Standard meta-analytic theory assumes that study outcomes are normally distributed with known variances. However, methods derived from this theory are often applied to effect sizes having skewed distributions with estimated variances. Both shortcomings can be largely overcome by first applying a variance stabilizing transformation. Here we concentrate on study outcomes with Student t-distributions and show that we can better estimate parameters of fixed or random effects models with confidence intervals using stable weights or with profile approximate likelihood intervals following stabilization. We achieve even better coverage with a finite sample bias correction. Further, a simple t-interval provides very good coverage of an overall effect size without estimation of the inter-study variance. We illustrate the methodology on two meta-analytic studies from the medical literature, the effect of salt reduction on systolic blood pressure and the effect of opioids for the relief of breathlessness. Substantial simulation studies compare traditional methods with those newly proposed. We can apply the theoretical results to other study outcomes for which an effective variance stabilizer is available.