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Week 3 lecture note

Week 3 lecture note - Let's get started Here is what you...

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Let's get started! Here is what you will learn in this lesson. Learning objectives for this lesson Upon completion of this lesson, you should be: Understand the relationship between the slope of the regression line and correlation Comprehend the meaning of the Coefficient of Determination, R 2 Now how to determine which variable is a response and which is an explanatory in a regression equation Understand that correlation measures the strength of a linear relationship between two variables Realize how outliers can influence a regression equation Be able to reasonably estimate the correlation from a scatterplot Correlation and Regression Correlation and regression is concerned with examining the relationship between two (or more) quantitative variables. Three tools will be used to describe, picture, and quantify the relationship between quantitative variables: 1. Scatterplot , a two-dimensional graph of data values for two quantitative variables. 2. Correlation , a statistic that measures the strength and direction of a linear relationship between two quantitative variables. 3. Regression equation, an equation that describes the average relationship between a quantitative response variable and a quantitative explanatory variable. Equations of Straight Lines: Review The equation of a straight line is given by y = a + bx. When x = 0, y = a, the intercept of the line; b is the slope of the line: it measures the change in y per unit change in x. Two examples: Data 1 Data 2 x y x y 0 3 0 13 1

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5 1 11 2 7 2 9 3 9 3 7 4 11 4 5 5 13 5 3 For the 'Data 1' the equation is y = 3 + 2x ; the intercept is 3 and the slope is 2. The line slopes upward, indicating a positive relationship between x and y. For the 'Data 2' the equation is y = 13 - 2x ; the intercept is 13 and the slope is -2. The line slopes downward, indicating a negative relationship between x and y. Plot for Data 1 Plot for Data 2 y = 3 + 2 x
y = 13 - 2 x The relationship between x and y is 'perfect' for these two examples—the points fall exactly on a straight line or the value of y is determined exactly by the value of x. Our interest will be concerned with relationships between two variables which are not perfect. The 'Correlation' between x and y is r = 1.00 for the values of x and y on the left and r = -1.00 for the values of x and y on the right. Regression analysis is concerned with finding the 'best' fitting line for predicting the average value of a response variable y using a predictor variable x.

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