables, but the results are useful only if the scatterplot shows a
More informationCORRELATION AND REGRESSION
CORRELATION AND REGRESSION CORRELATION Introduction CORRELATION problems which involve measuring the strength of a relationship. Correlation Analysis involves various methods and techniques used for studying
More informationSection 9.5. Testing the Difference Between Two Variances. Bluman, Chapter 9 1
Section 9.5 Testing the Difference Between Two Variances Bluman, Chapter 9 1 This the last day the class meets before spring break starts. Please make sure to be present for the test or make appropriate
More informationQuantitative Bivariate Data
Statistics 211 (L02) - Linear Regression Quantitative Bivariate Data Consider two quantitative variables, defined in the following way: X i - the observed value of Variable X from subject i, i = 1, 2,,
More informationσ. We further know that if the sample is from a normal distribution then the sampling STAT 2507 Assignment # 3 (Chapters 7 & 8)
STAT 2507 Assignment # 3 (Chapters 7 & 8) DUE: Sections E, F Section G Section H Monday, March 16, in class Tuesday, March 17, in class Wednesday, March 18, in class Last Name Student # First Name Your
More informationy n 1 ( x i x )( y y i n 1 i y 2
STP3 Brief Class Notes Instructor: Ela Jackiewicz Chapter Regression and Correlation In this chapter we will explore the relationship between two quantitative variables, X an Y. We will consider n ordered
More informationCan you tell the relationship between students SAT scores and their college grades?
Correlation One Challenge Can you tell the relationship between students SAT scores and their college grades? A: The higher SAT scores are, the better GPA may be. B: The higher SAT scores are, the lower
More informationStatistics Introductory Correlation
Statistics Introductory Correlation Session 10 oscardavid.barrerarodriguez@sciencespo.fr April 9, 2018 Outline 1 Statistics are not used only to describe central tendency and variability for a single variable.
More information(ii) Scan your answer sheets INTO ONE FILE only, and submit it in the drop-box.
FINAL EXAM ** Two different ways to submit your answer sheet (i) Use MS-Word and place it in a drop-box. (ii) Scan your answer sheets INTO ONE FILE only, and submit it in the drop-box. Deadline: December
More informationFinal Exam - Solutions
Ecn 102 - Analysis of Economic Data University of California - Davis March 17, 2010 Instructor: John Parman Final Exam - Solutions You have until 12:30pm to complete this exam. Please remember to put your
More informationBusiness Mathematics and Statistics (MATH0203) Chapter 1: Correlation & Regression
Business Mathematics and Statistics (MATH0203) Chapter 1: Correlation & Regression Dependent and independent variables The independent variable (x) is the one that is chosen freely or occur naturally.
More informationInferences for Regression
Inferences for Regression An Example: Body Fat and Waist Size Looking at the relationship between % body fat and waist size (in inches). Here is a scatterplot of our data set: Remembering Regression In
More information79 Wyner Math Academy I Spring 2016
79 Wyner Math Academy I Spring 2016 CHAPTER NINE: HYPOTHESIS TESTING Review May 11 Test May 17 Research requires an understanding of underlying mathematical distributions as well as of the research methods
More informationCorrelation and Linear Regression
Correlation and Linear Regression Correlation: Relationships between Variables So far, nearly all of our discussion of inferential statistics has focused on testing for differences between group means
More informationChapter 12 : Linear Correlation and Linear Regression
Chapter 1 : Linear Correlation and Linear Regression Determining whether a linear relationship exists between two quantitative variables, and modeling the relationship with a line, if the linear relationship
More informationLecture Slides. Elementary Statistics Tenth Edition. by Mario F. Triola. and the Triola Statistics Series. Slide 1
Lecture Slides Elementary Statistics Tenth Edition and the Triola Statistics Series by Mario F. Triola Slide 1 Chapter 10 Correlation and Regression 10-1 Overview 10-2 Correlation 10-3 Regression 10-4
More informationBusiness Statistics. Lecture 10: Correlation and Linear Regression
Business Statistics Lecture 10: Correlation and Linear Regression Scatterplot A scatterplot shows the relationship between two quantitative variables measured on the same individuals. It displays the Form
More informationSec Coefficient of Determination and Standard Error of the Estimate.
