The value for #alpha# can be thought of as the chance of rejecting #H_0# (the null hypothesis) when, in fact, #H_0# is actually true.
Let's say we're testing #H_0: mu=0# against #H_1: mu != 0#. Then assuming the mean is actually 0, our chosen #alpha# is the chance that our data "accidentally" convinces us that the mean is far enough away that we should say it isn't 0.
The value for #alpha# is also called the #"Type I error"#, and it's the probability that a random value from the standard normal distribution #Z# will be in our rejection region.
The rejection region bounds (those 1.65 numbers in your answer) are the #z#-coordinates that produce a total tail area equal to #alpha#. For a two-tailed test, we seek a #z# that satisfies #"P"(Z < –z) = "P"(Z>z) = alpha/2# (the area of the two tails sums to #alpha#). For a one-tailed test, we seek a #z# that satisfies #"P"(Z > z) = alpha# (right-tailed test) or #"P"(Z< z)= alpha# (left-tailed test).
For example, for a right-tailed test with #alpha=0.05#, we want only a 5% chance that a random value from #Z# will be in the upper tail, and the #z#-coordinate that gives us an upper-tail area of 0.05 is #z=1.65.#
For a two-tailed test at #alpha=0.10,# we want a 10% chance that a random value from #Z# is in either tail. Thus, each tail should have half of that 10%, so that there is a 5% chance of it falling in the left tail and a 5% chance it'll be in the right tail. And what's the #z#-coordinate that gives an upper-tail area of 5%? That's right, #z=1.65,# same as above.
These #z#-values are usually found by lookup in a #z#-table such as this one:
(This is just a small piece of a real table; a full table will be much larger.)
Quick example: if you're doing an upper-tail test and want a significance level of #alpha=0.33,# you find 0.33 in the table (or as close as you can) and see that it appears in row #0.4# and column #.04,# so your #z#-value is #z=0.44.#
