= Solution
Treatment was assigned by school, so observations within one school are <clustered data> and plausibly correlated. A <random intercept> models shared unobserved school characteristics and prevents the effective amount of treatment-level information from being exaggerated by treating all students as independent.
The fitted school standard deviation is $0.209$, so the estimated between-school variation is nonzero. Accounting for it raises the TV and SC standard errors from about $0.065$ in "lm1" to about $0.105$ in "lme1", reflecting that only 28 independently randomized schools identify those effects.
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