Welcome to lesson on how to use the ti-84 graphing calculator,
to find the probability of a randomly selected data value
being between above a given data value
in a normal distribution. And we'll also find the probability
of a randomly selected data value being between two data values
in a normal distribution. Let's look at our first example.
A set of data has a normal distribution. The mean is 100
and the standard deviation is 20. We want to find the probability of a data
value
being below 85 and the probability of a data value being
above 85. Which we can express in this way here
as the probability of X being less than 85
and the probability the X being greater than 85.
Notice how 85 is less than one standard deviation
below the mean of 100. So let's begin by modeling this distribution.
Mu or the mean is 100, which would be here in the middle
and again 85 would be
less than one standard deviation below the mean. Maybe somewhere in here.
Now if you weren't going to use technology to find these probabilities，
we would then find the corresponding z-score.
Then using the standard normal distribution, the area
below the z-score, would represent the probability
that X is less than 85. And the area above the z-score
would be the probability that X is greater than 85.
But because we'll be using the ti-84,
we don't actually have to find a z-score. We can use the normal CDF feature
and then enter A comma B comma mu comma sigma.
And this will give the value of the cumulative normal density function
occurring between X equals A and X equals B.
This would also be the cumulative area under the standard normal distribution
from the left. And notice how we do have to to enter a value for A and B,
were A would be the lower bound data value and B would be the upper bounded data value.
So B is equal to 85. We'll have to make up a value for A.
but as long as that several standard deviations away from the mean of 100
it'll work just fine. So we'll exaggerate the lower bound of the data value.
So we'll begin by pressing second
vars for distribution, option 2.
Now we're going to enter our lower bound data value.
Let's go ahead and use negative 99999
comma, the upper bound data value is 85.
Comma, mu,
which is the mean, which is 100. Comma,
standard deviation which is 20. Close parenthesis
and enter. This represents the cumulative
area under the standard normal distribution to the left of
X equals 85. And therefore the probability that X is
less than 85, is approximately 0.2266.
Which means the probability that X is grater than 85,
will be 1 minus 0.2266.
Which would be 0.7734
Now let's take a look at a second example.
Here, much of the information is the same.
We have a normal distribution. The mean is 100 and the standard deviation is still 20.
But now we wanna find the probability of a data value being between
75 and one hundred ten. So again, mu is equal to to 100.
One hundred and ten
is 1/2 a standard deviation above the mean.
That be somewhere in here . And X equals 75
is more than one standard deviation below the mean, maybe somewhere in here.
And again, if we were doing this by hand
we'd find the corresponding z-scores. We'll call this Z sub 1 and then Z sub 2.
And i we found the area bounded by these two Z scores,
it would give us the probability that were looking for or the probability
that X is between 75 in one hundred and ten.
But again using a ti-84, we don't actually have to find a Z scores
Again press second, vars, option
2.
our lower bound data values is now 75. So 75 comma
upper bounded data value is one hundred and ten.
comma, the mean of 100,
comma, the standard deviation. Close parenthesis
enter. So the probability that X is between 75 and one hundred ten,
is approximately, 0.5858.
I hope you found this helpful.