The top button
on the Normal Tool says "Normal Tool." This allows
us to find probabilities for any normal distribution. The
bottom button says "Standard Normal Tool." This
allows us to find probabilities for a special case of the
normal distribution (sometimes known as the z-distribution
or unit normal).
Instructions will
be given for the general case first (Normal Tool). Then we
will go on to instructions for the simpler case (Standard
Normal Tool).
**1.
Normal Tool: Finding probabilities for any normal distribution**
We will use an
example to illustrate the step by step use of the Normal Tool.
As a big picture overview, let's suppose that scientists doing
research take some interesting phenomena in nature, reduce
it to numbers by measurement operations, and then model those
numbers as a random variable. A frequently used random variable
is the normal probability distribution.
**Height Example**.
In this example we will be interested in the heights of northern
European males. We take such a person and reduce him to a
single number via the usual operations for measuring someone's
height. Then we model the height of northern European males
as a normal population with mu = 150 cm and sigma = 30 cm.
In other words, our model is **N(150, 30)**.
**Click on the
top button--Normal Tool**.
**What is the
Probability ***Between* 140 and 170? We have modeled
the heights of northern European males as N(150, 30). If that
model is true, and if we sample one man from that population,
what are the chances he has a height between 140 cm and 170
cm?
**Total Area
under the Normal curve**. Remember that we can interpret
the area below a normal curve as probability. The total area
below the normal curve (from negative infinity up to positive
infinity) is assumed to be 1. That is, the probability that
a man's height will fall between negative and positive infinity
is 1.
**Area Between**.
Since the total area under the curve is 1, the area between
140 and 170 must be some fraction of 1. On the Normal Tool
the first thing you must do is make sure that the little icon
indicating "area between" is clicked (see lecture
graphic). "Between" is the default setting for the
Normal Tool, so when you open it up it automatically gives
you the area between two values.
**NOTE: Do not
use the 'ENTER' key on your keyboard to enter values. **
DO enter values by clicking on the Buttons next to the field
where you enter the value.
**Set mu**.
On the lecture graphic, arrows point to little boxes where
you can set mu and sigma. First type in the mu which is relevant
to whatever example you are working on. Then **click the
"ENTER MU (50 - 500)" button** right next to the
box where you entered the value of mu. (Note: The Normal Probability
Tool only accepts values of mu between 50 and 500.) For our
height example, I have entered mu = 150.
**Set sigma**.
The lecture graphic also shows where to enter the value of
sigma (toward the lower right-hand corner of the tool). For
our height example, I have entered sigma = 30. You must type
in the value of sigma and then **press the "ENTER SIGMA"
button** next to it. DO NOT use the 'Enter' or 'Return'
key on your keyboard to enter scores.
**Set lower value**.
We are looking for the area (probability) between two values.
The lecture graphic shows you where you can enter the lower
of the two values. Once you type in the number, click on the
**button** which says "**ENTER LOWER SCORE**."
For the height example, the lower value is 140 cm, so on the
lecture graphic I have set the lower value to 140.
**Set upper value**.
Similarly, as you can see on the lecture graphic, there's
a box where you can enter the upper score. Once you type in
the upper value number, click on the **button** which says
"**ENTER UPPER SCORE**." Following the height
example, I have set the upper score to 170 on the lecture
graphic.
**Find probability**.
You have entered mu, sigma, upper score and lower score. Now
you are ready to find the answer to the question. The lecture
graphic points to a box where the probability will appear.
All you have to do is read it and record it. For the height
example, the probability that a northern European man's height
will fall between 140 and 170 cm is .3747.
**Black
Area**. Probability is represented by the black area under
the curve. Look at the normal distribution on Normal Probability
Tool. The black area between 140 and 170 represents a probability
of .3747.
We have set up
a correspondence between area on a picture we can see and
the concept of probability. This allows us to picture probability
clearly and simply.
**Click and drag**.
Play with the Normal Tool. You'll notice that there are two
blue pointers just below the normal curve. One is labeled
"lower score" and the other "upper score."
If you click on either of them, you can drag the black area
to whatever value you want. The upper or lower score changes
accordingly. The probability changes also accordingly. Try
it and watch how the black area and the probability change
together.
