Premaster Statistics Tutorial 4 Full solutions


 Arline Wilkinson
 1 years ago
 Views:
Transcription
1 Premaster Statistics Tutorial 4 Full solutions Regression analysis Q1 (based on Doane & Seward, 4/E, 12.7) a. Interpret the slope of the fitted regression = 125, b. What is the prediction for if = 2,000? c. Would the intercept be meaningful if this regression applies to home sales in a certain subdivision, different form the one used to find the regression equation? A1 a. Increasing the size of a home by 1 square foot increases the price by $150. b. = $125,000 + ($150 2,000) = $425,000. c. The intercept might be interpreted as the value of the lot without a home. But the range of values for does not include zero so it would be dangerous to extrapolate for = 0. Extra Observe the somewhat confusing habit in economic literarure of writing regression equations in the form = 125, , where is a variable, not a unit. Q2 (based on Doane & Seward, 4/E, 12.13) The regression equation = was estimated from a sample of 34 cities in the eastern United States. Both variables are in thousands of dollars. is the median selling price of homes in the city, and is median family income for the city. a. Interpret the slope. b. Is the intercept meaningful? Explain. c. Make a prediction of when = 50 and also when = 100. d. Given: = What is the meaning of that? (Data are from Money Magazine 32, no. 1 [January 2004], pp ) A2 a. Increasing the median income by $1,000 raises the median home price by $2,610; b. If median income is zero, then the model suggests that median home price is $51,300; c. $181,800 and $312,300; d. 34% of the variance of is explained by the model. Sol a. Increasing the median income by $1,000 raises the median home price by $2,610. b. If median income is zero, then the model suggests that median home price is $51,300. While it does not seem logical that the median family income for any city is zero, it is unclear what the lower bound would be. c. prediction HomePrice = $ (2.61 $50) = $181.8 (in $1000) or $181,800 prediction Homeprice = $ (2.61 $100) = $312.3 (in $1000) or $312,300 d. 34% of the variance of is explained by the model. That is quite low. And it might be due to chance: perhaps a lucky sample. Fortunately, the latter can be judged by statistical significance. The model is significant if the slope is significantly different from zero: this seems to be the case looking at the value (see later). Q3 (based on Doane & Seward, 4/E, 12.26) A regression was performed using data on 16 randomly selected charities in The variables were = expenses (millions of dollars) and = revenue (millions of dollars). a. Write the fitted regression equation. b. Construct a 95 percent confidence interval for the slope. c. Perform a righttailed test for zero slope at =.05. State the hypotheses clearly. (Data are from Forbes 172, no. 12, p. 248, and PM_STAT 1 Tutorial 4
2 SUMMARY OUTPUT Regression Statistics Multiple R 0, R Square 0, Adjusted R Square 0, Standard Error 14, Observations 16 ANOVA df SS MS F Significance F Regression , , , ,07289E08 Residual , ,13245 Total Coefficients Standard Error t Stat Pvalue Lower 95% Upper 95% Intercept 7, ,0403 Revenue 0,9467 0,0936 A3 a. = ; b ; c. reject Sol a. Define : expenses ($1,000,000); : revenue ($1,000,000) a. = b. For a 95% confidence level use ;. =2.145 The 95% confidence interval is ±( ) or c. Use our 5steps procedure: Model ( step 0 ): = + + where ~ (0, ) (i) : 0 versus : >0 ( =0.05) (ii) Sample statistic: ; reject for large values. (iii) Distribution test statistic under : = ~ Assumptions: see model (iv) Calculated test statistic: =. = Critical value: = ;. =1.761 (v) Because >, reject. There is evidence that the slope is positive; increased revenue is correlated with increased expenses. Q4 Use a linear regression model to explain the height (Dutch: lengte ) of female premaster students ( ) in terms of their shoe size (Dutch: schoenmaat ). Below you find some computer output, based on a random sample of these students. PM_STAT 2 Tutorial 4
3 Predicted values for: Lengtecm 95% Confidence Interval 95% Prediction Interval Schoenmaat Predicted lower upper lower upper A4 Sol a. Determine the theoretical and the estimated model belonging to the given output. b. It is claimed that the slope in this model is larger than 2. Test this hypothesis ( =1%). c. Is this a useful model in order to predict the height of female premaster students? (Perhaps you have seen a footprint in the snow; is it useful (using this model) to predict the height of the person concerned?) d. You see a footprint of size 38 in the snow and looking up you see in the distance a (female) premaster student just walking away. Give a relevant 95% interval for the height of this (female) premaster student. e. The next day you see another footprint of size 38. Give a relevant 95% interval for the average height of all (female) premaster students with shoe size 38. f. Calculate a 90% confidence interval for the constant in the regression model. a. Theoretical model: = + +, with ~ (0, ); Estimated model: = + = ; b. reject ; c. not very useful; d , ; e , ; f , a. Theoretical model: = + +, with ~ (0, ). = height in cm, = shoe size (may be stated for individual observations with or without the subscript ). Estimated model: = + = b. Use the 5 steps procedure! step 0 (model): see a. (i) : 2; : >2; =1% (ii) Sample statistic: ; reject for large values (iii) Distribution test statistic under : = ~ = ( =91) Assumptions: see model. We do not really need normality because is so large. (iv) Calculated test statistic: =. = PM_STAT 3 Tutorial 4
