SOLUTION: The mean mass of game fish in a certain locale is determined to be 2.5 kg and the standard deviation is 0.75 kg. a. What percent have a mass between 1 kg and 4 kg? b. Suppose

 

 
Now we convert everything from from x-scores to z-scores. The x-scores are
the actual values of the fishes' mass in kg.  The z-scores will be the
coded values of those x-scores.
 
So we find the z-scores of the x-endpoints of the
desired area, 1 and 4, using
 

 
For left endpoint x=1,
 
, which
means that left endpoint x=1 on the x-axis will 
corresponds to the new left endpoint z=-2 on the new
z-axis.
 
For right endpoint x=4,
 
, which
means that right endpoint x=4 on the x-axis will 
corresponds to the new right endpoint x=2.
 
Now we take away the x-axis, put in the
new z-axis, and move the y-axis to the middle of the graph:
 

 
We can find the right half of the area by looking in
the table.  Find 2.0 in the z-column and .00 across the
top, and we see 0.4772.
 
By the symmetry of the normal curve, the left half of 
that area is the same, so we double 0.4772 and the
answer is .9544
 
--------------------
 
You can also do the whole thing immediately on a TI-83
or TI-84:
 
Clear the screen
 
PRESS     2nd VARS 2
 
you see this on the screen:
   
                normalcdf(
 
Then type after it so that you have this on the screen:
 
                normalcdf(1,4,2.5,.75)
 
Press ENTER and read  .954499876
 
Notice that is rounds to .9545, not .9544 as we found
using the table in the book, but this can be expected 
because the table is not as accurate as the calculator. 
 
----------------------------------------------------

On the graph with the x-axis, we want the area marked A:
 

 

It's the area marked A, the area to the left of the point 1.75 on the
x-axis that we're looking for.   
 
So now we 'code' that x-score cut-off point 1.75 as a z-scores, for
this is our new cut-off point, x=1.75,
 
, which
means that the cut-off point x=1.75 on the x-axis will 
corresponds to the new cut-off point z=-1 on the new
z-axis.
 

 
Now we look up z=1.00 in the table and read 0.3413. However that's the
area marked B, not the area marked A.  The table only gives the area
starting at the middle of the graph (left of the y-axis).



However we know that the whole left side is .5.  So we subtract the area
marked B, which is 0.3413, from .5000 and get 0.1587.

So the percent is found by moving the decimal two places right and putting
on a percent mark.

Answer:  15.87% of the fish must be thrown back. 

----

As before can also do the whole thing immediately on a TI-83
or TI-84:
 
Clear the screen
 
PRESS     2nd VARS 2
 
you see this on the screen:
   
                normalcdf(
 
Then type after it so that you have this on the screen:
 
                normalcdf(-99999999,1.75,2.5,0.75) 

[that's a negative sign, not a minus sign, located at the very
 bottom right of the calculator. It looks like this (-).

Press ENTER and read  0.1586552596

which this time rounds to the same answer 0.1587 as when we 
used the table.  


normalcdf(LEFTBOUND, RIGHTBOUND, MEAN, STANDARDDEVIATION) then press ENTER.

If the area goes all the way to the left, use -99999999 for the LEFTBOUND.
If the area goes all the way to the right, use 99999999 for the RIGHTBOUND.

Edwin