Question

conception:

for the null hypothesis test problem why we can use standard normal distribution table?

by the central limited theorem and law of large number we have that when n is getting large, then the sample distribution will be normal distribution but it doesn't mean it is standard normal distribution. But why we can still use the table to find the value

Answer 1

Let's consider X has a bell shaped distribution with mean and standard deviation

According to the central limit, as sample size(n) becomes sufficiently large say n>30, the sampling distribution of the sample mean is approximately normal.

i.e

By using the invariance property of the asymptotic normality,

That is:

Since this is the test statistic in the null hypothesis test problem, therefore we used standard normal distribution table to find the z critical value.