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    12.5 The Regression Equation – Introduction to Statistics

    12.5 THE REGRESSION EQUATION

    12.5 THE REGRESSION EQUATION LEARNING OBJECTIVES

    Find the equation of the line-of-best fit.

    Use the line-of-best-fit to make predictions.

    We often want to use values of the independent variable to make predictions about the value of the dependent variable.  For example, we might want to use the amount a business spends on advertising each quarter to make a prediction about the revenue the business will generate that quarter.  When a linear relationship exists between an independent and dependent variable, we can build a linear model of that relationship, and then we can use that model to make predictions about the dependent variable.

    Simple linear regression is a modeling technique in which the linear relationship between one independent variable

    x x

    and one dependent variable

    y y

    is approximated by a straight line, called the line-of-best-fit or least squares line.  It is important to note that the line-of-best-fit only models the linear relationship between the independent and dependent variables.

    The equation for the regression line is:

    ^ y = b 0 + b 1 x ^ y = predicted value of y x =

    value of the independent variable

    b 0 = y

    -interecept of the line

    b 1 = slope of the line

    y^=b0+b1xy^=predicted value of yx=value of the independent variableb0=y-interecept of the lineb1=slope of the line

    The value of ^ y y^

    is the estimated value of

    .  It is the value of

    y y y y

    obtained using the regression line. It is not generally equal to the value of

    y y

    from the sample data.  The values for the slope

    b 1 b1 and the y y -intercept b 0 b0

    in the line-of-best-fit are calculated using the sample data and the least squares method.  Although there are formulas to calculate the values of the slope and

    y y

    -intercept in the regression line, we will calculate the slope and

    y y

    -intercept using the built-in functions in Excel.

    The slope of the linear regression equation:

    The slope of the line-of-best-fit

    b 1 b1

    and the correlation coefficient

    r r

    have the same sign.  That is,

    b 1 b1 and r r

    are either both positive or both negative.

    The slope b 1 b1

    of the regression equation tells us how the dependent variable

    y y

    changes for a one unit increase in the independent variable

    x x .

    When interpreting the slope, be specific to the context of the question, using the actual names of the variable and correct units.

    The

    y y

    -intercept of the linear regression equation:

    The y y -intercept b 0 b0

    of the line-of-best-fit is the predicted value of the dependent variable

    y y when x = 0 x=0 .

    When interpreting the

    y y

    -intercept, be specific to the context of the question, using the actual names of the variable and correct units.

    CALCULATING THE SLOPE AND

    y y

    -INTERCEPT OF THE LINEAR REGRESSION EQUATION IN EXCEL

    To calculate the slope of the linear regression equation, use the slope(array for y’s,array for x’s) function.

    For array for y’s, enter the cell array containing the dependent variable

    y y data.

    For array for x’s, enter the cell array containing the independent variable

    x x data.

    Visit the Microsoft page for more information about the slope function.

    To calculate the y y

    -intercept of the linear regression equation, use the intercept(array for y’s,array for x’s) function.

    For array for y’s, enter the cell array containing the dependent variable

    y y data.

    For array for x’s, enter the cell array containing the independent variable

    x x data.

    Visit the Microsoft page for more information about the intercept function.

    NOTE

    The order in which the data is entered into these functions is important.  In both the slope and intercept functions, the data for the dependent variable is entered in the first array and the data for the independent variable is entered in the second array.  The output from the slope and intercept function will be different when the order of the inputs are switched.

    EXAMPLE

    A statistics professor wants to study the relationship between a student’s score on the third exam in the course and their final exam score.  The professor took a random sample of 11 students and recorded their third exam score (out of 80) and their final exam score (out of 200).  The results are recorded in the table below.  The professor wants to develop a linear regression model to predict a student’s final exam score from the third exam score.

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    MCQ Regression analysis

    ec1011: data analysis ii multiple choice questions topic regression analysis the process of constructing mathematical model or function that can be used to

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