# Are group effects in a mixed effects model assumed to have been picked from a normal distribution?

Say we're interested in how student exam grades are affected by the number of hours that those students study. We sample students from several different schools. We run the following mixed effects model: $$\text{exam.grades}_i = a + \beta_1 \times \text{hours.studied}_i + \text{school}_j + e_i$$

Am I right in saying that, in this model, each school is assumed to have been picked from a larger population of schools, and that the effect of school is normally distributed? Therefore, can we do all the 'usual' normal distribution-type procedures for the group effect of school? Can we say things like 68% of schools will be within 1 standard deviation of mean group effect of school? And can we calculate a 95% confidence interval for the overall mean group effect of school?

Am I also right in saying that linear regression with fixed effect of school cannot calculate these normal distribution statistics because they use a reference group and dummy variables?