A variable is normally distributed
with a mean of 20 and a standard deviation of four.
Determine the areas below.
Write your answers in decimal form,
round to the nearest thousandth as needed.
The thousandth place value
means three decimal places.
Notice I've already set up
the normal distribution curve here,
where we have the mean of 20 in the middle
and the standard deviation's marked off
to the right and left of 20.
For part A,
we're asked to find that area to the left of 19.
Looking at the graph,
notice how 19 would be here,
to the left of 20.
And because we're looking for the area
to the left of 20,
we're looking for this area here.
This area is equal to the probability
that X, the value is, less than 19.
So, we can say the area
is equal to the probability
that the data value is less than 19.
Again, less than 19
because we want the area to the left.
And now we'll use
the free online normal distribution calculator,
listed here below,
to determine the probability which will give us the area.
So, going to the calculator,
[clicks]
we first select "Normal Distribution."
Then enter the "Mean" and "Standard Deviation."
Again, the mean is 20
and the standard deviation is four.
Next, under "Probability",
we select the first option.
We want the probability that X is less than 19,
and therefore we enter 19 and click calculate.
Notice how the graph below,
it does show the area that we want.
Which is also the same as the probability
that X is less than 19,
which to three decimal places
is approximately 0.401.
Next we're asked to find the area
to the left of 13.
Which is the same as determining the probability
that X is less than 13.
So, we'll say the area
is equal to the probability that X is less than 13.
And now we'll go back to the calculator
and change the value of 19 to 13.
And when we click calculate,
we'll see the desired to area graphed below,
but is equal to
the probability that X is less than 13,
which to three decimal places,
is approximately 0.040.
For part C,
find the area to the right of 17.
So, going back to our graph here on the side,
notice X equals 17 would be
to the right of 16 here.
Because we're looking for the area to the right,
this area here.
This would be the same
as the probability that X is greater than 17.
So, the area
is equal to the probability that X is greater than 17.
Going back to our calculator,
we now select the second option,
where we have the probability X greater than.
And then we enter the value of 17, click calculate.
The graph shows the area that we were looking for,
which is equal to the given probability,
which to three decimal places,
is approximately 0.773.
For part D,
we're asked to find the area to the right of 22.
Which is equal to the probability
that X is greater than 22.
Going back to the calculator,
we change the value of 17 to 22, click calculate.
The graph does show the area to the right of 22,
under the normal distribution curve,
which is the area that we want.
Which is the same as the given probability,
which to three decimal places is 0.309.
And finally, for part E,
we're asked to find the area between 13 and 26.
Going back to our graph,
here we have X equals 13,
here we have X equals 26.
We're looking for the area between these values.
This area here.
And this area
is equal to the probability
that X is greater than 13
and less than 26.
So, going back to the calculator one last time.
Now we select the third option
where we have an interval for the X values,
and we enter 13 + tab + 26,
click calculate.
That is the area is shown below,
which to three decimal places
is approximately 0.893,
shown here in red.
I hope you found this helpful.