Sec 10.3 Coefficient of Determination and Standard Error of the Estimate. Review concepts x 1 3 4 5 y 10 8 1 16 0 y fill in the third row of the table for each x value. Review concepts x 1 3 4 5 y 10 8
More informationPredicted Y Scores. The symbol stands for a predicted Y score
REGRESSION 1 Linear Regression Linear regression is a statistical procedure that uses relationships to predict unknown Y scores based on the X scores from a correlated variable. 2 Predicted Y Scores Y
More informationCorrelation. A statistics method to measure the relationship between two variables. Three characteristics
Correlation Correlation A statistics method to measure the relationship between two variables Three characteristics Direction of the relationship Form of the relationship Strength/Consistency Direction
More informationExtra Exam Empirical Methods VU University Amsterdam, Faculty of Exact Sciences , July 2, 2015
Extra Exam Empirical Methods VU University Amsterdam, Faculty of Exact Sciences 12.00 14.45, July 2, 2015 Also hand in this exam and your scrap paper. Always motivate your answers. Write your answers in
More informationTest 3 Practice Test A. NOTE: Ignore Q10 (not covered)
Test 3 Practice Test A NOTE: Ignore Q10 (not covered) MA 180/418 Midterm Test 3, Version A Fall 2010 Student Name (PRINT):............................................. Student Signature:...................................................
More informationSampling Distributions: Central Limit Theorem
Review for Exam 2 Sampling Distributions: Central Limit Theorem Conceptually, we can break up the theorem into three parts: 1. The mean (µ M ) of a population of sample means (M) is equal to the mean (µ)
More informationPsychology 282 Lecture #4 Outline Inferences in SLR
Psychology 282 Lecture #4 Outline Inferences in SLR Assumptions To this point we have not had to make any distributional assumptions. Principle of least squares requires no assumptions. Can use correlations
More informationregression analysis is a type of inferential statistics which tells us whether relationships between two or more variables exist
regression analysis is a type of inferential statistics which tells us whether relationships between two or more variables exist sales $ (y - dependent variable) advertising $ (x - independent variable)
More informationThis gives us an upper and lower bound that capture our population mean.
Confidence Intervals Critical Values Practice Problems 1 Estimation 1.1 Confidence Intervals Definition 1.1 Margin of error. The margin of error of a distribution is the amount of error we predict when
More informationChapter 10. Regression. Understandable Statistics Ninth Edition By Brase and Brase Prepared by Yixun Shi Bloomsburg University of Pennsylvania
Chapter 10 Regression Understandable Statistics Ninth Edition By Brase and Brase Prepared by Yixun Shi Bloomsburg University of Pennsylvania Scatter Diagrams A graph in which pairs of points, (x, y), are
More informationBusiness Statistics. Chapter 14 Introduction to Linear Regression and Correlation Analysis QMIS 220. Dr. Mohammad Zainal
Department of Quantitative Methods & Information Systems Business Statistics Chapter 14 Introduction to Linear Regression and Correlation Analysis QMIS 220 Dr. Mohammad Zainal Chapter Goals After completing
More informationy = a + bx 12.1: Inference for Linear Regression Review: General Form of Linear Regression Equation Review: Interpreting Computer Regression Output
12.1: Inference for Linear Regression Review: General Form of Linear Regression Equation y = a + bx y = dependent variable a = intercept b = slope x = independent variable Section 12.1 Inference for Linear
More informationStats Review Chapter 14. Mary Stangler Center for Academic Success Revised 8/16
