Each of the distributions used has population mean 0 and variance 1, as indicated on the scale at the bottom of the left panel. The

-value is for testing

versus

. Of the three non-normal distributions considered, the

methods work almost as well for the uniform and Laplace distributions as the normal for sample sizes as small as 10, but for such small samples the exponential distribution generates

intervals and

-tests that are not conservative. As

increases, there is an improvement.

Snapshot 1: setting

and using the

-test method with the normal distribution and

, we find after some simulations that the empirical values for the estimates

and

are close to their theoretical values of 0.95 and 0.05

Snapshot 2: as in Snapshot 1, but using the

-test instead of the

-test; after a large number of simulations, we see the empirical coverage probability is about 92%, which means the confidence intervals are too narrow; similarly the type I error at 8% is well above its nominal 5% value; this demonstrates that the

-test method is not conservative in small samples

Snapshot 3: as in Snapshot 1, but with the uniform distribution; no observable difference from the normal population case

Snapshot 4: as in Snapshot 1, but with the Laplace distribution; this distribution has thicker tails than the normal, but even here the empirical coverage is not significantly different from the nominal 95% level, illustrating the robustness of the

-distribution method

Snapshot 5: as in Snapshot 1, but with the exponential distribution; this distribution is right-skewed and very non-normal and in this case the

method yields an empirical coverage rate of about

% instead of the nominal 95% value; the

method is not even conservative in this case

Snapshot 6: as in Snapshot 5, but the sample size is increased to

; the

methods have improved and only slightly overstate the statistical significance of the test

Snapshot 7: as in Snapshot 6, but the sample size is increased to

; now the approximation is accurate