Ch 14 - Regression - Chapters3and14:RegressionAnalysis...

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183 Stat 250 Gunderson Lecture Notes Chapters 3 and 14: Regression Analysis The invalid assumption that correlation implies cause is probably among the two or three most serious and common errors of human reasoning. Stephen Jay Gould, The Mismeasure of Man Describing and assessing the significance of relationships between variables is very important in research. We will first learn how to do this in the case when the two variables are quantitative. Quantitative variables have numerical values that can be ordered according to those values. We will study the material from Chapters 3 and 14 together. We will merge the two chapters together into one overall discussion of these ideas. Main idea We wish to study the relationship between two quantitative variables. Generally one variable is the ____ RESPONSE ______ variable , denoted by y . This variable measures the outcome of the study and is also called the ______ DEPENDENT _______ variable. (thought to depend on x) The other variable is the ____ EXPLANATORY ____ variable , denoted by x . It is the variable that is thought to explain the changes we see in the response variable. The explanatory variable is also called the ____ INDEPENDENT __ variable. The first step in examining the relationship is to use a graph a scatterplot to display the relationship. We will look for an overall pattern and see if there are any departures from this overall pattern. If a linear relationship appears to be reasonable from the scatterplot, we will take the next step of finding a model (an equation of a line) to summarize the relationship. The resulting equation may be used for predicting the response for various values of the explanatory variable. If certain assumptions hold, we can assess the significance of the linear relationship and make some confidence intervals for our estimations and predictions. Let's begin with an example that we will carry throughout our discussions.
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184 Graphing the Relationship: Exam 2 versus Final Scores How well does the exam 2 score for a Stats 350 student predict their final exam score? Below are the scores for a random sample of n = 6 students from a previous term. Exam 2 Score 33 65 44 64 60 40 Final Exam Score 53 80 78 93 88 58 Response (dependent) variable y = FINAL EXAM SCORE . Explanatory (independent) variable x = ___ EXAM 2 SCORE . Step 1: Examine the data graphically with a scatterplot. Add the points to the scatterplot below: Interpret the scatterplot in terms of ... overall form (is the average pattern look like a straight line or is it curved?) direction of association (positive or negative) strength of association (how much do the points vary around the average pattern?) any deviations from the overall form? None here! x = y =
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185 Describing a Linear Relationship with a Regression Line Regression analysis is the area of statistics used to examine the relationship between a quantitative response variable and one or more explanatory variables. A key element is the estimation of an equation that describes how, on average, the response variable is related to the explanatory variables. A regression equation can also be used to make predictions.
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