Since 90% of the distribution area is between -z and +z, that means
that the 90% is in the middle and the remaining 10% is split equally
as 5% below the left green line in the graph below and 5% is above
the right green line.
What we are looking for is the values on the z-axis (the horizontal axis)
where the two red question marks are located.
To do this we can either use a TI graphing calculator or a normal table.
I'll show you both ways:
On the TI-84 calculator,
press 2ND
press VARS
press 3
next, depending on what model you have, if you see
invnorm
area:
m:
s:
Paste
Then make it read like this
invnorm
area:0.05
m:0
s:1
Paste
Then scroll down to Paste and press ENTER
and read
invNorm(.05,0,1)
Or if you have an older model TI, you just see invNorm(
so type the rest in make it read like the above, that
is, invNorm(.05,0,1)
Then press ENTER
and read:
-1.644853626
and round to -1.645
That's -z = -1.645, so +z = +1.645.
----------------------------
But if you're using tables, go to this website
https://www.easycalculation.com/statistics/normal-ztable.php
where there is a table that reads from the y-axis over to
the right green line in the graph above. Notice that only
half of the 90% or 45% is between the y-axis and the right
green line on the graph above. So we look through the body
of the table for the closest entry to 0.4500. We don't find
that. The closest we can find are the two numbers 0.4495 and
0.4505 which are located side by side in the table. 0.4500
is exactly half-way between 0.4495 and 0.4505. So we see
that the z-value on the far left on the row they are on is
1.6. We see that the heading of the vertical column that
0.4495 is on is 0.04, and the vertical column that 0.4505 is
on is 0.05. Since 0.4500 is exactly half-way between those,
we add 0.045 to 1.6 and get 1.645. So +z = 1.645, abd so
-z = -1.645.
Edwin
