This video will show how
to determine probabilities
of a normal distribution using desmos.com.
A data set of test scores is normally distributed
with a mean of 72 and a standard deviation of 8.2.
Determine the probability
that a randomly selected test score is
in the given interval.
Give the answer as a decimal to 3 decimal places
and a percent to 1 decimal place.
For part a we're asked to find the probability
that x, the randomly selected test score is greater than 80.
The first step is to go to desmos.com
and click start graphing.
Notice how by default the cursor is in cell one.
From here we need to open the Desmos keyboard
by clicking the show keyboard button
in the lower left hand corner.
Then click functions on the right.
Then click distributions.
Then click normal distribution.
Now looking at cell one,
notice how we have normal distribution.
By default the mean is zero,
and the standard deviation is one.
This is only true for the standard normal distribution.
And therefore we need to enter the mean
as well as the standard deviation.
The mean is 72 and the standard deviation is 8.2
and therefore we enter 72, 8.2.
Next we are going to click the zoom fit button
which is the magnifying glass shown here on the left.
Now looking at the graph notice how we have a nice graph
of our normal distribution
and the horizontal axis is labeled with the test scores.
Next we click the box to the left
of find cumulative probability or CDF.
Notice this shades all of the area
under the normal distribution curve
and the minimum shows as negative infinity,
and the maximum shows as positive infinity.
To approach negative infinity along the horizontal axis,
we would move left.
To approach positive infinity along the horizontal axis,
we would move right.
We need to change these to find the probability
that x is greater than 80.
Because you want the probability that x is greater than 80,
we are going to change the minimum to 80.
And leave the maximum at infinity.
Looking at the graph notice how this shades the area
below the curve to the right of the test score of 80.
This area represents the probability that x is greater
than 80 which is shown here in decimal form.
To three decimal places the value is approximately 0.165.
Now we know the probability that x is greater than 80
is approximately 0.165 which as a percent is 16.5%.
Notice how we have the probability as a decimal
to three decimal places
and a percent and one decimal place.
Next we want the probability that x is less than 65.
Which means now 65 is going to be the maximum.
And we need to change the minimum back to negative infinity.
So going back to our graph,
again find the probability that x is less than 65.
Let's change the maximum to 65.
We need to change the minimum back to negative infinity.
The easiest way to do this is just to clear the 80.
And then click out of the cell
and it automatically changes back to negative infinity.
Looking at the graph we do have the area shaded
below the curve to the left of 65
which represents where x is less than 65.
The probability that x is less than 65 is shown here
in decimal form rounded to three decimal places,
we have approximately 0.197.
Which is equal to 19.7%.
And finally for part c we want to find the probability
that x is greater than 72 and less than 92.
Which means now the minimum is going to be 72,
and the maximum is going to be 92.
And therefore we change the minimum to 72.
The maximum to 92.
And again we can always check our work
by analyzing the graph.
Notice how we do have the area shaded below the curve
between the test scores of 72 and 92
which is what we want.
This area represents the probability a
randomly selected test score would be in this interval.
To three decimal places, we have approximately 0.493.
Which is equal to 49.3%.
I hope you found this helpful.