I am a statistician, and you're right, for this kind of thing we would normally use a binary response model such as a logit or probit model that constrains the response variable to be between 0 & 1.
However in this case it doesn't matter since there's only one independent variable (state), and it's binary so there's only 2 different predictions the model could make (which will be the correct probabilities of 0.45 & 0.55, even with a linear model).
The normal R^2 formula can't be applied to a logit/probit model; instead you use an alternative such as McFadden's or Cox & Snell pseudo R-squared. I'd be interested to see what value they take for this example.
Linear models are sometimes used even in models with many independent variables since it can be shown that the coefficients in a linear model are unbiased estimators for the average partial effects of any non-linear binary response model.
The normal R^2 formula can't be applied to a logit/probit model; instead you use an alternative such as McFadden's or Cox & Snell pseudo R-squared. I'd be interested to see what value they take for this example.
Linear models are sometimes used even in models with many independent variables since it can be shown that the coefficients in a linear model are unbiased estimators for the average partial effects of any non-linear binary response model.