Question

For Central Limite Theorem, if n>30, we say the sampling distribution is normal. However, most of the time, with population standard deviation unknown, we still have to use t value to compute a confidence interval. But I wonder for normal distribution(z distribution), even though we do not know population sd, why cannot we use z value directly to compute confidence interval, as it has stated in central limit theorem that the distribution is normal.

Answer 1

According to central limit theorem,

If distribution of x is normally distributed then the sampling distribution of sample mean is approximately normally distributed. If distribution of x is not normally distributed but n > 30 then we can say that the sampling distribution of sample mean is approximately normally distributed.

If population standard deviation is known then we can use z interval

If population standard deviation is not known then we can use t interval. But if sample size is large (n > 30) and population standard deviation is not known then we can use z interval.