Transcript Errors in Hypothesis Tests

Errors in Hypothesis Tests
When you perform a hypothesis test
you make a decision:
reject H0 or fail to reject H0
When you make one
of these decisions,
there is a possibility
that you could be
wrong!
That you made an
error!
There are two decisions that we
make; reject or fail to reject.
Each could possibly be a wrong
decision; therefore, there are
two types of errors.
Type I error
• When you reject a null hypothesis
that is really true
• Denoted by a
– Is the level of significance of the test
a
m0
Type II error
• When you fail to reject the null
hypothesis when it is false
• Denoted by b
Fail to reject
Reject
a
b
Type I –
reject a
Thetrue
x-barHis
0 really
The x-bar is really
part of the H0 curve,
part of the Ha curve,
but we mistake it as
but we mistake it as
being part of the Ha
being part of the H0
Type II – curve
curve
fail to reject
a false H0
H0 True H0 False
Reject
Fail to
reject
Type I
error a
Correct
Suppose H0 is false
& we reject it,
what
Suppose
typeHH
of0isistrue
false
Suppose
0
b
Suppose
H0 is true
decision
&
we
was
fail
made?
to
reject
& we &
failwetoreject
rejectit, what
typewas
of
typeit,
ofwhat
decision
it, what
type
of
decision was made?
decision was made?
made?
Correct
Type II
error
How do we word statements of
type I & type II errors?
“We say this decision
when in reality this is true.”
You replace the red,
underlined words
with words from
context!
Consequences – are NOT the
definitions of type I & II errors.
They are what happens as a result
of making that incorrect decision.
I - that is why
there must be
Consider aType
murder
trial:
evidence beyond a reasonable
What
areto send
doubt! We don’t
want
HH0: :defendant is
What are the hypotheses?
innocent people0 to jail!
innocent
Ha:
Hais: defendant
What
a type I is guilty
guilty when really innocent
Type I error – Say the defendant iserror?
Consequence:
is a consequence
of a to
type
AnWhat
innocent
person goes
prison
I error?
Which of these errors does our society believe to be
worse?
a type
IIreally guilty
is notis
guilty
when
Type II error – Say defendantWhat
error?
person goes free
Consequence: A guilty
What is a consequence of a type
II error?
Facts:
As a
As
a have
increases,
b
• Every time you make a decision,
you
decreasesb
potentially made an error. decreases,
increases
• a & b are inversely related
Fail to reject H0
Reject H0
a
m0
b
ma
Facts continued:
• The seriousness of the error types is
determined by the specific situations.
– Depending upon the situation type I or type II
may be the more serious.
• We often DO NOT know if anSomeone
error ismade
an error with
made in real life.
– Except for cases like
• Firestone tires
• Drugs like: Phen-phen & Vioxx
these products
Lay’s Chip Company decides to accept a
truckload of potatoes based upon results from a
sample of potatoes from the truckload.
What are the hypotheses?
Type I error?
H0: potatoes good
Ha: potatoes bad
Say the potatoes are bad when they really are good
Type II error? Say the potatoes are good when they really are bad
From the supplier’s viewpoint,
which is more serious? A type I error
From the chip company’s viewpoint,
which is more serious? A type II error
Sometimes, the
seriousness
depends upon the
person’s point-ofview
Water samples are taken from water used for cooling as it
is being discharged from a power plant into a river. It
has been determined that as long as the mean temperature
of the discharged water is at most 150 degrees F, there
will be no negative effects on the river’s ecosystem. To
investigate whether the plant is in compliance with
regulations that prohibit a mean discharge above 150
degrees F, fifty water samples will be taken at randomly
selected times, and the temperature of each sample
recorded.
Type I : Say the
H0: m = 150
temperature
is that
Most
people
Type
would
II:
Say
agree
the
What are the hypotheses? Ha: m >150
150°
when
the typeabove
temperature
II error
would
isn’tbe
it’s
really
above
because
150°below.
when
it
What are the Type I and II errors? more serious
would endanger
it’s reallythe
above.
river’s
Which is more serious?
ecosystem.
A doctor is considering a new medication
to help fight infections. However, the
medication has the possibility of being
highly toxic to the patient. You will test the
medication to determine toxicity.
What are the hypotheses?
What are the Type I
& II errors?
H0: medicine is not toxic
Ha: medicine is toxic
Type I: say medicine is toxic when it
really isn’t
Type II : say medicine isn’t toxic
when it really is
Which is more serious?
Most would consider a type II error
more serious since people could be
harmed.
How does one decide what a
level to use?
After assessing the consequences of
type I and type II errors, identify the
largest a that is tolerable for the
problem. Use that a level for your
level of significance.