4 Critical value: = ;. = using Excel. With the table, you may take a conservative value ;. = value: (using Excel) (v) Decision: do reject, because value smaller than 1% or because >. Conclude that the slope is larger than 2. c. It is a statistically significant model, so the question about practically relevant is meaningful. We have = which is quite low. It would have quite limited value in predicting the height of a thief if the police found a footprint in the snow. d. This is individual prediction : e. This is mean prediction : f. ± ;. =51.386±( ), so (Excel: ;. = ) Q5 A consumer products company wants to measure the effectiveness of different types of advertising media in the promotion of its products. Specifically, the company is interested in the effectiveness of radio advertising and newspaper advertising (including the cost of discount coupons). A sample of 22 cities with approximately equal populations is selected for study during a test period of one month. Each city is allocated a specific expenditure level both for radio advertising and for newspaper advertising. The sales of the product (in thousands of dollars) and also the levels of media expenditure (in thousands of dollars) during the test month are recorded, with the following results: SPSS results: PM_STAT 4 Tutorial 4
5 a. State the multiple regression equation (description of the model including assumptions and the estimated model). b. Interpret the meaning of the slopes, and, in this problem. c. Interpret the meaning of the regression coefficient,. d. Which type of advertising is more effective? Explain. e. Determine whether there is a significant relationship between sales and the two independent variables (radio advertising and newspaper advertising) at the 0.05 level of significance. f. Interpret the meaning of the value. g. Compute the coefficient of multiple determination,, and interpret its meaning. h. Find the adjusted and interpret its meaning. i. Is there evidence that the slope coefficient for Radio advertisements is more than 10 at = 0.05? A5 A. = with ~ (0, ) and = + + = ; d. newspaper advertising is more effective; e. there is evidence of a significant linear relationship; i. there is evidence that the coefficient for radio advertisements is larger than 10. Sol a. Statistical model: = with ~ (0, ^2) where =Sales, =Radio Advertising, =Newspaper Advertising Estimated model: = + + = where =Estimated Sales b. For a given amount of newspaper advertising, each increase of $1000 in radio advertising is estimated to result in a mean increase in sales of $13,081. For a given amount of radio advertising, each increase of $1000 in newspaper advertising is estimated to result in the mean increase in sales of $16,795. c. When there is no money spent on radio advertising and newspaper advertising, the estimated mean amount of sales is $156, PM_STAT 5 Tutorial 4
6 d. The slope of newspaper advertising is higher than the slope of radio advertising, so newspaper advertising is more effective. e. Model: see a. (i) : = =0; :not ; =0.05 (ii) Sample statistic: = ; reject for large values (iii) Under : ~, Assumptions: see model formulation (error term normally distributed with constant variance) (iv) = =40.16;. =3.522; value=0.000 (v) reject because > or equivalently because <. Conclude that there is evidence of a significant linear relationship. f. value<0.0005: the probability of obtaining an of or even larger is less than if is true. g. = = =0.8087, or rather directly from SPSS output. So, 80.87% of the variation in sales can be explained by variation in radio advertising and variation in newspaper advertising. Note: model is significant and =0.81, so practically it is a useful model. h. =0.789 from computer output. This is the proportion of explained variance, but taking into account the number of variables and number of observations. i. This test is not provided by SPSS, but is not difficult to derive from it. (i) : 10 against : >10 ( =0.05) (ii) Sample statistic: ; reject for large values. (iii) Under : ~ ; assumptions: see earlier. (iv) =. =1.7516;. = ;. = (v) Reject because >. There is evidence that the coefficient for radio advertisements is larger than 10. Regression diagnostics Q1 (Doane & Seward, 4/E, 12.37) An estimated regression for a random sample of vehicles is = , where is miles per gallon and is the engine s horsepower. The standard error is =2.03. Suppose an engine has 200 horsepower and its actual (observed) fuel efficiency is = a. Calculate the predicted. b. Calculate the residual. c. Standardize the residual using. d. Is this engine an outlier? A1 a ; b.5.13; c ; d. an unusual observation, not an outlier Sol a. = = b. = = =5.13 c. = =. = d. 2< <3, so we refer to this engine as an unusual observation, not as an outlier. Q2 (Doane & Seward, 4/E, 12.38) A sample of season performance measures for 29 NBA teams was collected for a season. A regression analysis was performed on two of the variables with = total number of free throws made and = total number of free throws attempted. Calculate the leverage statistic for the PM_STAT 6 Tutorial 4
