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 the 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
true H0
The
Thesample
samplestatistic
statistic(p(phat)
hat)isisreally
reallypart
partof
ofthe
the
H0 curve,
but we
mistake
Ha curve,
but
we
it mistake
as beingitpart
of the Ha
Type II as
– being part
curve
of the
fail
to H0 curve
reject a
false H0
H0
True
H0
False
Reject Type I
Correct
error a
Fail to
reject
Type
II
Correct
error
btrue
Suppose
Suppose
H H
is H
Suppose
is is
0
0
0
Suppose
H
is
true
0 reject
&false
we
false
fail
&
to
we
& we failreject
to
&
we
reject
it,
it,
it,
what
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type
type
of of
reject
it,
what
whatwas
type
of
decision
made?
was
typedecision
of decision
decision
was
was made?
made?
made?
How do we word statements
of type I & type II errors?
“We decide 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
ConsiderType
a murder
trial:
must be evidence beyond a
WhatWe
aredon’t
reasonable doubt!
H0: defendant is
H
:
What arewant
the to
hypotheses?
send innocent0 people
innocent to
jail! Ha: H : defendant is guilty
a
What is a type
is guilty when really innocent
I error?
Type I error –Decide the defendant
Consequence:
What is person
a consequence
of a
An innocent
goes to prison
type I error?
Which of these errors does our society believe to
be worse?
guilty II
when really guilty
Type II error – Decide defendant
What isis anottype
error?
What is a consequence
of a
person goes free
Consequence: A guilty type
II error?
Facts:
As a
As a you
increases,
• Every time you make a decision,
b
have potentially made an decreases,
error.
bdecreases
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
an error
Someone
made
an error with
is 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?
H0: potatoes good
Ha: potatoes bad
Type I error?Decide the potatoes are bad when they really are
good
they really
Type II error? Decide the potatoes are good whenSometimes,
are bad
the
seriousness
From the supplier’s viewpoint,
depends upon
A
type
I
error
which is more serious?
the person’s
point-of-view
From the chip company’s viewpoint,
which is more serious? A type II error
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
Type I :
the temperature of each sample recorded.
Type II:
Decidewould
the agree
H0: m =Most
150 people
Decide
temperature
is
What are the hypotheses?
that the
type the
II error
Ha: m >150 temperature
above
150°
would be
more
serious
isn’t
above
150°
What are the Type I and II errors?
when
it’s really
because
it would
when
it’s
really
below.
endanger
the
river’s
Which is more serious?
above.
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.
H : medicine is not
0
What are the hypotheses?
What are the Type I
& II errors?
toxic
Ha: medicine is toxic
Type I: decide
medicine is toxic
when it really isn’t
Type II : decide 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.
A new flu vaccine claims to prevent a
certain type of flu in 70% of the
people who are vaccinated. In a test,
vaccinated people were exposed to
the flu. Is this claim too high?
Identify the following decisions: Correct
We decide the true proportion
of vaccinated people who do
not get the flu is less than
70% when in fact it really is
less.
decision!
A new flu vaccine claims to prevent a
certain type of flu in 70% of the
people who are vaccinated. In a test,
vaccinated people were exposed to
the flu. Is this claim too high?
Identify the following decisions: Type II
We decide the true proportion
of vaccinated people who do
not get the flu is not less
than 70% when in fact it
really is less.
error
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.
Suppose a blood test was used to detect a
serious illness (like HIV).
State the hypotheses.
State Type I & Type II errors and their
consequences.
Which is more serious?