  In this work, we propose differentially private methods for hypothesis
testing, model averaging, and model selection for normal linear models. We
consider Bayesian methods based on mixtures of $g$-priors and non-Bayesian
methods based on likelihood-ratio statistics and information criteria. The
procedures are asymptotically consistent and straightforward to implement with
existing software. We focus on practical issues such as adjusting critical
values so that hypothesis tests have adequate type I error rates and
quantifying the uncertainty introduced by the privacy-ensuring mechanisms.
