Turn the Dial: Who Is the Curve About?

CS130 · After the scavenger hunt on King, Tomz & Wittenberg (2000), Figure 1. Each bar is the model's expected probability of voting for one hypothetical person at that age, with its confidence interval. You decide who that person is.

Term	value x	coefficient β̂	x · β̂
Sum: log-odds z
p = 1 / (1 + e^−z)

Work through these in order

Before touching anything, guess: at age 18, is a high-school graduate's chance of voting above or below one half? Now set the age dial to 18. Were you right? Notice the model is already answering for a specific person: 12 years of education, income 3.96, and "white = 0.86".
Slide the age dial slowly from 18 up to 95 and watch the highlighted bar. Where is it widest? There are 1,905 respondents aged 18–25 in the data and 124 aged 85 or older. What does the bar's width measure?
Pin the curve, then move education from 12 to 16. Read the pinned-vs-new gap at age 20 and again at age 60. Is the gap the same size in probability? Is it the same size in log-odds? Why do the two disagree?
Set "white" to 0, then to 1, then back to the sample mean. How much does the curve move? What kind of respondent does white = 0.86 describe, and would a bank, a campaign or a court accept that description?
Change the confidence level from 99% to 90%. What changes and what stays put? Then switch to "Simulate (KTW)" and reseed a few times: how close is the simulated interval to the one from standard errors?
Tick "Break it". Where does the curve go, and what is the one number the scavenger hunt asked you about (at age 38) that explains it?

The curve is not the share of people who voted at each age. It is E[turnout | age, and the other values you fixed]: the model's expected probability for one made-up respondent, recomputed at each age. Change who that respondent is and the whole curve moves. The bars come from a different place: the sampling distribution of the estimated coefficients β̂. They are uncertainty about the model, not variation among people, which is why they widen exactly where the data run thin and barely change when you move education or income.

z = x·β̂ p = 1/(1+e^−z) se(z) = √(xᵀ V̂ x) interval = logistic(z ± q·se(z)), q = 2.576 for 99%

What this page assumes and leaves out

The coefficients and their covariance matrix are the logistic regression turnout ~ age + agesqrd + educate + income + white fit to the 15,837 respondents in turnout.dta. They are fixed; nothing is re-estimated here.
agesqrd is age²/100 unless "Break it" is ticked.
"Standard errors" builds the interval on the log-odds scale (delta method) and maps the ends through the logistic function, as the scavenger hunt's hint does. "Simulate" draws 2,000 coefficient vectors from a multivariate normal with mean β̂ and covariance V̂ and takes percentiles, as King, Tomz & Wittenberg do. KTW's "expected value" averages over fundamental uncertainty too; for a binary outcome that average equals the probability, so nothing is lost by skipping it.
Setting white to its mean of 0.86 follows KTW. The page lets you compare it with 0 and 1 so you can judge whether a fractional person is a sensible thing to describe.
Left out: estimation from data, model fit, and whether age² is the right functional form. Those are separate lessons.