Errors in Hypothesis Tests

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Errors in Hypothesis Tests
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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!
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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.
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Type I error

• When you reject the null hypothesis that is
  really true
• Denoted by a
• Is the level of significance of the test

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Type II error

• When you fail to reject the null hypothesis when
  it is false
• Denoted by b

Type I reject a true H0
The x-bar is really part of the H0 curve, but we
mistake it as being part of the Ha curve
The x-bar is really part of the Ha curve, but we
mistake it as being part of the H0 curve
a
Type II fail to reject a false H0
b
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Type I error
Correct
a
Type II error
Correct
Suppose H0 is true we fail to reject it, what
type of decision was made?
Suppose H0 is false we reject it, what type of
decision was made?
Suppose H0 is false we fail to reject it, what
type of decision was made?
b
Suppose H0 is true we reject it, what type of
decision was made?
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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!
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Consequences are NOT the definitions of type I
II errors. They are what happens as a result
of making that incorrect decision.
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Consider a murder trial
Type I - that is why there must be evidence
beyond a reasonable doubt! We dont want to send
innocent people to jail!
What are H0 Ha
H0 defendant is innocent Ha defendant is guilty

• What are the hypotheses?
• Type I error
• Consequence
• Type II error
• Consequence

What is a type I error?
Decide the defendant is guilty when really
innocent
What is a consequence of a type I error?
An innocent person goes to prison
Which of these errors does our society believe to
be worse?
What is a type II error?
Decide defendant is not guilty when really guilty
What is a consequence of a type II error?
A guilty person goes free
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Facts
As a increases, b decreases
As a decreases, b increases

• Every time you make a decision, you have
  potentially made an error.
• a b are inversely related

Fail to reject H0
Reject H0
a
m0
b
ma
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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 is made in real
  life.
• Except for cases like
• Firestone tires
• Drugs like Phen-phen Vioxx

Someone made an error with these products
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Lays 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? Type II error? From the
suppliers viewpoint, which is more
serious? From the chip companys viewpoint,
which is more serious?
H0 potatoes good Ha potatoes bad
Decide the potatoes are bad when they really are
good
Sometimes, the seriousness depends upon the
persons point-of-view
Decide the potatoes are good when they really are
bad
A type I error
A type II error
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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 rivers 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. What are the hypotheses? What
are the Type I and II errors? Which is more
serious?
Type I Decide the temperature is above 150
when its really below.
Type II Decide the temperature isnt above 150
when its really above.
Most people would agree that the type II error
would be more serious because it would endanger
the rivers ecosystem.
H0 m 150 Ha m 150
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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? Which
is more serious?
H0 medicine is not toxic Ha medicine is toxic
Type I decide medicine is toxic when it really
isnt Type II decide medicine isnt toxic when
it really is
Most would consider a type II error more serious
since people could be harmed.
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• A government agency has received numerous
  complaints that a particular restaurant has been
  selling underweight hamburgers. The restaurant
  advertises that its patties are a quarter
  pound (4 ounces).
• Identify the following decisions
• We decide the mean weight of hamburgers is less
  than 4 ounces when in fact they really are less.

Correct decision!
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• A government agency has received numerous
  complaints that a particular restaurant has been
  selling underweight hamburgers. The restaurant
  advertises that its patties are a quarter
  pound (4 ounces).
• Identify the following decisions
• We decide the mean weight of hamburgers is not
  less than 4 ounces when in fact they really are
  less.

Type II error
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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.