In this video,
we will look at three examples
of determining the probability
that a randomly selected data value
from a normal distribution has a Z-score
in the given interval.
We'll be determining the probabilities using a table
that shows a probability left of a Z-score.
For part a, we're asked to find the probability
the Z-score is less than negative 1.5.
Let's first locate the Z-score of negative 1.5
on the standard normal distribution shown below,
which is approximately here.
Because we want the probability the Z-score
is less than negative 1.5,
we want to find the probability on the left,
or the probability the data value
would have a Z-score in this region.
So going to our table,
the percentile in the table gives the percent
less than the Z-score.
So to find the probability the Z-score
is less than negative 1.5,
we simply locate the Z-score of negative 1.5 in the table.
The corresponding percentage
will give us the probability.
Well, notice the Z-score of negative 1.5 is located here.
The corresponding probability is 6.68%.
And therefore, the probability the Z-score
is less than negative 1.5 is approximately 6.68%.
For part b, we're asked to find that probability
the Z-score is greater than negative 1.5.
Now we want the probability to the right,
or the probability the data value
would have a Z-score
in this region here.
When we're trying to find the probability on the right
or to find the percent more than the Z-score,
we take 100% and then subtract the percent from the table.
So again, we just found
that probability to the left,
or when Z is less than negative 1.5,
is approximately 6.68%.
So it should make sense
if you want to find the probability
that the Z-score is greater than negative 1.5,
or the probability on the right,
you would take 100%
and then subtract 6.68%,
which gives us 93.32%.
So the probability the Z-score is greater than negative 1.5
is approximately 93.32%.
And then for the third example,
we're asked to find the probability
that the Z-score is greater than negative 0.4
and less than 1.8.
So let's go ahead and model this
on the standard normal distribution.
The Z-score of negative 0.4 is approximately here.
And the Z-score of 1.8 is approximately here.
So we're trying to find the probability
the data value will have a Z-score in this region here.
To find the percent
or probability between two Z-scores,
we subtract the smaller percent
from the larger percent given in the table.
So let's go back to our table
and locate the Z-scores of negative 0.4 and 1.8.
Negative 0.4 or negative 0.40 is here.
The probability to the left is approximately 34.46%,
and the Z-score of 1.8 is here.
The probability to the left is approximately 96.41%.
So to find the probability between the Z-scores,
we subtract 34.46% from 96.41%.
So we have approximately 96.41% minus 34.46%,
which is equal to 61.95%.
And now let's look at our graph
to see why we are subtracting these percentages.
Remember, the percent or probabilities from the table
give the probabilities or percent to the left.
So to find the probability
the Z-score is in this region here or in this interval,
we took 96.41%
and subtracted 34.46%
because the 96.41% gave us the probability
to the left of 1.8,
or the probability of the Z-score is in this region here,
again to the left of the Z-score of 1.8.
Then we subtracted 34.46%,
which was the probability the Z-score was
to the left of negative 0.4,
or less than negative 0.4,
which should be the probability the Z-score is
in this region here.
And notice how that leaves us with the probability
the Z-score is between negative 0.4 and 1.8.
I hope you found this helpful.