7 A2 Sol Q3 following three teams and state whether or not the leverage would be considered high. Given: =999,603 and =2004. a. The Golden State Warriors attempted 2,382 free throws. b. The New Jersey Nets attempted 2,125 free throws. c. The New York Knicks attempted 1,620 free throws. a. Yes; b. No; c. Yes a. h= +( ) = +( ) =0.18. The value of = =0.14, so this observation has a high leverage statistic. b. h= +( ) = +( ) =0.05. The value of = =0.14, so this observation does not have a high leverage statistic. c. h= +( ) = +( ) =0.18. See a. A consumer products company wants to measure the effectiveness of different types of advertising media in the promotion of its products. Specifically, the company is interested in the effectiveness of radio advertising and newspaper advertising (including the cost of discount coupons). A sample of 22 cities with approximately equal populations is selected for study during a test period of one month. Each city is allocated a specific expenditure level both for radio advertising and for newspaper advertising. The sales of the product (in thousands of dollars) and also the levels of media expenditure (in thousands of dollars) during the test month are recorded, with the following results: SPSS output is given below: PM_STAT 7 Tutorial 4
8 a. State the multiple regression equation (description of the the model including assumptions and the estimated model). b. Interpret the meaning of the slopes, and, in this problem. c. Interpret the meaning of the regression coefficient,. d. Which type of advertising is more effective? Explain. e. Determine whether there is a significant relationship between sales and the two independent variables (radio advertising and newspaper advertising) at the 0.05 level of significance. f. Interpret the meaning of the value. g. Compute the coefficient of multiple determination,, and interpret its meaning. h. Find the adjusted and interpret its meaning. i. Perform a residual analysis on your results. PM_STAT 8 Tutorial 4
9 j. If appropriate, perform the DurbinWatson test using =0.05. k. Are the regression assumptions valid for these data? l. Construct a 95% confidence interval estimate of the population slope between sales and radio advertising. m. At the 0.05 level of significance, determine whether each independent variable makes a significant contribution to the regression model. On the basis of these results, indicate the independent variables to include in this model, using statistical significance as the only criterion. n. Test : 10 against : >10 (or is there evidence that the slope coefficient for Radio is more than 10?) o. Is there serious collinearity? p. Some might argue that there is a pattern in the residuals, suggesting a quadratic relation between sales and both advertising variables. We computed the variables and and included these variables in the regression model. State the model and the estimated model. Compare the two models. PM_STAT 9 Tutorial 4
10 q. Any comments on the residual for this model? A3 See full solutions. Sol a. Model: = + + +, ~ (0, ), =Sales, =Radio Advertising, =Newspaper Advertising; = b. For a given amount of newspaper advertising, each increase of $1000 in radio advertising is estimated to result in a mean increase in sales of $13,081. For a given amount of radio advertising, each increase of $1000 in newspaper advertising is estimated to result in the mean increase in sales of $16,795. c. When there is no money spent on radio advertising and newspaper advertising, the estimated mean amount of sales is $156, PM_STAT 10 Tutorial 4
11 d. Technically you would need : = which is outside our scope. Just look at magnitude of slope coefficient: newspaper is more effective. e. Five (six?) steps procedure: (0) Model: = + + +, ~ (0, ), =Sales, =Radio Advertising, =Newspaper Advertising (1) : = =0; : ( = =0); =0.05 (2) Sample statistic: = ; reject for large values (3) Under : ~ ; ; assumptions: see model formulation in step (0) (4) = =40.158;. = ; ;. =3.522; value=0.000 (5) > or value<, so reject and conclude that there is evidence of a significant linear relationship. f. value< The probability of obtaining an test statistic of or larger is less than if is true. g. = =,, =0.8087, or rather directly from SPSS output. So, 80.87% of the,, variation in sales can be explained by variation in radio advertising and variation in newspaper advertising. Note: model is significant and =0.81, so practically it is a useful model. h. =0.789 from computer output. This is proportion of explained variance, but taking into account the number of variables and number of observations. (In samples squared will be somewhat inflated (biased upward), while squared adjusted is not). i. There appears (not very clear!) to be a quadratic relationship