Stats Review Chapter 14 Revised 8/16 Note: This review is meant to highlight basic concepts from the course. It does not cover all concepts presented by your instructor. Refer back to your notes, unit
More informationBNAD 276 Lecture 10 Simple Linear Regression Model
1 / 27 BNAD 276 Lecture 10 Simple Linear Regression Model Phuong Ho May 30, 2017 2 / 27 Outline 1 Introduction 2 3 / 27 Outline 1 Introduction 2 4 / 27 Simple Linear Regression Model Managerial decisions
More informationUNIT 12 ~ More About Regression
***SECTION 15.1*** The Regression Model When a scatterplot shows a relationship between a variable x and a y, we can use the fitted to the data to predict y for a given value of x. Now we want to do tests
More informationTesting a Claim about the Difference in 2 Population Means Independent Samples. (there is no difference in Population Means µ 1 µ 2 = 0) against
Section 9 2A Lecture Testing a Claim about the Difference i Population Means Independent Samples Test H 0 : µ 1 = µ 2 (there is no difference in Population Means µ 1 µ 2 = 0) against H 1 : µ 1 > µ 2 or
More informationAMS7: WEEK 7. CLASS 1. More on Hypothesis Testing Monday May 11th, 2015
AMS7: WEEK 7. CLASS 1 More on Hypothesis Testing Monday May 11th, 2015 Testing a Claim about a Standard Deviation or a Variance We want to test claims about or 2 Example: Newborn babies from mothers taking
More informationMultiple Linear Regression
1. Purpose To Model Dependent Variables Multiple Linear Regression Purpose of multiple and simple regression is the same, to model a DV using one or more predictors (IVs) and perhaps also to obtain a prediction
More informationChapter 14 Simple Linear Regression (A)
Chapter 14 Simple Linear Regression (A) 1. Characteristics Managerial decisions often are based on the relationship between two or more variables. can be used to develop an equation showing how the variables
More informationCorrelation: Relationships between Variables
Correlation Correlation: Relationships between Variables So far, nearly all of our discussion of inferential statistics has focused on testing for differences between group means However, researchers are
More information1 Descriptive statistics. 2 Scores and probability distributions. 3 Hypothesis testing and one-sample t-test. 4 More on t-tests
Overall Overview INFOWO Statistics lecture S3: Hypothesis testing Peter de Waal Department of Information and Computing Sciences Faculty of Science, Universiteit Utrecht 1 Descriptive statistics 2 Scores
More informationCh. 16: Correlation and Regression
Ch. 1: Correlation and Regression With the shift to correlational analyses, we change the very nature of the question we are asking of our data. Heretofore, we were asking if a difference was likely to
More informationBasic Statistics Exercises 66
Basic Statistics Exercises 66 42. Suppose we are interested in predicting a person's height from the person's length of stride (distance between footprints). The following data is recorded for a random
More informationSimple Linear Regression: One Quantitative IV
Simple Linear Regression: One Quantitative IV Linear regression is frequently used to explain variation observed in a dependent variable (DV) with theoretically linked independent variables (IV). For example,
More informationAMS 7 Correlation and Regression Lecture 8
AMS 7 Correlation and Regression Lecture 8 Department of Applied Mathematics and Statistics, University of California, Santa Cruz Suumer 2014 1 / 18 Correlation pairs of continuous observations. Correlation
More informationNotebook Tab 6 Pages 183 to ConteSolutions
Notebook Tab 6 Pages 183 to 196 When the assumed relationship best fits a straight line model (r (Pearson s correlation coefficient) is close to 1 ), this approach is known as Linear Regression Analysis.
More informationdetermine whether or not this relationship is.
