Question

what determines the shape of t distribution? what makes the “best” t distrubution? why is it the “best”?

Answer 1

The shape of t distribution is depends on the degrees on freedom (DF).

the T-distrubution is similar to normal distribution with its bell shape, but T- distribution has heavier tails.

This tail heaviness is determined by the degrees of freedom.

The smaller values gives heavier tails.

Higher values , the t-distribution resembles Standard normal distribution. I.e. high degrees of freedom.

T-distribution is best when the sample size is less than 30, n<30. and also when we dont know the population standard deviation.

That is in most pratical situations,population standard deviation is unknown and here we use a sample estimate to scale.

Under these assumptions about the sample distribution: (the sample mean minus the population mean )divided by (its estimated standard error) has a t distribution with (n-1) degrees of freedom.

Now, when the population variance is unknown the t distribution is used to construct confidence intervals for the population mean. The standard error of the mean is the sample standard deviation divided by n^1/2

and as the sample size increases, we can use again normal distribution.