Question

Pretend like someone in another field who uses statistics (e.g. a psychologist, etc.) asks you to explain the difference between the t distribution and the normal distribution. Explain to them in a way a non-expert would understand, making sure to highlight when you would use each distribution during a hypothesis test.

Answer 1

Let's say for a psychologist he want to study on the mental health issue for particular city .

The data for that city is normally distributed. We know the mean and standard deviation of that data

Now the psychologist decides to take n no. Of people out of that city who have mental issue ( n -sample) .

This psychologist does not know the standard deviation of the population . Then he has to calculate the standard deviation of the sample . For any hypothesis test for a psychologist on this sample will be t distribution test statistics.  

In general the tail of t distribution is bell shape like normal distribution but it has heavier tails , meeaning it can produce value that fall far from its mean .

In non statistical language a t distribution carries more spread values than normal distribution .

Now for hypothesis test if population standard deviation is know we will use normal distribution .

And if population standard deviation is unkown but sample standard deviation is know we will use t distribution