Inferences for Simple Linear Regression

Here's what you'll find in this section:

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• The Simple Linear Regression

  The Simple Linear Regression

  Regression analysis is a statistical tool that utilizes the relation between two or more quantitative variables so that one variable (dependent variable) can be predicted from the others (independent variables). For example, if one knows the relationship between advertising expenditures and sales, one can predict sales by regression analysis once the level of advertising expenditures has been set. In this chapter, we specifically consider the case when a single independent variable is used for predicting the dependent variable and the dependent variable and the independent variable are linearly related.
  

   

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  • The Model

  • The Model

    The model can be stated as: [ Y_i;=; _0 + _1 X_i + _i, i=1,2,..., n ; ] where

    • Y is the response variable (also called dependent variable),
    • X is the predictor (also called independent variable),
    • and are the unknown parameters,
    • is the error (also called random deviation).
    Note that, Y = + X is the population regression line; while the least squares line Y = + X introduced in Chapter 2 is an estimate of this population line.

     

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• Point Estimates of Parameters of the Model

• Point Estimates of Parameters of the Model

  Recall the following notations:

  

  Then

  • = point estimate of = .
  • = point estimate of = .
  • The estimated regression line is: Y = + X.
  • Point estimate of the mean y value when x = is + .
  • Point prediction of a new observation when x = is + .
  Also recall that the i-th residual is defined to be [ e_i=y_i-y_i, where y_i= b_0 +b_1 x_i, ] and the residual sum of squares, SSRes is

   

  

  We then have

  • = point estimate of = SSRes/(n-2).

  Note that

  1. One should only do prediction within the range where you observed X.
  2. Sum of the residuals equals zero.
  3. The estimated slope and the sample correlation coefficient (Week 2) have the same sign. In fact = r .
  EXAMPLE:\ Eddie's Restaurant decides to invest some money on advertising. To have a better idea of what the sales amount would be if they invest 5 to 7 thousand dollars on advertising, they collect the following data on the relationship between advertising X and sales Y from a sample of five restaurants (all numbers are in thousands of dollars)

  

   

  • , . Therefore, .
  • .
  • The estimated regression line is: Y = 35.042 + 1.394 X.
  • Suppose Eddie is going to put down 6 (thousand) dollars as the advertising expense, what do you predict the restaurant will earn?

    Ans: (thousand).

  
  

   

• Inferences for Regression Parameters

• Inferences for Regression Parameters

  In many situations, a general form for a % confidence interval for a parameter is [ ^ (critical value)SE(^ ), ] where is a sample statistic used to estimate , SE [the standard error of ] gives the variation of and the critical value is a value such as or . Thus, all we have to do is (1) find the formula for SE and (2) ``plug-in'' numbers into this general equation to get a confidence interval. The following confidence (prediction) intervals all follow this rule.

  
  

   

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