Who voted for the Nazis? Researchers attempted to answer this question by analyzing aggregate election data from the 1932 German election during the Weimar Republic. We analyze a simplified version of the election outcome data, which records, for each precinct, the number of eligible voters as well as the number of votes for the Nazi party. In addition, the data set contains aggregate occupation statistics for each precinct. Table 7.7 (p. 395) presents the variable names and descriptions of the CSV data file `nazis.csv`. Each observation represents a German precinct.

The goal of the analysis is to investigate which types of voters (based on their occupation category) cast ballots for the Nazi party in 1932. One hypothesis says that the Nazis received much support from blue-collar workers. Since the data do not directly tell us how many blue-collar workers voted for the Nazis, we must infer this information using a statistical analysis with certain assumptions. Such an analysis, where researchers try to infer individual behaviors from aggregate data, is called *ecological inference*.

To think about ecological inference more carefully in this context, consider the following simplified table for each precinct $i$. (Note: Table is only reproduced in the knitted file if using pdf output)

\begin{center}

\begin{tabular}{l|c c|c}

& \multicolumn{2}{c|}{Occupation} & \\

& Blue-collar & Non-blue-collar & \\

\hline

Vote choice & & & \\

\hspace{5pt} Nazis & $W_{i1}$ & $W_{i2}$ & $Y_i$ \\

\hspace{5pt} Other parties & $1 - W_{i1}$ & $1 - W_{i2}$ & $1 - Y_i$ \\

\hspace{5pt} or abstention & & & \\

& $X_i$ & $1 - X_i$ & \\

\hline

\end{tabular}

\end{center}

The data at hand tells us only the proportion of blue-collar voters $X_i$ and the vote share for the Nazis $Y_i$ in each precinct, but we would like to know the Nazi vote share among the blue-collar voters $W_{i1}$ and among the non-blue-collar voters $W_{i2}$. Then, there is a deterministic relationship between $X$, $Y$, and $\{W_1,W_2\}$. Indeed, for each precinct $i$, we can express the overall Nazi vote share as the weighted average of the Nazi vote share of each occupation:

$Y_i = X_i W_{i1} + (1 - X_i)W_{i2}$.

1. We exploit the linear relationship between the Nazi vote share $Y_i$ and the proportion of blue-collar voters $X_i$ given in equation (7.35) by regressing the former on the latter. That is, fit the following linear regression model:

$\mathbb{E}(Y_i|X_i) = \alpha + \beta X_i$.

Compute the estimated slope coefficient, its standard error, and the 95% confidence interval. Give a substantive interpretation of each quantity.