In this video we will look
at three examples of determining the probability
that a randomly selected data value
from a normal distribution is in a given interval.
We'll be determining the probabilities using
a table that gives the probability left of a Z-score.
So again, the set of data is normally distributed.
The mean is 78, and the standard deviation is six.
We're asked to find the following probabilities.
And I've already modeled the distribution, shown below,
where again the mean is 78
and the standard deviation is six.
Because we're using a table that uses Z-scores,
the first step is to find the corresponding Z-scores
for the data values of 70, 75 and 88.
Remember the Z-score is equal to the data value minus
the mean divided by the standard deviation.
So the data value of 70 has a Z-score
of approximately negative 1.3.
The data value of 75 has a Z-score of negative 0.5.
And the data value of 88 has a Z-score
of approximately 1.7.
So now we can rewrite the probabilities using Z-scores
rather than data values.
So the probability that x is less than 70
is approximately the probability
the Z-score is less than negative 1.3.
The probability that x is greater
than 70 is approximately equal
to the probability the Z-score is greater than negative 1.3.
And the probability that x is greater than 75
and less than 88 is approximately equal
to the probability the Z-score is greater
than negative 0.5 and less than 1.7.
Now we can use the table to determine the probabilities.
Looking at part a though, let's first find
the data value of 70 on the normal distribution.
Notice that 70 is approximately here.
And the corresponding Z-score is approximately negative 1.3.
Because we're looking for the probability
that x is less than 70 or z is less than negative 1.3,
we're looking for the probability on the left,
or the probability that the Z-score
is less than negative 1.3 or the data value is less than 70,
which would be this region here.
Notice how this will be the probability to the left.
So going to our table,
we locate the Z-score of negative 1.3,
which is here.
Notice how the corresponding probability
is approximately 9.68%.
And this percent is the percent to the left
or probability to the left.
And therefore, this is the probability we are looking for.
So the probability that x is less than 70
or that the Z-score is less than negative 1.3
is approximately 9.68%.
Which again, is a probability the data value
or Z-score would be in this region here.
Notice for part b we're asked
to find the probability that x is greater than 70
or that the Z-score is greater than negative 1.3.
So now we're looking for the probability to the right
or the probability the data value
or Z-score is in this region here.
To find the percent or probability more than a Z-score,
we take 100% and subtract the percent from the table.
And that should make sense,
because the probability to the left
is approximately 9.68%.
And therefore the probability to the right would have
to be 100% minus 9.68%.
Which is 90.32%.
So the probability that x is greater than 70
or the Z-score is greater than negative 1.3
is approximately 90.32%.
And now for the last example,
we need to find the probability
the Z-score is between negative 0.5 and 1.7.
To find the percent or probability between two Z-scores,
we subtract the smaller percent
from the larger percent given from the table.
So going back to our table,
we need to locate the Z-scores of negative 0.5 and 1.7.
The Z-score of negative 0.5 is here.
The probability of 30.85 though is a percent to the left.
And the Z-score of 1.7 is here
where the probability to the left is 95.54.
To find the probability between the two Z-scores,
we subtract 30.85% from 95.54%.
So again, we have 95.54% minus
30.85%, which gives us 64.49%.
Let's go back to our normal distribution
and show why we are subtracting these percentages.
The data value of 75 is approximately here,
where the Z-score is negative 0.5.
And the data value of 88 is approximately here,
where the Z-score is approximately 1.7.
And again, we're looking for the probability
the data value is between these two data values
or these two Z-scores.
Which would be the probability in this region here.
The 95.54 percent gave us the percent
or probability that the data value was less than 88,
or the Z-score was less than 1.7.
Which was the probability that the data value
or Z-score was in this region here.
Again, to the left.
And then we subtracted 30.85%, which was the probability
that the data value was less than 75,
or the Z-score was less than negative 0.5.
Which was the probability the data value
or Z-score was in this region here.
Leaving us with the probability of the data value being
between 75 and 88, or the Z-score being
between negative 0.5 and 1.7.
I hope you found this helpful.