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101 Concepts for the Level I Exam

Essential Concept 4: Confidence Interval of Regression Coefficient, Predicted Value of the Dependent Variable (Y)


The confidence interval for a regression coefficient is given by:

     \[\mathrm{Confidence\ interval=}\widehat{{\mathrm{b}}_{\mathrm{1}}}\mathrm{\pm }{\mathrm{t}}_{\mathrm{c}}{\mathrm{s}}_{\widehat{{\mathrm{b}}_{\mathrm{1}}}}\]

where:
\noindent t${}_{c}$   is the critical t value for a given level of significance and degrees of freedom

\noindent ${\mathrm{s}}_{\widehat{{\mathrm{b}}_{\mathrm{1}}}}$   is the standard error of the correlation coefficient
To calculate the predicted value of the dependent variable Y we use a three-step process:

  • Calculate estimates of the regression coefficients
  • Assume values for the independent variables
  • Use the regression equation:
    \item ${\widehat{\mathrm{Y}}}_{\mathrm{i}}\mathrm{=\ }\widehat{{\mathrm{b}}_0}\mathrm{+\ }\widehat{{\mathrm{b}}_{\mathrm{1}}}{\widehat{\mathrm{X}}}_{\mathrm{1i}}\mathrm{+}{\widehat{\mathrm{b}}}_2{\widehat{\mathrm{X}}}_{\mathrm{2i}}\mathrm{+\dots \dots +}\widehat{{\mathrm{b}}_{\mathrm{k}}}{\widehat{\mathrm{X}}}_{\mathrm{ki}}$ }