in the plot of the residuals against both radio and newspaper advertising. Thus, quadratic terms for each of these explanatory models should be considered for inclusion in the model. j. DurbinWatson has no meaning here, as there is no natural ordering of the 22 cases. k. The skewness and kurtosis of the residual are pretty well between 1 and 1. This l. 95% confidence interval on : ± = ± Alternative notation , of = 9.398, m. First test : (1) : =0; : 0; =0.05 (2) Sample statistic: ; reject for large and small values (3) Under : = ~ = ; assumptions: see previous model formulation (4) =. =7.43;. = ;. =2.093; value=0.000 (5) Reject because value= = There is evidence that the variable contributes to a model already containing. Now for : (4) =. =5.67;. = ;. =2.093; value=0.000 (5) Reject because value= = There is evidence that the variable contributes to a model already containing. n. (1) : 10; : >10; =0.05 (4) = normal calculator. =.. =1.7516; = A value cannot be computed with a (5) Reject because >. There is evidence that the coefficient is larger than 10. o. = , = =1.009 or directly from output. = There is no serious collinearity as the VIF is smaller than 5 (even close to 1, the minimum value!). For the results are identical. can be computed by regressing on all remaining explanatory PM_STAT 11 Tutorial 4
12 variables ( ). Both VIFs are equal because we have only two explanatory variables in this model. p. Model: = , ~ (0, ), =Sales, =Radio Advertising, =Newspaper Advertising Note =0.907, considerably larger than =0.789 in smaller model. Not all variables are significant anymore, perhaps due to the substantial multicollinearity (VIFs are 13.4, 9.1, 13.4 and 9.45, all considerably larger than 5). You perhaps could eliminate the variables and (the linear, nonsignificant terms, and you end up with the model below, which is very good. q. The skewness and kurtosis statistics look fine, well between 1 and 1. There is no sign of heteroscedasticity. PM_STAT 12 Tutorial 4
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 1propZint 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
More informationSELFTEST: SIMPLE REGRESSION
ECO 22000 McRAE SELFTEST: SIMPLE REGRESSION Note: Those questions indicated with an (N) are unlikely to appear in this form on an inclass examination, but you should be able to describe the procedures
More information12/31/2016. PSY 512: Advanced Statistics for Psychological and Behavioral Research 2
PSY 512: Advanced Statistics for Psychological and Behavioral Research 2 Understand when to use multiple Understand the multiple equation and what the coefficients represent Understand different methods
More information, then the form of the model is given by: which comprises a deterministic component involving the three regression coefficients (
Multiple regression Introduction Multiple regression is a logical extension of the principles of simple linear regression to situations in which there are several predictor variables. For instance if we
More informationMultiple Regression in SPSS STAT 314
Multiple Regression in SPSS STAT 314 I. The accompanying data is on y = profit margin of savings and loan companies in a given year, x 1 = net revenues in that year, and x 2 = number of savings and loan
More informationUsing SPSS for Multiple Regression. UDP 520 Lab 7 Lin Lin December 4 th, 2007
Using SPSS for Multiple Regression UDP 520 Lab 7 Lin Lin December 4 th, 2007 Step 1 Define Research Question What factors are associated with BMI? Predict BMI. Step 2 Conceptualizing Problem (Theory) Individual
More informationModule 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
More informatione = random error, assumed to be normally distributed with mean 0 and standard deviation σ
1 Linear Regression 1.1 Simple Linear Regression Model The linear regression model is applied if we want to model a numeric response variable and its dependency on at least one numeric factor variable.
More information" 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
More information, has mean A) 0.3. B) the smaller of 0.8 and 0.5. C) 0.15. D) which cannot be determined without knowing the sample results.
BA 275 Review Problems  Week 9 (11/20/0611/24/06) CD Lessons: 69, 70, 1620 Textbook: pp. 520528, 111124, 133141 An SRS of size 100 is taken from a population having proportion 0.8 of successes. An
More informationMultiple Regression Analysis in Minitab 1
Multiple Regression Analysis in Minitab 1 Suppose we are interested in how the exercise and body mass index affect the blood pressure. A random sample of 10 males 50 years of age is selected and their
More informationSimple Linear Regression
Inference for Regression Simple Linear Regression IPS Chapter 10.1 2009 W.H. Freeman and Company Objectives (IPS Chapter 10.1) Simple linear regression Statistical model for linear regression Estimating
More informationFinal 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
More informationA. Karpinski
Chapter 3 Multiple Linear Regression Page 1. Overview of multiple regression 32 2. Considering relationships among variables 33 3. Extending the simple regression model to multiple predictors 34 4.
