Conditions for Inference About a Mean

Inference for a mean uses a t-distribution built from your sample, and that distribution only behaves correctly if three things are true. Random means the data came from random sampling or a randomized experiment, so your results aren't biased by how the sample was chosen. The 10 percent condition means your sample size is at most 10 percent of the population size, which keeps observations close enough to independent even though you sampled without replacement. Normal-or-large means the sampling distribution of the sample mean is close enough to normal, either because the population itself is roughly normal or because your sample is large enough for the central limit theorem to smooth out any skew.

That third condition is why sample size matters so much. A strongly skewed population produces a skewed sampling distribution for small samples, which throws off the t-procedures. But as n grows, the sampling distribution of the mean pulls toward normal regardless of the population's shape, which is exactly what the central limit theorem guarantees. So skewness stops being a dealbreaker once the sample is big enough, usually judged by n being around 30 or more, though smaller samples can still pass if the data themselves show no strong skew or outliers.

This topic sets up every t-test and t-interval that follows in the unit. Once you've confirmed these three conditions, you can trust that the t-distribution with n minus 1 degrees of freedom actually models your standardized sample mean, which is what makes the rest of the inference machinery valid.