# 35 - Chapter 12 Analyzing Association Between Quantitative...

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1 Chapter 12: Analyzing Association Between Quantitative Variables: Regression Analysis Section 12.1: How Can We Model How Two Variables Are Related?

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2 Learning Objectives 1. Regression Analysis 2. The Scatterplot 3. The Regression Line Equation 4. Outliers 5. Influential Points 6. Residuals are Prediction Errors 7. Regression Model: A Line Describes How the Mean of y Depends on x 8. The Population Regression Equation 9. Variability about the Line 10. A Statistical Model
3 Learning Objective 1 : Regression Analysis The first step of a regression analysis is to identify the response and explanatory variables We use y to denote the response variable We use x to denote the explanatory variable

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4 Learning Objective 2 : The Scatterplot The first step in answering the question of association is to look at the data A scatterplot is a graphical display of the relationship between the response variable (y-axis) and the explanatory variable (x-axis)
5 Learning Objective 2: Example: What Do We Learn from a Scatterplot in the Strength Study? An experiment was designed to measure the strength of female athletes The goal of the experiment was to find the maximum number of pounds that each individual athlete could bench press

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6 57 high school female athletes participated in the study The data consisted of the following variables: x: the number of 60-pound bench presses an athlete could do y: maximum bench press Learning Objective 2: Example: What Do We Learn from a Scatterplot in the Strength Study?
7 For the 57 girls in this study, these variable are summarized by: x: mean = 11.0, st.deviation = 7.1 y: mean = 79.9 lbs, st.dev. = 13.3 lbs Learning Objective 2: Example: What Do We Learn from a Scatterplot in the Strength Study?

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8 Learning Objective 2: Example: What Do We Learn from a Scatterplot in the Strength Study?
9 Learning Objective 3: The Regression Line Equation When the scatterplot shows a linear trend, a straight line can be fitted through the data points to describe that trend The regression line is: is the predicted value of the response variable y is the y-intercept and is the slope bx a y + = ˆ y ˆ a b

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10 Learning Objective 3: Example: What Do We Learn from a Scatterplot in the Strength Study?
11 The MINITAB output shows the following regression equation: BP = 63.5 + 1.49 (BP_60) The y-intercept is 63.5 and the slope is 1.49 The slope of 1.49 tells us that predicted maximum bench press increases by about 1.5 pounds for every additional 60-pound bench press an athlete can do Learning Objective 3: Example: What Do We Learn from a Scatterplot in the Strength Study?

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12 Learning Objective 4: Outliers Check for outliers by plotting the data The regression line can be pulled toward an outlier and away from the general trend of points
13 Learning Objective 5 : Influential Points An observation can be influential in affecting the regression line when two thing happen: Its x value is low or high compared to the rest of the data It does not fall in the straight-line pattern that the rest of the data have

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