What condition must be met for sample mean x̅ to be normally distributed when using t distribution?

For the sample mean to be normally distributed when using the t distribution, the condition that must be satisfied is that the sample size must be greater than or equal to 30. This is derived from the Central Limit Theorem, which states that as the sample size increases, the distribution of the sample means will tend to be normal, regardless of the shape of the population distribution, provided that the sample size is sufficiently large.

When the sample size reaches 30 or more, the effects of any population non-normality diminish, and the sampling distribution of the mean approximates a normal distribution. This condition allows statisticians to use the t distribution effectively, which is particularly useful when dealing with smaller sample sizes or when the population standard deviation is unknown.

While it is advantageous for the population to be normally distributed for smaller sample sizes, the threshold of 30 is well-recognized in statistical practice as a rule of thumb for the normality of the sample mean. Hence, the requirement of having a sample size of 30 or greater is fundamental in ensuring that the sample mean can be adequately described using the t distribution.