Why Assume Normal Distribution for Small Sample Mean (n=8)?

As per the Central Limit Theorem, when the sample size is large enough (typically greater than 30), the sample mean tends to follow a normal distribution. However, for smaller sample sizes, the distribution of the sample mean may not be normal.

In such cases, we assume an approximate normal distribution for the sample mean if the population from which the sample is drawn is normal or if the sample size is sufficiently large (greater than or equal to 30).

For small samples (n=8), we cannot be sure that the sample mean follows a normal distribution. However, we can make the assumption based on the fact that the sample size is not too small and that the distribution of the population is not too skewed.

Assuming an approximate normal distribution allows us to use the standard normal distribution to calculate the confidence interval for the sample mean.