More informationWeek 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
More information12/31/2016. PSY 512: Advanced Statistics for Psychological and Behavioral Research 2
PSY 512: Advanced Statistics for Psychological and Behavioral Research 2 Understand linear regression with a single predictor Understand how we assess the fit of a regression model Total Sum of Squares
More informationInterpreting Multiple Regression
Fall Semester, 2001 Statistics 621 Lecture 5 Robert Stine 1 Preliminaries Interpreting Multiple Regression Project and assignments Hope to have some further information on project soon. Due date for Assignment
More informationSupplement 13A: Partial F Test
Supplement 13A: Partial F Test Purpose of the Partial F Test For a given regression model, could some of the predictors be eliminated without sacrificing too much in the way of fit? Conversely, would it
More informationRegression stepbystep using Microsoft Excel
Step 1: Regression stepbystep using Microsoft Excel Notes prepared by Pamela Peterson Drake, James Madison University Type the data into the spreadsheet The example used throughout this How to is a regression
More informationResiduals. Residuals = ª Department of ISM, University of Alabama, ST 260, M23 Residuals & Minitab. ^ e i = y i  y i
A continuation of regression analysis Lesson Objectives Continue to build on regression analysis. Learn how residual plots help identify problems with the analysis. M231 M232 Example 1: continued Case
More informationRegression. Name: Class: Date: Multiple Choice Identify the choice that best completes the statement or answers the question.
Class: Date: Regression Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Given the least squares regression line y8 = 5 2x: a. the relationship between
More information1. 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
More informationSimple Linear Regression in SPSS STAT 314
Simple Linear Regression in SPSS STAT 314 1. Ten Corvettes between 1 and 6 years old were randomly selected from last year s sales records in Virginia Beach, Virginia. The following data were obtained,
More informationST 311 Evening Problem Session Solutions Week 11
1. p. 175, Question 32 (Modules 10.110.4) [Learning Objectives J1, J3, J9, J1114, J17] Since 1980, average mortgage rates have fluctuated from a low of under 6% to a high of over 14%. Is there a relationship
More informationCHAPTER 2 AND 10: Least Squares Regression
CHAPTER 2 AND 0: Least Squares Regression In chapter 2 and 0 we will be looking at the relationship between two quantitative variables measured on the same individual. General Procedure:. Make a scatterplot
More informationSimple 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
More informationE205 Final: Version B
Name: Class: Date: E205 Final: Version B Multiple Choice Identify the choice that best completes the statement or answers the question. 1. The owner of a local nightclub has recently surveyed a random
More informationName: Student ID#: Serial #:
STAT 22 Business Statistics II Term3 KING FAHD UNIVERSITY OF PETROLEUM & MINERALS Department Of Mathematics & Statistics DHAHRAN, SAUDI ARABIA STAT 22: BUSINESS STATISTICS II Third Exam July, 202 9:20
More informationCHAPTER 9: SERIAL CORRELATION
Serial correlation (or autocorrelation) is the violation of Assumption 4 (observations of the error term are uncorrelated with each other). Pure Serial Correlation This type of correlation tends to be
More informationHomework 8 Solutions
Math 17, Section 2 Spring 2011 Homework 8 Solutions Assignment Chapter 7: 7.36, 7.40 Chapter 8: 8.14, 8.16, 8.28, 8.36 (ad), 8.38, 8.62 Chapter 9: 9.4, 9.14 Chapter 7 7.36] a) A scatterplot is given below.
More information0.1 Multiple Regression Models
0.1 Multiple Regression Models We will introduce the multiple Regression model as a mean of relating one numerical response variable y to two or more independent (or predictor variables. We will see different
More informationSimple Linear Regression Chapter 11
Simple Linear Regression Chapter 11 Rationale Frequently decisionmaking situations require modeling of relationships among business variables. For instance, the amount of sale of a product may be related
More informationRegression 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
More informationSydney Roberts Predicting Age Group Swimmers 50 Freestyle Time 1. 1. Introduction p. 2. 2. Statistical Methods Used p. 5. 3. 10 and under Males p.