Section 9-1 Correlation A correlation is a between two. The data can be represented by ordered pairs (x,y) where x is the (or ) variable and y is the (or ) variable. There are several types of correlations
More informationMathematics for Economics MA course
Mathematics for Economics MA course Simple Linear Regression Dr. Seetha Bandara Simple Regression Simple linear regression is a statistical method that allows us to summarize and study relationships between
More informationProb/Stats Questions? /32
Prob/Stats 10.4 Questions? 1 /32 Prob/Stats 10.4 Homework Apply p551 Ex 10-4 p 551 7, 8, 9, 10, 12, 13, 28 2 /32 Prob/Stats 10.4 Objective Compute the equation of the least squares 3 /32 Regression A scatter
More informationInference for Regression
Inference for Regression Section 9.4 Cathy Poliak, Ph.D. cathy@math.uh.edu Office in Fleming 11c Department of Mathematics University of Houston Lecture 13b - 3339 Cathy Poliak, Ph.D. cathy@math.uh.edu
More informationCorrelation and Regression
Correlation and Regression October 25, 2017 STAT 151 Class 9 Slide 1 Outline of Topics 1 Associations 2 Scatter plot 3 Correlation 4 Regression 5 Testing and estimation 6 Goodness-of-fit STAT 151 Class
More informationDSST Principles of Statistics
DSST Principles of Statistics Time 10 Minutes 98 Questions Each incomplete statement is followed by four suggested completions. Select the one that is best in each case. 1. Which of the following variables
More informationExam: practice test 1 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Exam: practice test MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem. ) Using the information in the table on home sale prices in
More informationLecture 15: Chapter 10
Lecture 15: Chapter 10 C C Moxley UAB Mathematics 20 July 15 10.1 Pairing Data In Chapter 9, we talked about pairing data in a natural way. In this Chapter, we will essentially be discussing whether these
More informationCIVL /8904 T R A F F I C F L O W T H E O R Y L E C T U R E - 8
CIVL - 7904/8904 T R A F F I C F L O W T H E O R Y L E C T U R E - 8 Chi-square Test How to determine the interval from a continuous distribution I = Range 1 + 3.322(logN) I-> Range of the class interval
More informationBusiness Statistics 41000: Homework # 5
Business Statistics 41000: Homework # 5 Drew Creal Due date: Beginning of class in week # 10 Remarks: These questions cover Lectures #7, 8, and 9. Question # 1. Condence intervals and plug-in predictive
More informationSTAT 515 fa 2016 Lec Statistical inference - hypothesis testing
STAT 515 fa 2016 Lec 20-21 Statistical inference - hypothesis testing Karl B. Gregory Wednesday, Oct 12th Contents 1 Statistical inference 1 1.1 Forms of the null and alternate hypothesis for µ and p....................
More informationReminder: Student Instructional Rating Surveys
Reminder: Student Instructional Rating Surveys You have until May 7 th to fill out the student instructional rating surveys at https://sakai.rutgers.edu/portal/site/sirs The survey should be available
More informationChapter 4. Regression Models. Learning Objectives
Chapter 4 Regression Models To accompany Quantitative Analysis for Management, Eleventh Edition, by Render, Stair, and Hanna Power Point slides created by Brian Peterson Learning Objectives After completing
More informationCorrelation Analysis
Simple Regression Correlation Analysis Correlation analysis is used to measure strength of the association (linear relationship) between two variables Correlation is only concerned with strength of the
More informationData Analysis and Statistical Methods Statistics 651
y 1 2 3 4 5 6 7 x Data Analysis and Statistical Methods Statistics 651 http://www.stat.tamu.edu/~suhasini/teaching.html Lecture 32 Suhasini Subba Rao Previous lecture We are interested in whether a dependent
More informationAnalysis of Variance ANOVA. What We Will Cover in This Section. Situation
Analysis of Variance ANOVA 8//007 P7 Analysis of Variance What We Will Cover in This Section Introduction. Overview. Simple ANOVA. Repeated Measures ANOVA. Factorial ANOVA 8//007 P7 Analysis of Variance
More informationMathematical Notation Math Introduction to Applied Statistics
Mathematical Notation Math 113 - Introduction to Applied Statistics Name : Use Word or WordPerfect to recreate the following documents. Each article is worth 10 points and should be emailed to the instructor
More informationInferences for Correlation
Inferences for Correlation Quantitative Methods II Plan for Today Recall: correlation coefficient Bivariate normal distributions Hypotheses testing for population correlation Confidence intervals for population
More informationAnswer Key. 9.1 Scatter Plots and Linear Correlation. Chapter 9 Regression and Correlation. CK-12 Advanced Probability and Statistics Concepts 1
9.1 Scatter Plots and Linear Correlation Answers 1. A high school psychologist wants to conduct a survey to answer the question: Is there a relationship between a student s athletic ability and his/her
More informationStudent s t-distribution. The t-distribution, t-tests, & Measures of Effect Size
Student s t-distribution The t-distribution, t-tests, & Measures of Effect Size Sampling Distributions Redux Chapter 7 opens with a return to the concept of sampling distributions from chapter 4 Sampling
More information2.57 when the critical value is 1.96, what decision should be made?
