When we choose whether a value of the mean (x bar) is "unusually" far from the mean (mu) is a matter of choosing whether:

(x bar - mu)/(sigma/root(n)) is "unusually" far from zero.

The concern is, what is "unusual"?

It it were defined as "occuring on less than 5 % of occassions"

- what range of values of (x bar - mu)/(sigma/root(n)) would be regarded as unusually far from zero?

- what would this range be if we replace 5% by 1%?