$8.$ FILE Consumption. The consumption function, first developed by John Maynard Keynes, captures one of the key relationships in economics. It expresses consumption as a function of disposable income, where disposable income is income after taxes. The accompanying data file shows quarterly U$.$S$.$ consumption and disposable income $($Consumption and Income, respectively, in $\backslash$$$)$ for the years $2000-2019.$

a$.$ Find the sample regression equation for the model: Consumption $\backslash (=\backslash$beta$\_\{0\}+\backslash$beta$\_\{1\}\backslash )$ Income $\backslash (+\backslash$varepsilon $\backslash ).$

b$.$ In this model, the slope coefficient is called the marginal propensity to consume. Interpret its meaning.

c$.$ Construct and interpret the $\backslash (95\backslash \%\backslash )$ confidence interval for mean consumption if disposable income is $\backslash$$$15,000.$