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

As per the central limit theorem, the sampling distribution of the mean becomes approximately normal as the sample size increases. However, for small sample sizes (n<30), the assumption of normality is necessary to construct a confidence interval for the mean. This is because the t-distribution, which is used to calculate confidence intervals for small sample sizes, assumes normality of the underlying population.

In addition, assuming normality allows us to use known properties of the normal distribution, such as the mean and standard deviation, to calculate the confidence interval. Without this assumption, it would be challenging to determine the appropriate distribution to use and how to calculate the confidence interval.

Therefore, to construct a 95% confidence interval for the mean of small samples, we assume an approximate normal distribution to use the t-distribution and its associated properties.