**Positive and
Negative Infinity**. Play with the Normal Tool some more.
You'll notice that to the right of the white boxes where you
enter the upper and lower scores there are buttons labeled
"-oo" and "+oo." This is as close as we
could get to the symbols for negative infinity (-oo) and positive
infinity (+oo). If you click on the minus infinity button
(-oo) the lower score will become minus infinity. If you click
on the plus infinity button (+oo) the upper score will become
plus infinity. Try this out now. Find the probability that
a height will fall between minus and plus infinity. (Answer:
1.) What is the probability that a height will fall between
minus infinity and 150 cm? (Answer: .5.)
Now we will turn
to a related question--what is the area probability *outside*
of two values?
**What is the
Probability ***Outside* 140 and 170? If we sample one
northern European male, what's the probability that his height
will fall outside of 140 and 170? In other words, what are
the chances that he'll be either below 140, or he'll be above
170 in height? That's what we mean by the word "outside."
**Area Outside**.
The first thing you have to do is click the icon for "Area
Outside" on the Normal Tool. The Normal Tool will now
show you the area outside 140 and 170. It will also change
the probability.
And then you
do exactly the same thing that you did before. For our current
example, you **set mu **at** ** 150, **set sigma**
at 30, **set the lower value** at 140, **set the upper
value** at 170.
**Find probability**.
Then you simply read the probability. This time it is .6253.
The probability that a height will fall outside (above or
below) 140 and 170 cm is .6253.
Now we will turn
to finding the area* above* a certain value.
**What is the
Probability ***Above* 170? Perhaps a basketball coach
is interested in tall men. We have modeled the heights of
northern European males as N(150, 30). If that model is true,
and if we sample one man from that population, what are the
chances he has a height *above* 170 cm? This question
implies that the lower score will be 170 and the upper score
will be plus infinity. All scores above 170 will fall between
170 (on the low end) and plus infinity (on the high end).
**Set mu**:
150.
**Set sigma**:
30
**Click Between
Icon**.
**Set lower score**:
170
**Set upper score**:
+oo.
**Read probability**:
.2546. There's about a 25% chance that the man would have
a height above 170 cm. That's represented by the black area
under the normal curve.
Now we will turn
to a special case of the normal distribution: The Standard
Normal.
**Click on "Back
to Menu"** at the bottom of the tool.
**2.
Standard Normal Tool: Finding probabilities for N(0, 1)**
**N(0, 1)**:
There is a particular form of the normal distribution which
is very commonly used in statistics. It is called unit normal
or the standard normal or the z distribution. The unit normal
is simply a normal distribution which has a mean (mu) = 0,
and a standard deviation (sigma) = 1. In more compressed symbols
the unit normal is N(0, 1).
Everything works
exactly the same with the unit normal as it does for any normal.
So everything we've already learned applies to this topic.
We will just be using a particular member of the normal family
of distributions. This member of the family has mu = 0 and
sigma = 1 and is sometimes called the z distribution.
**z-Tables in
Stat Books**. The unit normal is the particular form of
the normal that is found in z-tables in the back of stat books.
"In the old days" before we had interactive programs
like Normal Tool, we had to convert all questions to z scores
and look up probabilities in z-tables.
[If you are still
using the Standard Normal Tool, click on "Back to Menu"
at the bottom so you see a simple page with two buttons.]
**Click on the
lower button--Standard Normal Tool.**
**Question**.
Suppose that we have N(0, 1) as our probability model. What
is the probability of a score between -1 and +1 on (N(0, 1)?
**Don't need
to set mu and sigma**. On the unit normal, N(0, 1), mu is
always 0 and sigma is always 1. So you don't need to set them.
**Click on the
Area Between Icon**.
**Set lower and
upper scores**. Set the lower and upper score as we did
above. In this case the lower score is -1 and the upper score
is +1. When you start the Unit Normal option, it will come
up with minus one and plus one as the lower and upper scores.
So we don't have to do anything to solve the particular question
we have asked.
**Read the probability**.
The answer is .6827. This should be familiar to you. If it's
not, it soon will be. |