183Stat 250 Gunderson Lecture NotesChapters 3 and 14: Regression AnalysisThe invalid assumption that correlation implies cause is probably among the two orthree most serious and common errors of human reasoning.‐‐Stephen Jay Gould, The Mismeasure of ManDescribing and assessing the significance ofrelationships between variablesis very importantin research. We will first learn how to do this in the case when the two variables arequantitative. Quantitative variables have numerical values that can be ordered according tothose values. We will study the material from Chapters 3 and 14 together. We will merge thetwo chapters together into one overall discussion of these ideas.Main ideaWe wish to study the relationship between two quantitative variables.Generally one variable is the ____RESPONSE______variable, denoted byy.This variable measures the outcome of the studyand is also called the ______DEPENDENT_______ variable.(thought to depend on x)The other variable is the ____EXPLANATORY____variable, denoted byx.It is the variable that is thought to explain the changes we see in the response variable. Theexplanatory variable is also called the ____INDEPENDENT__ variable.The first step in examining the relationship is to use a graph‐ascatterplot‐to display therelationship. We will look for an overall pattern and see if there are any departures from thisoverall pattern.If alinearrelationship appears to be reasonable from the scatterplot, we will take the next stepof finding a model (an equation of a line) to summarize the relationship. The resulting equationmay be used for predicting the response for various values of the explanatory variable. Ifcertain assumptions hold, we can assess the significance of the linear relationship and makesome confidence intervals for our estimations and predictions.Let's begin with an example that we will carry throughout our discussions.
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184Graphing the Relationship: Exam 2 versus Final ScoresHow well does the exam 2 score for a Stats 350 student predict their final exam score?Below are the scores for a random sample ofn= 6 students from a previous term.Exam 2 Score336544646040Final Exam Score538078938858Response(dependent) variabley=FINAL EXAM SCORE .Explanatory(independent) variablex=___EXAM 2 SCORE.Step 1: Examine the data graphically with a scatterplot.Add the points to the scatterplot below:Interpret the scatterplotin terms of ...overall form(is the average pattern look like a straight line or is it curved?)directionof association (positive or negative)strengthof association (how much do the points vary around the average pattern?)anydeviationsfrom the overall form?None here!x = y =
185Describing a Linear Relationship with a Regression LineRegression analysisis the area of statistics used to examine the relationship between aquantitative response variable and one or more explanatory variables. A key element is theestimation of an equationthat describes how, on average, the response variable is related tothe explanatory variables. A regression equation can also be used to make predictions.