Question 515085
You have a random variable x that is normally distributed, with mean {{{mu = 28}}} and standard deviation {{{sigma = 6.5}}} (these lowercase Greek letters are pronounced "mu" and "sigma" respectively.  You need to figure out the probability that x is less than or equal to 20.  You have the standard normal distribution table.  How do we put this all together?

The idea here is we can change this problem on a non-standard normal distribution to one on the standard normal distribution by computing the z-score of 20.  The z-score of 20 is simply the number of standard deviations 20 is above or below the mean.  The formula you have tells you the z-score:

z-score = {{{(x - mu)/sigma}}}

Let's apply this formula:

z-score of 20 = {{{(20 - 28)/6.5 = -1.23}}}

This makes sense: 20 is more than one, but less than two standard deviations below the mean of 28, and a z-score of -1.23 is consistent with this observation.

It turns out that, if x is normally distributed, then the probability that x <= 20 is exactly the same as the probability that z, a random variable on the standard normal distribution, is less than or equal to the z-score of 20, or -1.23:

{{{P(x <= 20) = P(z <= -1.23)}}}

You can now use your table to find the probability that z <= -1.23.  If your table gives probabilities for positive and negative z-scores, just look up -1.23 and read off the number.  You should get:

{{{P(z <= -1.23) =0.1093}}}

or something very close to it (depending on how the values were rounded).  If your table only gives probabilities for positive z-scores, then you can still get the right answer by observing that the standard normal distribution is symmetric: its bell curve shape is exactly the same on both sides of the mean.  That means that the probability that z <= -1.23 is exactly the same as the probability that z >= 1.23, its mirror image.  Since the probabilities that z <= 1.23 and z >= 1.23 should add to 1, we get that:

{{{P(z <= 1.23) + P(z >= 1.23) = 1}}}
{{{P(z >= 1.23) = 1 - P(z <= 1.23)}}} (solving for P(z >= 1.23))
{{{P(z <= -1.23) = 1 - P(z <= 1.23)}}} (remembering that {{{P(z <= -1.23) = P(z >= 1.23)}}})

So we look up the probability that z <= 1.23 in our table:

{{{P(z <= 1.23) = 0.8907}}}

and then subtract the value we got from the table from 1 to get our answer:

{{{P(z <= -1.23) = 1 - 0.8907 = 0.1093}}}

which is exactly what we got before.  Since the probability that x <= 20 is equal to the probability that z <= -1.23, this is our answer.

{{{P(x <= 20) = 0.1093}}}