Welcome to a lesson on how to use tables
to find the probability of a data value being below and above
a given Z-score in normal distribution
as well as how to find the probability of data value being between
two Z-scores. So for our first example,
we want to find the probability of a randomly selected data value from a
normal distribution
having a Z-score below one point four, and the probability
of the data value having a Z-score above one point four.
We can express this as the probability of
Z being less than one pint four, and the probability of Z being greater than one point four.
So if this is our standard normal distribution, Z equals one point four would be somewhere
in here.
And therefore the area to the left under the curve represents the
probability that Z
is less than one point four. And the area to the right end of the curve
represents the probability that Z is greater than one point four.
We can determine these probabilities by using the Z-score table
for the normal distribution. There are typically two types have tables.
One type is the cumulative table
which gives area to the left of the given Z-score,
and there's the zero through Z table that gives the area
between Z equals zero and a given Z-score.
Some textbooks have tables for both positive and negative Z-scores.
But for our examples, we'll be using tables with only positive Z-values,
and we'll have to use symmetry if Z is negative. So going back to our example,
we want to locate Z equals one point four
in the Z-score table. We'll take a look at both types of tables.
For the cumulative table, we locate
Z equals one point four. Because we don't have a hundreds place value,
we're looking for one point four,
zero, which would be here. If we did have a hundreds place value,
we'd have to find the correct column. So this value gives us the area
to the left of Z equals one point four, which is a probability
that Z is less than one point four. If we use the
zero to Z table, notice how one point four gives us this value here.
This is only the area between Z equals zero,
and Z equals one piont four. The area to the left of Z equals zero would be
zero point five. Notice if we add zero point five, two, to this value,
we get the value in the cumulative table. So again, this is the probability
that Z is
less than one point four. So this would be
zero point nine, one, nine, two.
And therefore, the probability that Z is greater than one point four,
this area here would be one minus
zero point nine, one, nine, two. Which would be
zero point zero, eight, zero eight.
Now let's take a look at a second example.
Here we want to find the probability
of a randomly selected data value from a normal distribution
having a Z-score between negative zero point five and two point one.
So again, if this is the standard normal distribution:
Z equals negative zero point five might be here.
And Z equals two point one
may be somewhere over here. So the probability that Z is between these
Z-scores would be the area under the curve
added by these two Z-scores. So to find this area,
we won't use the cumulative Z-score table, we'll use just a zero to Z table.
And again, our two Z-scores were
negative zero point five, and
two point one. Again our tables don't have negative Z-values,
so we'll use the Z-score of zero point five to determine the area
between Z equals zero and Z equals negative point five.
Notice when Z equals
zero point five, the area between Z equals zero and zero point five
is approximately zero point one, nine,
one, four, six. This would be the same as the area
between Z equals zero, and Z equals negative
zero point five. So this is the part of the probability,
so we'd have zero point one, nine,
let's just call it one, five. Plus
now we'll find Z equals two point one, which is here.
This tells us the area between Z equals zero,
and Z equals two point one would be approximately zero point four, eight,
two, one four. So we'll add this to find the desired probability.
So we have zero point, let's call it four, eight, two, one.
Giving us the probability of zero point six,
seven, three, six.
If we go back to the previous slide just for a moment, we found this area
here to be
approximately zero point one, nine, one, five.
And this area to be approximately zero point four, eight,
two one. Again giving the probability that Z is between
negative zero point five and two point one has zero point
six, seven, three, six.
I hope you found this helpful.