Sydney Roberts Predicting Age Group Swimmers 50 Freestyle Time 1 Table of Contents 1. Introduction p. 2 2. Statistical Methods Used p. 5 3. 10 and under Males p. 8 4. 11 and up Males p. 10 5. 10 and under
More informationInteraction between quantitative predictors
Interaction between quantitative predictors In a firstorder model like the ones we have discussed, the association between E(y) and a predictor x j does not depend on the value of the other predictors
More informationSTAT 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
More informationMath 62 Statistics Sample Exam Questions
Math 62 Statistics Sample Exam Questions 1. (10) Explain the difference between the distribution of a population and the sampling distribution of a statistic, such as the mean, of a sample randomly selected
More informationData and Regression Analysis. Lecturer: Prof. Duane S. Boning. Rev 10
Data and Regression Analysis Lecturer: Prof. Duane S. Boning Rev 10 1 Agenda 1. Comparison of Treatments (One Variable) Analysis of Variance (ANOVA) 2. Multivariate Analysis of Variance Model forms 3.
More informationThe general form of the PROC GLM statement is
Linear Regression Analysis using PROC GLM Regression analysis is a statistical method of obtaining an equation that represents a linear relationship between two variables (simple linear regression), or
More informationRegression in SPSS. Workshop offered by the Mississippi Center for Supercomputing Research and the UM Office of Information Technology
Regression in SPSS Workshop offered by the Mississippi Center for Supercomputing Research and the UM Office of Information Technology John P. Bentley Department of Pharmacy Administration University of
More informationStatistics II Final Exam  January Use the University stationery to give your answers to the following questions.
Statistics II Final Exam  January 2012 Use the University stationery to give your answers to the following questions. Do not forget to write down your name and class group in each page. Indicate clearly
More informationChapter 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
More informationMultiple 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
More informationChapter 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
More informationDEPARTMENT OF PSYCHOLOGY UNIVERSITY OF LANCASTER MSC IN PSYCHOLOGICAL RESEARCH METHODS ANALYSING AND INTERPRETING DATA 2 PART 1 WEEK 9
DEPARTMENT OF PSYCHOLOGY UNIVERSITY OF LANCASTER MSC IN PSYCHOLOGICAL RESEARCH METHODS ANALYSING AND INTERPRETING DATA 2 PART 1 WEEK 9 Analysis of covariance and multiple regression So far in this course,
More informationStatistics 100 Sample Final Questions (Note: These are mostly multiple choice, for extra practice. Your Final Exam will NOT have any multiple choice!
Statistics 100 Sample Final Questions (Note: These are mostly multiple choice, for extra practice. Your Final Exam will NOT have any multiple choice!) Part A  Multiple Choice Indicate the best choice
More information4. Multiple Regression in Practice
30 Multiple Regression in Practice 4. Multiple Regression in Practice The preceding chapters have helped define the broad principles on which regression analysis is based. What features one should look
More informationChapter Seven. Multiple regression An introduction to multiple regression Performing a multiple regression on SPSS
Chapter Seven Multiple regression An introduction to multiple regression Performing a multiple regression on SPSS Section : An introduction to multiple regression WHAT IS MULTIPLE REGRESSION? Multiple
More informationA Primer on Forecasting Business Performance
A Primer on Forecasting Business Performance There are two common approaches to forecasting: qualitative and quantitative. Qualitative forecasting methods are important when historical data is not available.
More informationwhere b is the slope of the line and a is the intercept i.e. where the line cuts the y axis.
Least Squares Introduction We have mentioned that one should not always conclude that because two variables are correlated that one variable is causing the other to behave a certain way. However, sometimes
More informationUCLA STAT 13 Statistical Methods  Final Exam Review Solutions Chapter 7 Sampling Distributions of Estimates
UCLA STAT 13 Statistical Methods  Final Exam Review Solutions Chapter 7 Sampling Distributions of Estimates 1. (a) (i) µ µ (ii) σ σ n is exactly Normally distributed. (c) (i) is approximately Normally
More informationRegression III: Dummy Variable Regression
Regression III: Dummy Variable Regression Tom Ilvento FREC 408 Linear Regression Assumptions about the error term Mean of Probability Distribution of the Error term is zero Probability Distribution of
More informationAnswers Investigation 4
Applications 1. a. Median height is 15.7 cm. Order the 1 heights from shortest to tallest. Since 1 is even, average the two middle numbers, 15.6 cm and 15.8 cm. b. Median stride distance is 124.8 cm. Order
More informationChapter 11: Two Variable Regression Analysis
Department of Mathematics Izmir University of Economics Week 1415 20142015 In this chapter, we will focus on linear models and extend our analysis to relationships between variables, the definitions
More informationAP * Statistics Review. Linear Regression
AP * Statistics Review Linear Regression Teacher Packet Advanced Placement and AP are registered trademark of the College Entrance Examination Board. The College Board was not involved in the production
More informationChapter 4 and 5 solutions
Chapter 4 and 5 solutions 4.4. Three different washing solutions are being compared to study their effectiveness in retarding bacteria growth in five gallon milk containers. The analysis is done in a laboratory,
More informationSimple 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
More informationCollinearity of independent variables. Collinearity is a condition in which some of the independent variables are highly correlated.
