Least-Squares Regression. Unit 3 Exploring Data
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1 Least-Squares Regression Unit 3 Exploring Data
2 Regression Line A straight line that describes how a variable,, changes as an variable,, changes unlike, requires an and variable used to predict the value of y for a given x
3 Notation = actual value = the predicted value Residual = = The residual is the.
4 Using the graph below, write an estimated equation for the line of best fit.
5 Least-Squares Regression Line (LSRL) The line that makes the of the of the distances of the points from the line as small as possible it the squares.
6 The LSRL: ALWAYS passes through has equation - slope: -intercept: We use in the regression line to emphasis that the line gives a response for any x.
7 So, what does this mean? How do we interpret this equation?
8 BEWARE!! The data is helps us create a line of best fit to make. It is not, so we need when interpreting the slope and y-intercept. According to the model. For slope: increases/decreases on average
9 Your Mantras: For Slope: For y-intercept:
10 fat (kg) = (NEA cal ) NEA = non-exercise activity Interpret slope: Interpret y-int.:
11 shrimp = (guests) Interpret slope: Interpret y-int How many shrimp would be needed for 100 guests?
12 Facts about Linear Regression The distinction between explanatory and response variables is. Reversing the roles gives a regression line. The regression line goes through the point
13 Finding LSRL First way: If Given Statistics --Means, Standard Deviation, Correlation --Find --Find --Write the equation.
14 Example: In Professor Palmer s Statistics course the correlation between the students total scores prior to the final examination and their final examination scores is r =.6. The pre-exam totals for all students in the course have a mean of 280 and a standard deviation of 30. The final exam scores have a mean of 75 and a standard deviation of 8. Professor Palmer has lost Julie s final exam but knows that her total before the exam was 300. She predicts her final exam score from her pre-exam total.
15 What is the slope of the LSRL of final exam scores on pre-exam total scores? Find the equation of the LSRL and use it to predict Julie s final exam score.
16 2 nd WAY: given data Enter data into two lists STAT CALC LinReg Enjoy!
17 POWERBOATS vs MANATEES P M
18 3 RD WAY: given computer output
19 Def: Using the regression line for prediction the range of values of the variable used to obtain the line.
20 Residual Plot The of the least-squares residuals is always. A residual plot is a of the regression against the variable ( ), which help assess the fit of a regression line. Any in the residuals indicate that a line is an appropriate model for the set of data. can have a profound effect on the leastsquares regression line and will appear as in the residual plot as well.
21 On the Calculator Calculating LSRL: (1) Enter the x-values in L1 and the y-values in L2 (2) Press STAT > to get to the Calc menu. (3) Press 8:LinReg(a+bx) and then finish the command by typing L1, L2, Y1. (Y1 is found under VARS/Y-VARS/1:Function) This will store the equation in Y1 to graph. Calculating residuals: once you have graphed the LSRL (1) Highlight L3 and define it as L2-Y1(L1). (2) Go to the Y= and deselect Y1. (3) Define Plot2 with L1 as the x-variable and L3 as the y-variable.
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23 If the line is a the residual plot should be and to y = 0. There should be to the residual plot.
24 If the residuals have a pattern the regression line is. If the residuals have or spread then prediction may be accurate for certain x-values
25
26 Outliers vs Influential Points An is an observation that lies far away from the other observations outliers in the direction have residuals Chapter 5 26
27 Outliers vs Influential Points Influential Point: an observation is influential if it would change the result of the. This is outliers in direction.
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29 Things to remember: An observation does not have to be an to be an point! An observation does not have to be an point to be an!
30 Impact on regression analyses Not every outlier influences the. Always determine if the regression analysis is influenced by or a data points.
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