errors in hypothesis tests. when you perform a hypothesis test you make a decision: when you make...
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Errors in Hypothesis Errors in Hypothesis TestsTests
When you perform a When you perform a hypothesis test you make hypothesis test you make
a decision:a decision:
When you make one of these
decisions, there is a possibility that you could
be wrong!That you made
an error!
reject H0 or fail to reject H0
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 errorType I error
•When you reject the null hypothesis that is really true
•Denoted by – Is the level of significance of the test
Fail to reject Reject
Type II errorType II error
• When you fail to reject the null hypothesis when it is false
• Denoted by
Type I – reject a true H0
Type II – fail to
reject a false H0
The x-bar is really part of the Ha curve, but we mistake it as being part of the H0
curve
The x-bar is really part of the H0 curve, but we mistake it as
being part of the Ha curve
H0 True
H0
FalseReject
Fail to reject
Type I error
Correct
Correct
Type II errorSuppose 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 true & we reject it, what type of
decision was made?
Suppose H0 is false & we fail
to reject it, what type of decision was
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.
Consider a murder trial:
What are the hypotheses?
Type I error –Consequence:
Type II error – Consequence:
Decide the defendant is guilty when really innocent
An innocent person goes to prison
Decide defendant is not guilty when really guilty
A guilty person goes free
H0: defendant is innocent
Ha: defendant is guilty
Which of these errors does our society believe to be worse?
Type I - that is why there must be evidence beyond a reasonable doubt! We don’t
want to send innocent people to jail!
What are H0:Ha:
What is a type I error?
What is a type II error?
What is a consequence of a type I error?
What is a consequence of a type II error?
Facts:Facts:• Every time you make a decision,
you have potentially made an error.
• & are inversely related
0
a
Fail to reject H0 Reject H0
As decreases
, increases
As increases,
decreases
Facts continued: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 NOTDO 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
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?
Type II error?
From the supplier’s viewpoint, which is more serious?
From the chip company’s viewpoint, which is more serious?
Decide the potatoes are bad when they really are good
Decide the potatoes are good when they really are bad
H0: potatoes good
Ha: potatoes bad
A type I error
A type II error
Sometimes, the
seriousness depends upon the person’s point-of-view
Most people would agree that the type II error would be more serious because it would endanger the river’s ecosystem.
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.
What are the hypotheses?
What are the Type I and II errors?
Which is more serious?
H0: = 150 Ha: >150
Type I : Decide the temperature is above 150° when it’s really below.Type II:Decide the temperature isn’t above 150° when it’s
really 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.
What are the hypotheses?
What are the Type I & II errors?
Which is more serious?
H0: medicine is not toxic
Ha: medicine is toxicType I: decide medicine is toxic when it really isn’t
Type II : decide medicine isn’t toxic when it really is
Most would consider a type II error more serious since people could be harmed.
A government agency has received numerous complaints that a particular restaurant has been selling underweight hamburgers. The restaurant advertises that it’s 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
!
A government agency has received numerous complaints that a particular restaurant has been selling underweight hamburgers. The restaurant advertises that it’s 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
How does one decide How does one decide what a level to use?what a level to use?
After assessing the consequences of type I and type II errors, identify the largest largest that is tolerable for the problem. Use that level for your level of significance.
Power of a testPower of a test
The powerpower of a test (against a specific alternative value)• Is the probability that the test
will reject the null hypothesis when the alternative is true
• In practice, we carry out the test in hope of showing that the null hypothesis is false, so high power is important
H0 True
H0
FalseReject
Fail to reject
Type I Correct
Correct Type II
Power
Suppose H0 is true – what if we decide to fail to reject
it?
Suppose H0 is false – what if we decide to
reject it?
Suppose H0 is true – what if we decide to
reject it?
Suppose H0 is false – what if we decide to
fail to reject it?
We correctly reject a We correctly reject a false Hfalse H00!!
A researcher selects a random sample of size 49 from a population with standard deviation = 35 in order to test at the 1% significance level the hypothesis:
H0: = 680
Ha: > 680
What is the probability of committing a Type I error?
= .01
H0: = 680
Ha: > 680
For what values of the sample mean would you reject the null hypothesis?
691.63
H0: = 680
Ha: > 680
If H0 is rejected, suppose that a is 700. What is the probability of committing a Type II error?
What is the power of the test?
=.0471
Power = 1 - .0471 = .9529
H0: = 680
Ha: > 680
If H0 is rejected, suppose that a is 695. What is the probability of committing a Type II error?
What is the power of the test?
Normalcdf(-10^99,691.63,695,5) =.2502
Power = 1 - .2502 = .7498
Reject H0Fail to Reject H0
Power = 1 - 0
a
What happens to What happens to , , , & , & power when the sample size power when the sample size
is increased?is increased?Reject H0Fail to Reject H0
0
a
Facts:Facts:• The researcher is free to determine the
value of .• The experimenter cannot control , since it
is dependent on the alternate value.• The ideal situation is to have as small as
possible and power close to 1. (Power > .8)• As increases, powerpower increases. (But also
the chance of a type I error has increased!)• Best way to increase power, without
increasing , is to increase the sample sizesample size
A water quality control board reports that water is unsafe for drinking if the mean nitrate concentration exceeds 30 ppm. Water specimens are taken from a well.
Identify the decision:
a) You decide that the water is not safe to drink when in fact it is safe.
Type I Error
A water quality control board reports that water is unsafe for drinking if the mean nitrate concentration exceeds 30 ppm. Water specimens are taken from a well.
Identify the decision:
b) You decide that the water is not safe to drink when in fact it is not safe.
Correct – Power!!
Bottles of a popular cola are suppose to contain 300 ml of cola. A consumer group believes the company is under-filling the bottles. (Assume = 50 with n = 30)
Find the power of this test against the alternative = 296 ml.