Welcome!
In this video, I’ll be constructing confidence intervals for a population mean, using the
Student t distribution.
A confidence interval, constructed from sample data, is a range of values
that is likely to include the population parameter, at some specified confidence level.
The confidence interval for a population mean is determined by taking the sample mean (the
point estimate) and subtracting and adding a margin of error to it.
As seen in an earlier video, if we assume the population standard deviation is known,
the margin of error is determined by this formula with a Z critical value.
Since the population standard deviation is usually unknown in practice, it is substituted
with the sample standard deviation, which requires we use the t-distribution.
The t-distribution is similar to the standard normal distribution in that it is also bell-shaped,
has a mean of 0, and has an area of 1 or 100% under the curve.
It however has a standard deviation greater than 1, which makes it flatter than the Z
distribution.
While there is only one Z distribution, there is a family of t distributions with the shape
of each determined by what we call degrees of freedom (df for short) which is based on
the sample size n.
For the confidence interval for a population mean,
that we’re concerned with here, the degrees of freedom is defined by n - 1
Here’s the t distribution curve for 2 degrees of freedom
for 5 and for 30.
We see that as degrees of freedom increases, the t distribution approaches the Z distribution.
Here is a sample of the t-distribution table.
We have the degrees of freedom on the left here.
At the top, we have the right tail area of the distribution.
In some tables, you will also find the two tails alpha.
And some will also include confidence levels.
The t-critical values are shown in the body part of the table.
The values of Z are also usually shown on most t-tables.
They reveal that as sample size (or degrees of freedom) increases,
the t-distribution approaches the Z distribution.
In constructing the confidence interval for a population mean
given the sample standard deviation, the margin of error is given by this formula
where t_α⁄2 is the critical value -that is the value we look up in the table
or obtain using software with degrees of freedom n – 1.
s is the sample standard deviation, and n is the sample size.
α is the significance level which is 1 minus the confidence level.
So, for a 90% confidence level α will be 0.10.
and that is divided equally into the two tails.
The values corresponding to these boundaries of the confidence interval are the critical
values.
Suppose we have a sample size n = 12.
Then the degrees of freedom n – 1 will be 11.
On the t-table, at 11 degrees of freedom, we can look up
0.05 in one tail or 0.10 in two tails
or 90% confidence if we have it.
The corresponding critical value will be 1.796 which will be negative for the left tail here
and positive on the right.
Let’s look at some examples : A random sample of 15 observations has a mean
of 20 and a standard deviation of 3.5.
To estimate the population mean with 95% confidence (that is, at the 95% confidence level) we
want to first determine: the margin of error
and then the confidence interval The sample size 15 here is considered small
because it is less than 30.
Sample sizes that are at least 30 are generally considered large enough for the central limit
theorem to apply.
Therefore, we have to assume that the population is normally distributed in order to safely
proceed with our calculations.
If the sample size is large (that is, at least 30),
this assumption is not necessary due to the central limit theorem.
Now, for n = 15, degrees of freedom is n – 1, is 14.
So, for 95% confidence or alpha (two tails) = 0.05
the t-critical value is 2.145 at 14 degrees of freedom.
The margin of error is therefore about 1.94
The lower limit of the confidence interval will be the sample mean minus the margin of
error and the upper limit will be the sample mean
plus the margin of error.
To interpret this, we say We are 95% confident that the true population
mean falls between 18.06 and 21.94.
In essence we are saying that, using this method, we expect to capture the population
mean 95% of the time.
This interval constructed however may or may not contain the true population mean.
Let’s look at another example.
Here, the sample size is increased to 85 and we want to construct a 99% confidence interval
for the population mean.
The sample size 85 here is considered large because it is larger than 30.
Therefore, according to the central limit theorem,
The distribution of sample means will be approximately normal for a sample size this large
and the assumption of a normal population is unnecessary.
In general, since sample size is in the denominator of the margin of error formula,
increasing the sample size alone will decrease the margin of error and consequently make
the confidence interval narrower.
On the other hand, reducing the sample size will increase the margin of error and make
the interval wider.
In contrast, a higher confidence level will produce a larger critical value, resulting
in a larger margin of error and thus, a wider confidence interval.
With n = 85 in this example, we have degrees of freedom n – 1 equal 84.
Now, note that most tables will not have degrees of freedom 84 on them,
so you can use software if applicable.
Otherwise, you can use the closest df available in the table.
At 99% confidence level with df = 84, the closest df here is 80 with a critical value
of 2.639 which we will use here.
The margin of error is therefore about 1.00.
And the 99% confidence interval will be from 19 to 21.
That is, we can be 99% confident that the true population mean lies within this interval.
Again, this means that if all possible samples of size 85 are taken from the same population,
we expect 99% of the confidence intervals constructed from them to capture the true
population mean within their interval.
This interval constructed here is just one of those many intervals.
And that’s confidence interval for the population mean using the t-distribution.
Thanks for watching.