Why Assume Normal Distribution for Small Samples (n=8) in Confidence Intervals?

As per the Central Limit Theorem, if the sample size is large enough (usually n > 30), the sampling distribution of the mean will be approximately normal, regardless of the underlying distribution of the population. However, for small sample sizes (n < 30), the distribution of the sample mean may not be normal, especially if the population is not normal.

In such cases, we assume an approximate normal distribution for the sample mean, provided that the sample size is not too small (n > 5). This assumption is based on the fact that the t-distribution approaches the normal distribution as the sample size increases. Therefore, for small samples, we use the t-distribution instead of the normal distribution to construct confidence intervals for the population mean.

Thus, assuming an approximate normal distribution for small sample sizes is a reasonable approximation, provided that the sample size is not too small and the underlying population distribution is not too skewed or heavy-tailed.