# How can I better interpret growth curve models?

I am doing growth curve model analysis in R, using lme4. I have doubts (well, many, but these are the most important).

(1) When I get the intercept result, it is not exact with the group mean in the initial measurement. What does that mean? For example, in this case, the intercept for the first group is 33.270, but the group's media in the first assessment was 31.000.

(2) When I include the summary function, it gives me p-values ​​that I don't know how to interpret. What exactly do they mean? For example, The p-value of the intercept, ola (different times for assessments), and Tipo-Colegioespecial (The first of my groups) are significant. What does it mean in a substantive interpretation?

Fixed effects:
Estimate Std. Error df t value Pr(>|t|)
(Intercept) 33.2704 3.6412 28.7453 9.137 0.000000000532 ***
Ola 2.9088 0.5192 44.9498 5.602 0.000001214894 ***
Tipo_Colegioespecial -11.9893 4.5025 28.8344 -2.663 0.0125 *
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Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1