Question

When, if ever, is it appropriate to use the standard normal distribution as a substitute for the t distribution with n – 1 degrees of freedom in estimating a population mean?

Answer 1

For a small sample size, the sample standard deviation is generally biased or under estimates the population standard deviation. This is the reason that we use t distribution instead of standard normal distribution. This is because The t distribution has lower peaks and fatter tails, which means that t distribution for n-1 degrees of freedom for sample size has a greater standard deviation and therefore we use t distribution in that case.

But not the standard normal distribution can be used as a substitute for t distribution with n - 1 degrees of freedom only for very high sample size n and hence very high degrees of freedom: n - 1. This is because for a high degrees of freedom, the t distribution comes closer to the standard normal distribution. Therefore a high sample size is an appropriate case for using the standard normal distribution in place of t distribution.