Chapter 10, Part B: Simple Linear Regression

Simple Linear Regression: Part B

Using the Estimated Regression Equation

• The estimated regression equation can be used for estimation and prediction.
• Key calculations involve:
  • Confidence Interval Estimate of $E(y_p)$
  • Prediction Interval Estimate of $y_p$
  • Where the confidence coefficient is $1 - \alpha$ and $t_{\alpha/2}$ is based on a t distribution with $n - 2$ degrees of freedom.

Point Estimation

• Example: If 3 TV ads are run prior to a sale, we expect the mean number of cars sold to be:
  $\hat{y} = 10 + 5(3) = 25 \text{ cars}$

Confidence Interval for $E(y_p)$

• Estimate of the Standard Deviation of Confidence Interval for $E(y_p)$
• Example: The 95% confidence interval estimate of the mean number of cars sold when 3 TV ads are run is:
  $25 \pm 3.1824(1.4491)$
  $25 \pm 4.61$
  $20.39 \text{ to } 29.61 \text{ cars}$

Prediction Interval for $y_p$

• Estimate of the Standard Deviation of an Individual Value of $y_p$
• Example: The 95% prediction interval estimate of the number of cars sold in one particular week when 3 TV ads are run is:
  $25 \pm 3.1824(2.6013)$
  $25 \pm 8.28$
  $16.72 \text{ to } 33.28 \text{ cars}$

Computer Solution

• Statistical software (e.g., Minitab) can be used to perform regression analysis.
• The independent variable was named "Ads" and the dependent variable was named "Cars" in the example.
• Performing the regression analysis computations without the help of a computer can be quite time-consuming.

Minitab Output

• Minitab prints the standard error of the estimate, s, as well as information about the goodness of fit.

• For each of the coefficients $b<em>0$ and $b</em>1$, the output shows its value, standard deviation, t value, and p-value.

• Minitab prints the estimated regression equation (e.g., Cars = 10.0 + 5.00 Ads).

• The standard ANOVA table is printed.

• Also provided are the 95% confidence interval estimate of the expected number of cars sold and the 95% prediction interval estimate of the number of cars sold for an individual weekend with 3 ads.

• Regression equation example from Minitab:

  The regression equation is
  Cars = 10 + 5.00 Ads
  
  Predictor      Coef   SE Coef      T      p
  Constant     10.000     2.366   4.23   0.024
  Ads          5.0000    1.0801   4.63   0.019
  
  S = 2.2      R-sq = 87.7%     R-sq(adj) = 83.6%
  
  Analysis of Variance
  
  SOURCE           DF      SS      MS      F      p
  Regression        1     100     100   21.43   0.019
  Residual Error    3      14   4.667
  Total             4     114
  
  Predicted Values for New Observations
  
  New Obs      Fit   SE Fit       95% C.I.           95% P.I.
        1   25.00     2.60  (20.39, 29.61)  (16.72, 33.28)
  

Residual Analysis

• Much of the residual analysis is based on an examination of graphical plots.
• Residual for Observation i.
• The residuals provide the best information about $\epsilon$.
• If the assumptions about the error term $\epsilon$ appear questionable, the hypothesis tests about the significance of the regression relationship and the interval estimation results may not be valid.

Residual Plot Against x

• If the assumption that the variance of $\epsilon$ is the same for all values of x is valid, and the assumed regression model is an adequate representation of the relationship between the variables, then:

  • The residual plot should give an overall impression of a horizontal band of points.

• Good Pattern Example

• Nonconstant Variance Example

• Model Form Not Adequate Example