Collinearity of independent variables Collinearity is a condition in which some of the independent variables are highly correlated. Why is this a problem? Collinearity tends to inflate the variance of
More informationRegression analysis in practice with GRETL
Regression analysis in practice with GRETL Prerequisites You will need the GNU econometrics software GRETL installed on your computer (http://gretl.sourceforge.net/), together with the sample files that
More informationMS&E 226: Small Data. Lecture 17: Additional topics in inference (v1) Ramesh Johari
MS&E 226: Small Data Lecture 17: Additional topics in inference (v1) Ramesh Johari ramesh.johari@stanford.edu 1 / 34 Warnings 2 / 34 Modeling assumptions: Regression Remember that most of the inference
More informationACTM State ExamStatistics
ACTM State ExamStatistics For the 25 multiplechoice questions, make your answer choice and record it on the answer sheet provided. Once you have completed that section of the test, proceed to the tiebreaker
More informationUnivariate 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
More information12.13 RESIDUAL ANALYSIS IN MULTIPLE REGRESSION (OPTIONAL)
12.13 Analysis in Multiple Regression (Optional) 1 Although Excel and MegaStat are emphasized in Business Statistics in Practice, Second Canadian Edition, some examples in the additional material on Connect
More informationStatistics for Management IISTAT 362Final Review
Statistics for Management IISTAT 362Final Review Multiple Choice Identify the letter of the choice that best completes the statement or answers the question. 1. The ability of an interval estimate to
More informationINCOME AND HAPPINESS 1. Income and Happiness
INCOME AND HAPPINESS 1 Income and Happiness Abstract Are wealthier people happier? The research study employed simple linear regression analysis to confirm the positive relationship between income and
More informationCourse 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
More informationUsing 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
More informationSimple Methods and Procedures Used in Forecasting
Simple Methods and Procedures Used in Forecasting The project prepared by : Sven Gingelmaier Michael Richter Under direction of the Maria JadamusHacura What Is Forecasting? Prediction of future events
More informationLecture 18 Linear Regression
Lecture 18 Statistics Unit Andrew Nunekpeku / Charles Jackson Fall 2011 Outline 1 1 Situation  used to model quantitative dependent variable using linear function of quantitative predictor(s). Situation
More informationSIMPLE REGRESSION ANALYSIS
SIMPLE REGRESSION ANALYSIS Introduction. Regression analysis is used when two or more variables are thought to be systematically connected by a linear relationship. In simple regression, we have only two
More information5. Multiple regression
5. Multiple regression QBUS6840 Predictive Analytics https://www.otexts.org/fpp/5 QBUS6840 Predictive Analytics 5. Multiple regression 2/39 Outline Introduction to multiple linear regression Some useful
More informationInferential Statistics
Inferential Statistics Sampling and the normal distribution Zscores Confidence levels and intervals Hypothesis testing Commonly used statistical methods Inferential Statistics Descriptive statistics are
More informationDO NOT TURN OVER UNTIL TOLD TO BEGIN
THIS PAPER IS NOT TO BE REMOVED FROM THE EXAMINATION HALLS University of London BSc Examination 2012 BA1040 (BBA0040) +Enc Business Administration Business Statistics Date tba: Time tba DO NOT TURN OVER
More informationEDUCATION AND VOCABULARY MULTIPLE REGRESSION IN ACTION
EDUCATION AND VOCABULARY MULTIPLE REGRESSION IN ACTION EDUCATION AND VOCABULARY 510 hours of input weekly is enough to pick up a new language (Schiff & Myers, 1988). Dutch children spend 5.5 hours/day
More informationRegression Analysis (Spring, 2000)
Regression Analysis (Spring, 2000) By Wonjae Purposes: a. Explaining the relationship between Y and X variables with a model (Explain a variable Y in terms of Xs) b. Estimating and testing the intensity
More information1) Write the following as an algebraic expression using x as the variable: Triple a number subtracted from the number
1) Write the following as an algebraic expression using x as the variable: Triple a number subtracted from the number A. 3(x  x) B. x 3 x C. 3x  x D. x  3x 2) Write the following as an algebraic expression
More informationElementary 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
More information121 Multiple Linear Regression Models
121.1 Introduction Many applications of regression analysis involve situations in which there are more than one regressor variable. A regression model that contains more than one regressor variable is
More informationUNDERSTANDING MULTIPLE REGRESSION
UNDERSTANDING Multiple regression analysis (MRA) is any of several related statistical methods for evaluating the effects of more than one independent (or predictor) variable on a dependent (or outcome)
More informationX 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.
