this lesson will show
how to determine a confidence interval of a population mean
from data using Desmos
with an unknown population standard deviation.
You're interested in finding a 98% confidence interval
for the mean number of visits for physical therapy patients.
The data below show the number of visits
for 12 randomly selected physical therapy patients.
To find the confidence interval,
we first need to determine the error E,
indicated by the formula shown below.
Once we determine E,
we can determine the confidence interval
by determining x-bar minus E and x-bar plus E.
Before we do this, though,
we need to use the data and determine the sample mean,
the sample standard deviation, and the sample size,
as well as the t-score that corresponds
to the 98% confidence interval.
We know the sample size n is 12
and the t score's indicated by t sub alpha divided by two,
where alpha divided by two is the area to the right
of the t score.
So looking at the distribution
in the lower right-hand corner here,
the area to the right is equal to 0.01,
which is 1% as a decimal, and therefore,
the t-score is indicated by t sub 0.01.
We need the area to the left though,
in order to find the t-score using Desmos.
Notice, the area to the left is 0.99.
So now let's enter the data into Desmos
and calculate the sample mean,
the sample standard deviation, as well as the t-score.
We'll be using the command shown here on the far right.
Let's go to Desmos. Click Graphing Calculator.
In cell one we enter the data.
Let's call it capital L equals,
and then we enter the data in square brackets.
So open bracket and then we enter the data.
Close bracket.
Notice how it tells us we have 12 elements,
and now let's go to cell two and determine the sample mean.
Let's click Show Keypad. Click functions, click Stats.
Click mean. Enter capital L, close parenthesis and Enter.
The mean is approximately 20.083.
And now let's calculate the sample standard deviation.
Click functions, click standard deviation,
which is S-T-D-E-V, capital L, close parenthesis, Enter.
The standard deviation is approximately 5.418,
and now let's determine the t-score.
Click functions. Click the distributions tab, click tdist.
Next, we ender the Degrees of Freedom,
which is n minus one.
12 minus one is 11. Close parenthesis, period.
Functions, inversecdf, and now we enter the area
to the left of the t-score, which is 0.99.
Close parenthesis. Enter.
We will use a t-score of approximately 2.718.
Let's go back and record the mean,
the sample standard deviation and the t-score.
Now that we have the information we need,
we will calculate the error
using the error formula in Desmos.
Going back to Desmos, in cell five, enter E equals t,
open parenthesis, s divided by square root n.
Notice how we have an option
to add sliders for t, s, and n, click "all."
Scroll down, and now we will enter the t-score,
the sample standard deviation and the sample size.
t is 2.718,
s is 5.418,
and n, the sample size, is 12.
Enter, and now to scroll back up.
Notice how the error is automatically calculated,
which is 4.251.
So now that we have the error,
we can determine the confidence interval.
So going back to our work,
we now know E is approximately 4.251,
and now we can determine the confidence interval.
We need to determine x-bar minus E and x-bar plus E.
Our x-bar minus E is 20.083
minus 4.251,
and x-bar plus E is equal to 20.083
plus 4.251.
To save some time I've already calculated these.
This comes out to 15.382
and 24.334.
Therefore part b, with 98% confidence,
the population mean number of visits
per physical therapy patient is between, again, 15.382
and 24.334.
To get a better understanding
of the 98% confidence interval for part c,
if many groups of 12 randomly selected
physical therapy patients are studied,
then a different confidence interval would be produced
from each group.
About 98% of these confidence intervals will contain
the true population mean number of visits per patient
and about 100% minus 98%, or 2%,
will not contain the true population mean number
of visits per patient.
I hope you found this helpful.