## Checking normality

I have a dataset for which I'm checking the assumption of normality. A normal probability plot shows a reasonably linear pattern in the center of the data. However, the tails show departures from the fitted line of the normal prob plot. At best, it seems the assumption of normality is plausible, particularly if a couple of outliers are removed, but on balance I would say that from the normal probability plot the assumption is questionable. I have calculated an exact 95% confidence interval using the chi-squared distribution, and this almost exactly matches the approximate confidence interval calculated based on the central limit theorem. Is it right to say that since these confidence intervals are identical, the assumption of normality appears reasonable, even though the probability plot is questionable ?