06-linear-regression-handouts.pdf - STAT 101 Module One...

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STAT 101 - Module One Page 1 of 23 Introduction to Linear Regression Review The measures the strength and direction of the linear relationship between two quantitative variables. If there is enough evidence of an association we may want to find a line that is a good representation of the linear relationship between our two quantitative variables. Such a line is called . We use linear regression to: Recall that we use to signify the explanatory variable and to signify the response variable. Regression A explains the average change in the re- sponse variable in relation to changes in the explanatory variable. Equation For a response variable y , and explanatory variable x , the linear regression line between y and x is given by: ˆ y = b o + b 1 * x where, ˆ y : b o : b 1 : 1
STAT 101 - Module One Page 2 of 23 This is known as the slope-intercept equation of a line. Recall from algebra, the slope of a line represents . The intercept refers to . To calculate the slope and the intercept of a regression line we use the following formulas: b o = ¯ y - b 1 * ¯ x b 1 = r * s y s x where, ¯ x : ¯ y : s x : s y : r : Example To illustrate the processing of creating a linear regression let’s look at some fire damage data. The explanatory variable in this is the distance between a fire and the nearest fire station in miles. The response variable is the damage in thousands of dollars caused by a fire. Data was collected from 15 homes in a major metropolitan area. From the scatterplot below, describe the form, strength, direction, and outliers of the data. Do you think a linear regression would be appropriate in this case? 2
STAT 101 - Module One Page 3 of 23 Given the above summary statistics information, calculate the equation of the regression equa- tion used to describe the relationship between distance from a fire station and fire damage. y versus ˆ y Correct linear equation: Incorrect linear equation: It is important to include the ”hat” in your linear equation! y : ˆ y : 3
STAT 101 - Module One Page 4 of 23 Regression Equation Using JMP Interpretation To interpret the results of our linear regressions into meaningful conclusions we can interpret the components of our equation in context. Recall: ˆ y = b o + b 1 * x To interpret b 1 , the slope: Interpretation One: For a one unit increase in the explanatory variable x , the pre- dicted value of the response variable y will change by the amount of the slope. Interpretation Two: For a one unit increase in the explanatory variable x , the value of the response variable y will change by the amount of the slope, on average . To interpret b o , the intercept: Interpretation: When the value of the explanatory variable is equal to 0, the predicted value of the response variable is equal to the intercept.

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