Ok, from your code I can see that you’re using RStan rather than cmdstanr. (I’d suggest moving to cmdstanr as it’s up to date with the current release of Stan and easier to install.)
First, you get posterior credible intervals, not confidence intervals using Bayesian inference. So I’d suggest checking that. The column you label “mean” I would label as “estimated value” (that is the posterior mean because we’r using a Bayesian estimator).
I’m not sure how the course titles relate to the parameters in your model. But let’s say alpha is the parameter (or transformed parameter or generated quantity) corresponding to microecon. To get the 95% posterior intervals in RStan, I think you can do this:
alpha_draws <- extract(fit, "alpha")
alpha_posterior_interval_95 <- quantile(c(0.025, 0.975), alpha_draws)
Here’s the doc for the extract function: