Chapter 3
Investigating Independence
Objectives
Students will be able to:
1) Understand what it means for attempts to be
independent
2) Determine when evidence in a hypothesis test
is statistically significant
3) Compare and contrast Type I and Type II errors
Terminology Time!!!
• An athlete’s attempts are independent if his or her
ABILITY to be successful is the same after a
successful PERFORMANCE and after an unsuccessful
PERFORMANCE.
• For example, Kyle is a basketball player. His shot
attempts are independent if the success of one shot
does not depend on the success of a previous shot.
• Another example: Michelle’s attempts at flipping a coin
and wanting it to land on heads are independent if the
success of one flip landing on heads does not depend
on the success of a previous flip.
Refresher…
• The significance level of a test is a
predetermined level of evidence that is required
to essentially rule out RANDOM CHANCE as a
plausible explanation.
• We say evidence in a hypothesis test is
statistically significant whenever the evidence is
convincing enough to reject the null hypothesis.
• “Statistically significant” is essentially a synonym
for “convincing.”
• Example: Suppose that we tested whether a
basketball player had a greater ABILITY to make a
free-throw following a made free-throw than
following a missed free-throw and that the results
of the test were not statistically significant. What
does it mean that the results of the test were not
statistically significant?
• This means we do not have convincing evidence to
support the claim that the player has a greater
ABILITY to make a free-throw following a made
free-throw than following a missed one.
• Note: If the results were statistically significant,
then we would have convincing evidence to
support the alternative hypothesis.
Type I and Type II Errors
• Even if we perform all of the correct procedures
in a hypothesis test, we can still make an error
with our conclusion.
• The different kinds of errors we can make are
known as Type I and Type II errors.
Type I Error
• A Type I error occurs when we find convincing
evidence that an alternative hypothesis is true, when
it reality it is not true. (This would lead us to reject
the null hypothesis).
• A Type I error is also known as a “false positive.”
• Examples of Type I errors:
• In a courtroom, the null hypothesis is that the
defendant is not guilty and the alternative hypothesis
is that the defendant is guilty. A Type I error would be
convicting the defendant, when in reality he is not
guilty.
• You go to the doctor for a checkup. The doctor tells
you that you are sick, when in reality you are not.
Type II Error
• A Type II error occurs when we do not find
convincing evidence that an alternative hypothesis
is true, when in reality it is true. (This would lead
us to fail to reject the null).
• A Type II error is also known as a “false negative.”
• Example of Type II errors:
• In a courtroom, a Type II error would be not
convicting a defendant, when in reality he is guilty.
• You go to the doctor for a checkup. The doctor tells
you that you are not sick, when in reality you are
sick.
• Mr. Chart best summarizes the types of errors
that can be made:
• Type I errors occur because we go into the hypothesis
testing process willing to accept a certain amount of
risk.
• If we have a 5% level of significance, then we can
expect to make a Type I error about 5% of the time.
• Type I errors can be reduced by using a smaller
significance level. This would require evidence to be
more convincing before concluding that the
alternative hypothesis is true.
• Caution: You do not want to make the significance
level too small. While it will decrease the likelihood
of causing a Type I error, it will also decrease the
likelihood that we decide to support the alternative
hypothesis (even when it is actually true!). That
would lead to an increase in causing Type II errors.
• Type II errors occur when the number of
PERFORMANCES is small. Remember that very
unusual PERFORMANCES can happen just by
RANDOM CHANCE.
• Type II errors can be reduced by increasing your
sample size. By gathering more data, we can be more
confident that an athlete’s PERFORMANCE will be
closer to his or her actual ABILITY.
• Super fun example time!!!
Suppose that we performed a hypothesis test to see
if the Los Angeles Dodgers had a greater ABILITY to
win following a win than following a loss.
a) State the hypotheses we are interested in testing.
b) Describe a Type I error and Type II error in the
context of this question.
Reminder:
A Type I error is finding convincing evidence that an
alternative hypothesis is true, when in reality it is not.
A Type II error is not finding convincing evidence that
an alternative hypothesis is true, when in reality it is.
A Type I error would be if we were convinced the
Dodgers had a greater ABILITY to win following a win
than following a loss when, in fact, the team’s ABILITY
was the same.
A Type II error would be if we aren’t convinced that the
Dodgers’ ABILITY to win was greater following a win
when, in fact, it was.
c) If the results of the test were statistically
significant, which type of error could we have
committed?
If the results of the test were statistically
significant, then we would have rejected the null
hypothesis.
Therefore, we could have committed a Type I error
(finding convincing evidence that an alternative
hypothesis is true, when in reality it is not; thus
leading us to reject the null).