Question 1155265
Find the​ Z-scores that separate the middle 47​% of the distribution from the area
in the tails of the standard normal distribution.
<pre>
We shade the middle 47% of the area between the curve and the horizontal axis
(in green below.  We want to know the values on the horizontal axis where the
question marks ("?") are.

{{{drawing(400,200,-5,5,-.5,1.5, graph(400,200,-5,5,-.5,1.5, exp(-x^2/2)),locate(4.8,-.01,z),locate(4.8,.2,z),

green(line(-0.628,0,-0.628,0.82103298),line(-0.6,0,-0.6,0.83527021),line(-0.572,0,-0.572,0.84908838),line(-0.572,0,-0.572,0.84908838),
line(-0.544,0,-0.544,0.86245871),line(-0.516,0,-0.516,0.87535304),line(-0.488,0,-0.488,0.88774388),line(-0.488,0,-0.488,0.88774388),
line(-0.46,0,-0.46,0.89960455),line(-0.432,0,-0.432,0.91090925),line(-0.404,0,-0.404,0.92163317),line(-0.404,0,-0.404,0.92163317),
line(-0.376,0,-0.376,0.93175255),line(-0.348,0,-0.348,0.94124482),line(-0.32,0,-0.32,0.95008863),line(-0.32,0,-0.32,0.95008863),
line(-0.292,0,-0.292,0.95826397),line(-0.264,0,-0.264,0.9657522),line(-0.236,0,-0.236,0.97253618),line(-0.236,0,-0.236,0.97253618),
line(-0.208,0,-0.208,0.97860029),line(-0.18,0,-0.18,0.98393051),line(-0.152,0,-0.152,0.98851447),line(-0.152,0,-0.152,0.98851447),
line(-0.124,0,-0.124,0.99234148),line(-0.096,0,-0.096,0.9954026),line(-0.068,0,-0.068,0.99769067),line(-0.068,0,-0.068,0.99769067),
line(-0.04,0,-0.04,0.99920032),line(-0.012,0,-0.012,0.999928),line(0.016,0,0.016,0.99987201),line(0.016,0,0.016,0.99987201),
line(0.044,0,0.044,0.99903247),line(0.072,0,0.072,0.99741136),line(0.1,0,0.1,0.99501248),line(0.1,0,0.1,0.99501248),
line(0.128,0,0.128,0.99184146),line(0.156,0,0.156,0.98790573),line(0.184,0,0.184,0.98321447),line(0.184,0,0.184,0.98321447),
line(0.212,0,0.212,0.97777861),line(0.24,0,0.24,0.97161077),line(0.268,0,0.268,0.96472519),line(0.268,0,0.268,0.96472519),
line(0.296,0,0.296,0.95713771),line(0.324,0,0.324,0.94886571),line(0.352,0,0.352,0.939928),line(0.352,0,0.352,0.939928),
line(0.38,0,0.38,0.93034481),line(0.408,0,0.408,0.92013765),line(0.436,0,0.436,0.90932929),line(0.436,0,0.436,0.90932929),
line(0.464,0,0.464,0.89794362),line(0.492,0,0.492,0.88600561),line(0.52,0,0.52,0.87354119),line(0.52,0,0.52,0.87354119),
line(0.548,0,0.548,0.86057716),line(0.576,0,0.576,0.84714111),line(0.604,0,0.604,0.8332613),line(0.604,0,0.604,0.8332613)),

locate(-.628-.1,0,"|"), locate(.628-.1,0,"|"),

locate(-.628-.1,-.15,"?"), locate(.628-.15,-.15,"?"),

graph(400,200,-5,5,-.5,1.5, exp(-x^2/2))

)}}}

There are two kinds of normal tables in use today. If your table is like the one
on this site:

https://www.mathsisfun.com/data/standard-normal-distribution-table.html

then take one-half of 0.47, which is 0.235.  Look through the body of the table
to find the nearest value to 0.235. The nearest is 0.2357 which has value in
the z-column 0.6 and the column heading 0.03, so add those together and get
0.63.  Then by symmetry, the z-score for the right question mark is z=0.63, and
the value for the left question mark is z=-0.63.