More informationCorrelation and Regression Analysis: SPSS
Correlation and Regression Analysis: SPSS Bivariate Analysis: Cyberloafing Predicted from Personality and Age These days many employees, during work hours, spend time on the Internet doing personal things,
More informationThe scatterplot indicates a positive linear relationship between waist size and body fat percentage:
STAT E150 Statistical Methods Multiple Regression Three percent of a man's body is essential fat, which is necessary for a healthy body. However, too much body fat can be dangerous. For men between the
More informationUnit 31 A Hypothesis Test about Correlation and Slope in a Simple Linear Regression
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
More informationHYPOTHESIS TESTING: CONFIDENCE INTERVALS, TTESTS, ANOVAS, AND REGRESSION
HYPOTHESIS TESTING: CONFIDENCE INTERVALS, TTESTS, ANOVAS, AND REGRESSION HOD 2990 10 November 2010 Lecture Background This is a lightning speed summary of introductory statistical methods for senior undergraduate
More informationGood 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
More informationFactors 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
More informationRegression in ANOVA. James H. Steiger. Department of Psychology and Human Development Vanderbilt University
Regression in ANOVA James H. Steiger Department of Psychology and Human Development Vanderbilt University James H. Steiger (Vanderbilt University) 1 / 30 Regression in ANOVA 1 Introduction 2 Basic Linear
More informationThe importance of graphing the data: Anscombe s regression examples
The importance of graphing the data: Anscombe s regression examples Bruce Weaver Northern Health Research Conference Nipissing University, North Bay May 3031, 2008 B. Weaver, NHRC 2008 1 The Objective
More information1. ε is normally distributed with a mean of 0 2. the variance, σ 2, is constant 3. All pairs of error terms are uncorrelated
STAT E150 Statistical Methods Residual Analysis; Data Transformations The validity of the inference methods (hypothesis testing, confidence intervals, and prediction intervals) depends on the error term,
More informationABSORBENCY OF PAPER TOWELS
ABSORBENCY OF PAPER TOWELS 15. Brief Version of the Case Study 15.1 Problem Formulation 15.2 Selection of Factors 15.3 Obtaining Random Samples of Paper Towels 15.4 How will the Absorbency be measured?
More informationPractice 3 SPSS. Partially based on Notes from the University of Reading:
Practice 3 SPSS Partially based on Notes from the University of Reading: http://www.reading.ac.uk Simple Linear Regression A simple linear regression model is fitted when you want to investigate whether
More informationECO220Y1Y Duration  3 hours Examination Aids: Calculator
Page 1 of 16 UNIVERSITY OF TORONTO Faculty of Arts and Science APRIL 2011 EXAMINATIONS ECO220Y1Y Duration  3 hours Examination Aids: Calculator Last Name: First Name: Student #: ENTER YOUR NAME AND STUDENT
More informationStatistics & Regression: Easier than SAS
ST003 Statistics & Regression: Easier than SAS Vincent Maffei, Anthem Blue Cross and Blue Shield, North Haven, CT Michael Davis, Bassett Consulting Services, Inc., North Haven, CT ABSTRACT In this paper
More informationSimple 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
More information(d) True or false? When the number of treatments a=9, the number of blocks b=10, and the other parameters r =10 and k=9, it is a BIBD design.
PhD Qualifying exam Methodology Jan 2014 Solutions 1. True or false question  only circle "true " or "false" (a) True or false? Fstatistic can be used for checking the equality of two population variances
More informationAP Statistics 2011 Scoring Guidelines
AP Statistics 2011 Scoring Guidelines The College Board The College Board is a notforprofit membership association whose mission is to connect students to college success and opportunity. Founded in
More information13: Additional ANOVA Topics. Post hoc Comparisons
13: Additional ANOVA Topics Post hoc Comparisons ANOVA Assumptions Assessing Group Variances When Distributional Assumptions are Severely Violated KruskalWallis Test Post hoc Comparisons In the prior
More informationCHAPTER 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
More informationChapter 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) 
More information