7: Time for Some Examples
- Page ID
- 57737
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We have covered a lot of theory and mathematics over the past several chapters. Here, we will apply what we have learned to help settle the theory into our minds. In other words, we will perform the analysis process with the information and skills we now have.
This means we will use data to answer our research questions. Of course, we will need to examine the research question to determine the appropriate model, check the assumptions — both statistically and graphically — and properly interpret the results.
That is a lot of summarizing to do!
✦•················• 👻 •··················•✦
And so, we have completed a majority of the important mathematics underlying ordinary least squares estimation. Be aware that OLS is how we estimate the parameters. The models itself is referred to as the classical linear model. It makes the usual four assumptions. The observations follow the equation
\begin{equation}
y_i = \beta_0 + \beta_1 x_{1,i} + \beta_2 x_{2,i} + \cdots + \beta_{k} x_{k,i} + \varepsilon_i
\end{equation}
and the residuals follow this distribution
\begin{equation}
\varepsilon_i \stackrel{\text{iid}}{\sim} N\left( 0;\ \sigma^2 \right)
\end{equation}
From those assumptions, we were able to use OLS to calculate formulas for the estimators of \(\beta_0, \beta_1, \ldots, \beta_{k}\). The next chapter used the distribution to determine the distribution of those estimators. This led to confidence intervals for the parameters and test statistics for testing hypotheses about the parameters. It also led to distributions and intervals and test statistics for estimated and predicted values of \(y\).
All of that from four small assumptions.
This chapter will apply these results to different research questions to illustrate the statistical research process. So, turn the page and begin seeing applications of what we have done.