Welcome to a lesson on the Student t-distribution,
sometimes referred to as just the t-distribution.
In practice, we rarely know
the population standard deviation.
In the past, when the sample size was large,
this did not present a problem to statisticians.
They used the sample standard deviation s
as an estimate for sigma, the population standard deviation,
and proceeded as before to calculate a confidence interval
with close enough results.
However, statisticians ran into problems
when the sample size was small.
A small sample size caused inaccuracies
in the confidence interval.
If you draw a simple random sample of size n
from a population
that has an approximately normal distribution with mean mu
and unknown population standard deviation sigma
and calculate the t-score
given by the difference of the sample mean
and the population mean divided by the quotient
of the sample standard deviation and the square root
of the sample size,
then the t-scores follow a Student's t-distribution
with n minus one degrees of freedom.
The t-score has the same interpretation as the z-score.
It measures how far x bar the sample mean
is from the population mean mu.
For each sample size n,
there is a different Student's t-distribution.
Here are some of the properties
of the Student's t-distribution.
And we will verify all these graphically in just a moment.
The graph for the Student's t-distribution
is similar to the standard normal curve.
The mean for the Student's t-distribution is zero
and the distribution is symmetric about zero.
The Student's t-distribution has more probability
in its tails than the standard normal distribution,
because the spread of the t-distribution
is greater than the spread
of the standard normal distribution.
The exact shape of the Student's t-distribution
depends on the degrees of freedom.
As the degrees of freedom increases,
the graph of the Student's t-distribution
becomes more like the graph
of the standard normal distribution.
The underlying population of individual observations
is assumed to be normally distributed
with an unknown population mean
and an unknown population standard deviation.
The size of the underlying population
is generally not relevant, unless it is very small.
If it is bell shaped or normal,
then the assumption is met and does not need discussion.
Random sampling is assumed,
but that is a completely separate assumption from normality.
Let's take a look at the standard normal distribution
and the Student's t-distribution using Desmos.
I've already set this up to save time.
In red we have the graph
of the standard normal distribution.
In blue we have the graph of the Student's t-distribution
right now where the degrees of freedom
is equal to n minus one, and since n is two,
here we see the graph of the t-distribution
with one degree of freedom.
And notice as the degrees of freedom increase,
the Student's t-distribution gets closer and closer
to the standard normal distribution.
So we go back down to one degree of freedom, notice how
the Student's t-distribution has more of a spread
than the standard normal distribution,
but both distributions are symmetrical about the y-axis.
The notation for the Student's t-distribution is shown below
where capital T is the random variable
and df stands for the degrees of freedom.
For example, if we have a sample size of n equals 20 items,
then we calculate the degrees of freedom as 20 minus one,
which is 19, and we use the notation shown here
to indicate the random variable T is represented
by a t-distribution with 19 degrees of freedom.
If the population standard deviation is not known,
the error bound for a population mean
is given by t sub alpha divided by two
times the quotient of s and the square root of n
where t sub alpha divided by two is the t-score
with the area to the right equal to alpha divided by two,
which means the area to the left
is equal to one minus alpha divided by two.
s equals the sample standard deviation,
and n is equal to the sample size.
The confidence interval is equal to x bar
minus the error bound to x bar plus the error bound.
Let's put all this information together
and look at an example.
You do a study of hypnotherapy
to determine how effective it is in increasing
the number of hours of sleep subjects get each night.
You measure hours of sleep for 12 subjects
with the following results shown below.
Construct a 95% confidence interval
for the mean number of hours slept for the population,
assumed normal, from which you took the data.
The first thing we need to do is find the mean
of the sample data,
as well as the sample standard deviation.
To do this, I've already entered the data in the TI-84.
From the home screen, if we press STAT and then ENTER,
again, I've already entered the data in L1.
So if we go back to the home screen, press STAT,
Right Arrow to calculate,
ENTER for one variable statistics,
the data is in L1, which is correct, ENTER.
Frequency List, we can leave this blank or enter one,
since we are not using a frequency table.
ENTER, ENTER.
Now we can see here again, x bar, the sample mean,
is approximately 8.9833
and Sx represents the sample standard deviation,
which is approximately 1.2904.
There are 12 data items in there
for the sample size n is 12.
The degrees of freedom are equal to n minus one,
which is 12 minus one or 11.
And the confidence level is 95%.
Next, we need to find alpha.
Alpha is equal to one minus the confidence level.
Alpha is equal to 0.05.
Alpha divided by two is equal to 0.025.
So now we need to find the t-score,
which in our case is t sub 0.025.
0.025 represents the area to the right of the t-score.
The area to the left would be one minus 0.025,
which is 0.975.
And let's make a note of this.
We need the area to the left
to find the t-score using technology.
Now we'll show how to find this t-score on the 84,
as well as using desmos.com.
On the TI-84, press 2ND, VARS for the distribution menu.
Next we select option four for inverse T.
Area means area to the left, which we now know is 0.975.
ENTER, degrees of freedom, 11.
ENTER, ENTER, ENTER.
And you can see we have, to four decimal places, 2.2010.
Notice how we have a nine in the fourth decimal place,
rounding up, we have 2.2010.
Or using desmos.com, go down to cell four,
open the Desmos keypad, click functions.
Under the distributions menu, click inversecdf.
Go back to functions, click t-distribution.
Enter the degrees of freedom, which is 11.
Right Arrow, comma, the area to the left,
which is 0.975, Enter.
And we do get the same result.
Now we have all the information we need
to find the error bound.
Or again, the error bound is equal to the t-score
times the quotient of the sample standard deviation
and the square root of the sample size,
which gives us 2.2010 times 1.2904
divided by the square root of 12,
which, to four decimal places, gives us 0.8199.
Let's go ahead and verify that.
To four decimal places, we are correct.
And now we can find the confidence interval
given by x bar minus the error bound to x bar
plus the error bound, which gives us 8.9833 minus 0.8199
to 8.9833 plus 0.8199,
which gives the confidence interval of 8.1634 to 9.8032,
which means we estimate with 95% confidence
that the true population mean number of hours slept
is between 8.1634 and 9.8032 hours.
I hope you've found this helpful.