Math 1342 Ch. 9-10 Review Name SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 9.1 1) If the test value for the difference between the means of two large
More informationAP Statistics Unit 6 Note Packet Linear Regression. Scatterplots and Correlation
Scatterplots and Correlation Name Hr A scatterplot shows the relationship between two quantitative variables measured on the same individuals. variable (y) measures an outcome of a study variable (x) may
More informationRegression Analysis and Forecasting Prof. Shalabh Department of Mathematics and Statistics Indian Institute of Technology-Kanpur
Regression Analysis and Forecasting Prof. Shalabh Department of Mathematics and Statistics Indian Institute of Technology-Kanpur Lecture 10 Software Implementation in Simple Linear Regression Model using
More informationInference for Proportions, Variance and Standard Deviation
Inference for Proportions, Variance and Standard Deviation Sections 7.10 & 7.6 Cathy Poliak, Ph.D. cathy@math.uh.edu Office Fleming 11c Department of Mathematics University of Houston Lecture 12 Cathy
More informationCorrelation & Simple Regression
Chapter 11 Correlation & Simple Regression The previous chapter dealt with inference for two categorical variables. In this chapter, we would like to examine the relationship between two quantitative variables.
More informationLab #12: Exam 3 Review Key
Psychological Statistics Practice Lab#1 Dr. M. Plonsky Page 1 of 7 Lab #1: Exam 3 Review Key 1) a. Probability - Refers to the likelihood that an event will occur. Ranges from 0 to 1. b. Sampling Distribution
More informationMidterm 2 - Solutions
Ecn 102 - Analysis of Economic Data University of California - Davis February 24, 2010 Instructor: John Parman Midterm 2 - Solutions You have until 10:20am to complete this exam. Please remember to put
More informationMarketing Research Session 10 Hypothesis Testing with Simple Random samples (Chapter 12)
Marketing Research Session 10 Hypothesis Testing with Simple Random samples (Chapter 12) Remember: Z.05 = 1.645, Z.01 = 2.33 We will only cover one-sided hypothesis testing (cases 12.3, 12.4.2, 12.5.2,
More informationCORELATION - Pearson-r - Spearman-rho
CORELATION - Pearson-r - Spearman-rho Scatter Diagram A scatter diagram is a graph that shows that the relationship between two variables measured on the same individual. Each individual in the set is
More informationAssociation Between Variables Measured at the Interval-Ratio Level: Bivariate Correlation and Regression
Association Between Variables Measured at the Interval-Ratio Level: Bivariate Correlation and Regression Last couple of classes: Measures of Association: Phi, Cramer s V and Lambda (nominal level of measurement)
More informationChapter 16. Simple Linear Regression and dcorrelation
Chapter 16 Simple Linear Regression and dcorrelation 16.1 Regression Analysis Our problem objective is to analyze the relationship between interval variables; regression analysis is the first tool we will
More information9. Linear Regression and Correlation
9. Linear Regression and Correlation Data: y a quantitative response variable x a quantitative explanatory variable (Chap. 8: Recall that both variables were categorical) For example, y = annual income,
More informationObjectives Simple linear regression. Statistical model for linear regression. Estimating the regression parameters
Objectives 10.1 Simple linear regression Statistical model for linear regression Estimating the regression parameters Confidence interval for regression parameters Significance test for the slope Confidence
More informationInference for the Regression Coefficient
Inference for the Regression Coefficient Recall, b 0 and b 1 are the estimates of the slope β 1 and intercept β 0 of population regression line. We can shows that b 0 and b 1 are the unbiased estimates
More informationCorrelation and Regression Analysis. Linear Regression and Correlation. Correlation and Linear Regression. Three Questions.
10/8/18 Correlation and Regression Analysis Correlation Analysis is the study of the relationship between variables. It is also defined as group of techniques to measure the association between two variables.
More information