MATH 80S Residuals and LSR Models October 3, 2011

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1 Ya A Pathway Through College Statistics- Open Source 2011 MATH 80S Residuals and LSR Models October 3, 2011 Statistical Vocabulary: A variable that is used to predict the value of another variable is called the predictor variable, also known as the independent variable or explanatory variable. The other variable, whose values you are predicting, is called the response variable, also known as the dependent variable. Our task is to predict the value of the response variable knowing the value of the explanatory variable (also known as predictor variable) using a best-fit line. So, how do we identify the line that is the best fit? We will now investigate this question with the goal of developing a method for determining which line is the best-fit line. Using Residuals to Determine If a Line Is a Good Fit- the Least Squares Regression Line as Line of Best Fit (LSR) A residual (or error) is the difference between the actual value of the response variable and the value predicted by the regression line. As a formula, residual = observed predicted. Analyzing residuals can help you assess the effectiveness of a least-squares regression (LSR) model for predicting values of the response variable. Note: The terms residual and error are used interchangeably in this lesson. More About the Size and Sign of Residuals Consider the scatterplot and its LSR line shown below. x y Fitted Line Plot Ya = Xa 3 4 Xa 6 7 8

2 1 The equation of the regression line is. Compute the predicted value of y for each x-value and fill in the following table. For each observation, locate on the regression line a point with coordinates Based on your predicted -values and the observed y-values in the original dataset, compute the residual (error) for each observation. Fill in the following table. (Hint: first, fill in -values from Question 1.) Residual

3 Ya A Pathway Through College Statistics- Open Source On the scatterplot below, draw a vertical dashed segment between each data point and the LSR line. These segments represent the residuals for the data points. (Note: The first residual segment is already drawn.) 30 Fitted Line Plot Ya = Xa Xa How is the sign of each residual (positive or negative) represented in this diagram? What does the length of each vertical dashed segment tell you about the corresponding residual?

4 6 Suppose an LSR model is created that predicts a subway fare based on miles traveled. Suppose an observation that represents the actual subway fare a person pays based on the miles traveled has a positive residual. A On a scatterplot, does the point representing this observation appear above or below the LSR line? B Is the actual fare the person paid more or less than the fare predicted by the model? 7 Suppose you have a scatterplot that shows sale price and acreage for 60 homes in a particular county, and an LSR model is created that predicts a home s sale price based on the home's acreage. One particular home is represented by a data point that is below the regression line. A Is the sale price of this home greater than the price predicted by the model or less than that price? B What is the sign of this data point s residual? C Another home has a sale price exactly equal to the price produced by the model. Is the data point for that home above the regression line, below the line, or on the line?

5 YOU NEED TO KNOW The LSR line is the line that minimizes the sum of the squared residuals. The acronym for sum of the squared residuals is SSE because residuals are also called errors (and the acronym SSR has another meaning in certain statistical analyses). As a formula, sum of the squared residuals = SSE = Σ(y ŷ) 2. 8 Compute the SSE for by completing the final column of the following table. Square the residual values you computed earlier and add up the squared residual values. (First, fill in -values and residual values from Question 2.) Residual Squared Residual Total: = SSE