Welcome to an introduction
to the chi-square distribution.
"Have you ever wondered if lottery numbers were
evenly distributed or if some numbers occurred
with a greater frequency?
How about if the types of movies people preferred were
different across different age groups?
Does the use of steroids in collegiate athletics differ
across the three NCAA divisions?
What about if a coffee machine was dispensing
approximately the same amount of coffee each time?
We can answer these questions
by conducting a hypothesis test.
The distribution used is called the chi-square distribution.
We will learn four applications
of the chi-square distribution.
Number one, the goodness-of-fit test,
which determines if data fit a particular distribution,
such as a lottery example;
Two, the test of independence, which determines
if two events are independent, such as the lottery example;
Three, the test of homogeneity, which determines
whether two populations have the same distribution,
such as the steroid example;
And number four, the test of a single variance,
which tests variability, such as in the coffee example.
We use the notation shown here
for the chi-square distribution,
where the random variable is
approximately the chi-square distribution,
where the subscript is equal to the degrees of freedom,
which depends on how a chi-square is being used.
For the chi-square distribution, the population mean is
equal to the degrees of freedom
and the population standard deviation is equal
to the square root of two times the degrees of freedom.
The random variable is shown as chi-square,
but can be any uppercase letter.
The random variable for a chi-square distribution
with k degrees or freedom is the sum of k independent,
squared standard normal variables.
If we randomly select independent samples of size n
from a normal distribution
with population standard deviation sigma
and sample standard deviation s, chi-square is equal
to the product of n minus one and the square
of the sample standard deviation,
or just the sample variance,
divided by the square of the population standard deviation,
or just the population variance.
When conducting a hypothesis test,
we will be using a right tailed test.
And therefore we're determining the area under the curve
to the right, as shown below.
To find the p-value on the TI-84,
we'll be using the ch-square cdf function
shown here on the left.
And now let's talk about the graph
of the chi-square distribution.
Number one, the curve is non-symmetrical
and it's skew to the right.
Number two, there's a different chi-square curve
for each degrees of freedom.
Let's take a look at an animation of this.
So right now, we see the graph
of the chi-square distribution,
where the degrees or freedom are equal to zero.
Now, we can see the graph change
as the degrees of freedom change,
where the degrees of freedom are equal to k here
in the lower left-hand corner.
Notice, as the degrees or freedom increase,
the curve approaches a normal distribution.
Going back to our notes.
Number three, the test statistic
for any test is always greater than or equal to zero.
Number four, when the degrees of freedom are greater
than 90, the chi-square curve
approximates the normal distribution.
For example, if we have a chi-square distribution
where the degrees of freedom are equal to 1,000,
the mean is 1,000, then the standard deviation is equal
to two times the square of 1,000,
which is approximately 44.7,
which is approximately a normal distribution.
And then finally the main mu is located
to the right of the peak.
In the next several videos, we'll be looking
at the goodness-of-fit test, the test of independence,
the test of homogeneity and the test of a single variance.
I hope you found this introduction helpful.