If your table is like the one on this site:

https://byjus.com/maths/z-score-table/

then subtract 0.47 from 1, which is 1-0.47 = 0.53. Take one-half of 0.53, which
is 0.265. Look through the body of the table to find the nearest value to 0.265.
The nearest is 0.2643 which has value in the z-column -0.6 and the column
heading 0.03, so add those together (in absolute value) and get -0.63. Then by
symmetry, the z-score for the left question mark is z=-0.63, and the value for
the right question mark is z=0.63.

{{{drawing(400,200,-5,5,-.5,1.5, graph(400,200,-5,5,-.5,1.5, exp(-x^2/2)),locate(4.8,-.01,z),locate(4.8,.2,z),

green(line(-0.628,0,-0.628,0.82103298),line(-0.6,0,-0.6,0.83527021),line(-0.572,0,-0.572,0.84908838),line(-0.572,0,-0.572,0.84908838),
line(-0.544,0,-0.544,0.86245871),line(-0.516,0,-0.516,0.87535304),line(-0.488,0,-0.488,0.88774388),line(-0.488,0,-0.488,0.88774388),
line(-0.46,0,-0.46,0.89960455),line(-0.432,0,-0.432,0.91090925),line(-0.404,0,-0.404,0.92163317),line(-0.404,0,-0.404,0.92163317),
line(-0.376,0,-0.376,0.93175255),line(-0.348,0,-0.348,0.94124482),line(-0.32,0,-0.32,0.95008863),line(-0.32,0,-0.32,0.95008863),
line(-0.292,0,-0.292,0.95826397),line(-0.264,0,-0.264,0.9657522),line(-0.236,0,-0.236,0.97253618),line(-0.236,0,-0.236,0.97253618),
line(-0.208,0,-0.208,0.97860029),line(-0.18,0,-0.18,0.98393051),line(-0.152,0,-0.152,0.98851447),line(-0.152,0,-0.152,0.98851447),
line(-0.124,0,-0.124,0.99234148),line(-0.096,0,-0.096,0.9954026),line(-0.068,0,-0.068,0.99769067),line(-0.068,0,-0.068,0.99769067),
line(-0.04,0,-0.04,0.99920032),line(-0.012,0,-0.012,0.999928),line(0.016,0,0.016,0.99987201),line(0.016,0,0.016,0.99987201),
line(0.044,0,0.044,0.99903247),line(0.072,0,0.072,0.99741136),line(0.1,0,0.1,0.99501248),line(0.1,0,0.1,0.99501248),
line(0.128,0,0.128,0.99184146),line(0.156,0,0.156,0.98790573),line(0.184,0,0.184,0.98321447),line(0.184,0,0.184,0.98321447),
line(0.212,0,0.212,0.97777861),line(0.24,0,0.24,0.97161077),line(0.268,0,0.268,0.96472519),line(0.268,0,0.268,0.96472519),
line(0.296,0,0.296,0.95713771),line(0.324,0,0.324,0.94886571),line(0.352,0,0.352,0.939928),line(0.352,0,0.352,0.939928),
line(0.38,0,0.38,0.93034481),line(0.408,0,0.408,0.92013765),line(0.436,0,0.436,0.90932929),line(0.436,0,0.436,0.90932929),
line(0.464,0,0.464,0.89794362),line(0.492,0,0.492,0.88600561),line(0.52,0,0.52,0.87354119),line(0.52,0,0.52,0.87354119),
line(0.548,0,0.548,0.86057716),line(0.576,0,0.576,0.84714111),line(0.604,0,0.604,0.8332613),line(0.604,0,0.604,0.8332613)),

locate(-.628-.1,0,"|"), locate(.628-.1,0,"|"),

locate(-.628-.1-.5,-.15,"-0.63"), locate(.628-.15-.2,-.15,"0.63"),

graph(400,200,-5,5,-.5,1.5, exp(-x^2/2))

)}}}

Edwin</pre>