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.
useruploads.socratic.org)
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:
i.stack.imgur.com
(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.
