Question

When testing the null hypothesis H0: E(Y)= μY,0 the appropriate t statistic has approximately a standard normal distribution in large samples if the null hypothesis is true. In a small sample, the same t statistic has a Student's t distribution with n−1 degrees of freedom under what additional assumption? and explain why

1) The distribution of Y is normal

2) The distribution of Y is Student's t with 1 degree of freedom

3) The distribution of Y is Student's t with n - 1 degrees of freedom

4) The sampling distribution of the sample mean is normal

Answer 1

Assumption needed for small sample is that the 1) The distribution of Y is normal.

The reason why this assumption is needed is due to the fact that :

The t-statistic consists of a ratio of two quantities, both random variables.

For the t-statistic to have the t-distribution, You also need: that the s in the denominator has chi square distribution and numerator and denominator be independent, and sample mean is normally distributed.

For those three things to be actually true, you need that the original data are normally distributed.