In this lesson, we will use desmos.com
to determine the probability of a sum from a sample
and also to determine a sum that is 1.5 standard deviations
above the mean of the sums.
To begin, let's review the central limit theorem for sums.
The central limit theorem for sums
says if you keep drawing more and more samples of size n
with a large enough n, and taking their sums,
the sums form their own distribution,
the sampling distribution,
which approaches a normal distribution
as the sample size increases.
The normal distribution has a mean equal to
the original mean multiplied by the sample size,
indicated by n times mu sub x,
and the standard deviation
is equal to the original standard deviation
multiplied by the square root of the sample size,
indicated by the squared of n times sigma sub x.
So going back to our example,
let's begin by determining the mean of the sums,
as well as the standard deviation of the sums.
We will indicate the mean of the sums
as mu sub the sum of x, which is equal to,
n, the sample size of 50,
times the mean of the original distribution,
which is 88.
50 times 88 is equal to 4400.
And now we will find the standard deviation of the sums,
indicated by sigma sub sum of x,
which is equal to the square root of n,
which is the square root of the sample size,
or the square root of 50,
times the standard deviation of the original distribution,
which is 12, which gives us 12 square root of 50,
which does simplify, but there's no need to simplify it.
Now, for part a, we're asked to find the probability
the sum of the 50 values is less than 4350,
which we can indicate as the probability
that the sum of x is less than 4350.
And now let's go to desmos.com,
graph the distribution of the sums,
then determine their probability.
To begin, go to desmos.com.
Click graphing calculator.
Click show key pad in the lower left hand corner.
Click functions.
Click the distributions tab.
Click normal distribution.
Now we enter the mean and standard deviation.
And again, the mean is 4400, comma,
the standard deviation is 12 square root 50.
Now we click the zoom fit button,
which is the magnifying glass in cell one.
And now we have a nice distribution of the sums.
Next we will click the Find Cumulative Probability box
in cell one.
That is by default, the min is negative infinity,
the max is positive infinity,
which results in all the area under the curve being shaded,
which is equal to one.
And now to find the probability
that the sum of x is less than 4350,
we leave the minimum at negative infinity
and change the maximum to 4350,
again, because we have the sum of x less than 4350.
Let's close the Desmos keypad.
Notice how the area under the curve
to the left of 4350 is shaded.
And the probability is shown
at the bottom of cell one to four decimal places.
We have 0.2778,
which is equal to 27.78%.
For B, we're asked to find the probability
that the sum is between 4300 and 4450,
which is the probability that the sum of x
is greater than 4300 and less than 4450.
So going back to the graph,
we changed the minimum to 4300,
and changed the maximum to 4450.
Close the Desmos keypad.
Notice how the area below the curve
is shaded between 4300 and 4450,
which represents a probability we are looking for.
The bottom of cell one to four decimal places,
we have 0.6029,
Which is equal to 60.29%.
For part c, we're asked to find the sum
that is 1.5 standard deviations above the mean of the sums.
So the sum we are looking for, the sum of the legs,
is equal to the mean of the sums, which is 4400.
Looking at our notes below,
this is indicated by n times mu sub x.
And then we have plus the Z-score, which is 1.5,
because the sum we are looking for
is 1.5 standard deviations above the mean of the sums.
And then we have times the standard deviation of the sums,
which is 12 square root 50,
which below is indicated by the square root of n
times sigma sub x.
Now, going back to the calculator,
you can click in cell two, and enter
4400 plus 1.5
times 12 square root 50, enter,
which to one decimal place is approximately 4527.3.
I hope you found this helpful.