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Unformatted text preview: 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 Final Exam Score 33 53 65 80 44 78 64 93 60 88 40 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: y= x= 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! 184 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. The simplest kind of relationship between two variables is a straight line, the analysis in this case is called linear regression. Regression Line for Exam 2 vs Final Remember the equation of a line? y = mx + b ˆ
In statistics we denote the regression line for a sample as: y b0 b1 x where: ˆ
y y-hat = the predicted y or estimated y value b0 y-intercept= estimated y when x=0 (not always meaningful) b1 slope = how much of an increase or decrease we expect
to see in y when x increases by 1 unit. Goal: To find a line that is “close” to the data points ‐ find the “best fitting” line. How? Observed
What do we mean by best? One measure of how good a line Predicted
fits is to look at the “observed errors” in prediction. Observed errors = A possible line Observed error if we used this
line to predict = y - yhat ˆ
______ y y _______ are called ____ residuals _____ So we want to choose the line for which the sum of squares of the observed errors (the sum of squared residuals) is the least. The line that does this is called: _____ Least Squares Regression Line _________ 185 The equations for the estimated slope and intercept are given by: b1...
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