**Normal Curve Calculations**

**Using the TI-83/TI-84 Graphing
Calculators**

**Dr. John C. Nardo **

**Division of
Mathematics & Computer Science **

**Oglethorpe****
****University**** **

**Atlanta****, ****GA**** **
**30319****
**

jnardo@oglethorpe.edu

**
www.oglethorpe.edu/faculty/~j_nardo/**

In
order to work with a normal curve, you must first identify its two
parameters: its mean_{} and its standard deviation _{}. These two pieces of
information uniquely identify the particular normal distribution, denoted _{}, with which you will work.
In traditional, pencil-and-paper statistics, each situation/problem is
transformed via z-scores to a situation/problem about the standard normal
distribution (_{} and _{}).

Such
work is unnecessary when using technology for normal curve calculations.

The
fall deer population in

You
may sketch a graph of this normal distribution via technology; however, in this
case, it is easier and quicker to sketch by hand.

__WARNING__: In
order to turn this screen dump into an appropriate sketch of this normal
distribution, we would need to: (1) mark
the mean, (2) note the standard deviation to the side, (3) put arrows on the
axis and the normal curve, (4) write the variable underneath the axis, and (5)
put several numbers/ values on the axis to orient the reader.

__The “normalcdf” Command__

The
normalcdf command calculates the area under the normal curve between two given
points. It requires four inputs/numbers
for it to function – **in this precise order** – minimum number/starting
point, maximum number/ending point, mean, and standard deviation. The normalcdf command is found in the
distribution menu. Press **[2 ^{nd}]
[VARS]** to access this menu. You
should see the following screen:

.

The
first command, normal**P**df, sketches **P**ictures of normal curves
(after much tedious window searching).
As mentioned above, sketch the curves by hand! The second command, normal**C**df,
performs **C**alculations. Press the
down arrow once to highlight/select the normalcdf command. Then press the **[ENTER]** key to paste it
back to the home screen.

The
calculator is now ready for you to input the four parameters/inputs – separated
by commas.

Fall Seasons in which the Deer Population will be between 4000 and 4500

The
four parameters/inputs for the normalcdf command in this case are: minimum=4000, maximum=4500. mean=4400, and
standard deviation=620. Type these four
numbers in order, separated by commas.
After pressing the **[ENTER]** key to execute the command, your
screen should look like this:

.

This number represents the area under this normal curve between 4000 and 4500. More importantly for this situation, it represents the proportion of Fall Seasons during which the deer population will be between 4000 and 4500. We may conclude that 30.47% of Fall Seasons will have a deer population between 4000 and 4500!

Fall Seasons in which the Deer Population will be above 5000

If
we wish to know the percentage of Fall Seasons for which the deer population is
above 5000, we must be careful. We still
must give **four** inputs to the calculator!
We envision starting at 5000 and ending at _{}, positive infinity.
We will represent positive infinity with the largest positive number
that the calculator can process: +1
followed by 99 zeros! We write this in
scientific notation by +1E99. The “E”
from scientific notation is accessed by shifting the comma key: pressing **[2 ^{nd}]**

Thus,
the command for this normal curve calculation is:

.

The proportion of Fall Seasons with a deer population above 5000 is 0.1665866273; the percentage is 16.66%.

Fall Seasons in which the Deer Population will be below 5000

Clearly,
if 16.66% of the time the population is above 5000, then we would deduce that
100% – 16.66% = 83.34% of the time, the population would be below 5000. Let’s check this deduction using similar reasoning
as above.

In
order to calculate the percentage of Fall Seasons with deer populations below
5000, we envision starting at negative infinity and stopping at 5000. We represent negative infinity by – 1
followed by 99 zeros, i.e. – 1 E 99.

Indeed, the direct command gives the same result as our deduction!

__WARNINGS__: (1) Be sure to use the
negative button beneath the number 3 key and not the subtraction button!

(2)
Be sure that the negative
sign goes in front of the one and not the 99!

So
far, we have been solving __direct__ problems. Given one or two numbers/values of the
variable, calculate the percentage of data associated with the number(s).

We
could turn this problem around to solve an __indirect__ problem. Given a percentage, find the number/value of
the variable associated with that percentage!

The
command for solving inverse problems, handily named “InvNorm,” is found in the
same menu as our previous command, the distributions menu. Press **[2 ^{nd}] [VARS]** to
access this menu. The third command from
the top is the inverse command. Press
down arrow twice to highlight/select this command.

Then
press **[ENTER]** to paste it to the home screen:

.

It
requires **three** parameters/inputs to make it work: percentage of area to the **LEFT**
(written as a decimal), mean, and standard deviation.

A
“lean” Fall Season is one in which the deer population is unnaturally low. We must define what we mean
mathematically. If the population is at
its lowest 5%, then we will define a Fall Season to be “lean.” Find the number associated with being a
“lean” Fall Season.

If
we are in the lowest 5%, then we would be on the left side of the normal
distribution; indeed, 5% of the curve will be to our LEFT. The InvNorm command will need these
parameters: 0.05, 4400, 620.

Thus, if we count 3380 or fewer deer in a particular Fall Season, we will be in a “lean” year!

__NOTE__: You could check that this works out! If you calculate the area under the curve
from negative infinity to 3380, you should get five percent. You can always check inverse problems via a
direct calculation.

An
“abundant” Fall Season is one in which the deer population is unnaturally
high. We must define what we mean
mathematically. If the population is at
its highest 5%, then we will define a Fall Season to be “abundant.” Find the number associated with being an
“abundant” Fall Season.

If we are in the highest 5%, then we would be on the right side of the normal distribution. Since 5% of the curve will be to our RIGHT, we know that 95% will be on our LEFT. The InvNorm command will need these parameters: 0.95, 4400, 620.

Thus, if we count 5420 deer or more in a particular Fall Season, we may classify it as an “abundant” year!

As before we can check this indirect calculation via a direct one. The percentage of Fall Seasons from 5420 to positive infinity should be 5%. Indeed it is!