More About Type I
and Type II Errors
O.J. Simpson trial: the situation
• O.J. is assumed innocent.
• Evidence collected:
size 12 Bruno Magli bloody footprint,
bloody glove, blood spots on white Ford
Bronco, the knock on the wall, DNA
evidence from above, motive(?), etc…
O.J. Simpson trial: jury decisions
• In criminal trial: The evidence does not
warrant rejecting the assumption of
innocence. Behave as if O.J. is innocent.
• In civil trial: The evidence warrants
rejecting the assumption of innocence.
Behave as if O.J. is guilty.
• Was an error made in either trial?
Errors in Trials
Truth
Jury
Decision
“Innocent”
Guilty
Innocent
Guilty
OK
ERROR
ERROR
OK
If O.J. is innocent, then an error was made in the civil trial.
If O.J. is guilty, then an error was made in the criminal trial.
Errors in Hypothesis Testing
Truth
Decision
Null
hypothesis
Do not
reject null
OK
Reject null
TYPE I
ERROR
Alternative
hypothesis
TYPE II
ERROR
OK
Definitions: Types of Errors
• Type I error: The null hypothesis is
rejected when it is true.
• Type II error: The null hypothesis is not
rejected when it is false.
• There is always a chance of making one of
these errors. We’ll want to minimize the
chance of doing so!
Example: Grade inflation?
Is there evidence to suggest the mean GPA of
college undergraduate students exceeds 2.7?
H0: μ = 2.7
HA: μ > 2.7
Random sample
of students
Data
n = 36
s = 0.6
and computeX
Decision Rule
Set significance level α = 0.05.
If p-value < 0.05, reject null hypothesis.
Let’s consider what our conclusion is based upon
different observed sample means…
If X  2.865
Reject null since p-value is (just barely!) smaller then 0.05.
If X  2.95
Reject null since p-value is smaller then 0.05.
If X  3.00
Reject null since p-value is smaller then 0.05.
Alternative Decision Rule
• “Reject if p-value  0.05” is equivalent to “Reject
if the sample average, X-bar, is
larger than 2.865”
• X  2.865 is called “rejection region.”
Type I Error
Minimize chance of
Type I error...
• … by making significance level  small.
• Common values are  = 0.01, 0.05, or 0.10.
• “How small” depends on seriousness of
Type I error.
• Decision is not a statistical one but a
practical one.
P(Type I Error) in trials
• Criminal trials: “Beyond a reasonable
doubt”. 12 of 12 jurors must unanimously
vote guilty. Significance level  set at
0.001, say.
• Civil trials: “Preponderance of evidence.”
9 out of 12 jurors must vote guilty.
Significance level  set at 0.10, say.
Example:
Serious Type I Error
• New Drug A is supposed to reduce diastolic
blood pressure by more than 15 mm Hg.
• H0: μ = 15 versus HA: μ > 15
• Drug A can have serious side effects, so
don’t want patients on it unless μ > 15.
• Implication of Type I error: Expose patients
to serious side effects without other benefit.
• Set  = P(Type I error) to be small  0.01
Example:
Not so serious Type I Error
• Grade inflation?
• H0: μ = 2.7 vs. HA: μ > 2.7
• Type I error: claim average GPA is more
than 2.7 when it really isn’t.
• Implication: Instructors grade harder.
Students get unhappy.
• Set  = P(Type I error) at, say, 0.10.
Type II Error and Power
• Type II Error is made when we fail to reject
the null when the alternative is true.
• Want to minimize P(Type II Error).
• Now, if alternative HA is true:
– P(reject|HA is true) + P(not reject|HA is true) =1
– “Power” + P(Type II error) = 1
– “Power” = 1 - P(Type II error)
Type II Error and Power
• “Power” of a test is the probability of
rejecting null when alternative is true.
• “Power” = 1 - P(Type II error)
• To minimize the P(Type II error), we
equivalently want to maximize power.
• But power depends on the value under the
alternative hypothesis ...
Type II Error and Power
(Alternative is true)
Power
• Power is probability, so number between 0
and 1.
• 0 is bad!
• 1 is good!
• Need to make power as high as possible.
Maximizing Power …
• The farther apart the actual mean is from
the mean specified in the null, the higher the
power.
• The higher the significance level , the
higher the P(Type I error), the higher the
power.
• The smaller the standard deviation, the
higher the power.
• The larger the sample, the higher the power.
That is, factors affecting power...
• Difference between value under the null and
the actual value
• P(Type I error) = 
• Standard deviation
• Sample size
Strategy for designing a
good hypothesis test
• Use pilot study to estimate std. deviation.
• Specify . Typically 0.01 to 0.10.
• Decide what a meaningful difference would
be between the mean in the null and the
actual mean.
• Decide power. Typically 0.80 to 0.99.
• Use software to determine sample size.
Using JMP to Determine Sample Size –
DOE > Sample Size and Power
Using JMP to Determine Sample Size –
One Sample Mean
P(Type I Error ) = 
Error Std Dev =
“guessimate” for standard
deviation (s or s)
Enter values for one or two of the
quantities:
1) Difference to detect
d  |Ho mean – HA mean|
= |m0 - m|
2) Sample Size = n
3) Power = P(Reject Ho|HA true)
=1-b
Using JMP to Determine Sample Size –
DOE > Sample Size and Power
For
  .05, d  .20, s = .60
and leaving Power and Sample
Size empty we obtain a plot of
Power vs. Sample Size (n).
Here we can see:
Power  .80  n  75 students
Power  .90  n  96 students
Power  .95  n  120students
Using JMP to Determine Sample Size –
DOE > Sample Size and Power (JMP Demo)
If sample is too small ...
• … the power can be too low to identify
even large meaningful differences between
the null and alternative values.
– Determine sample size in advance of
conducting study.
– Don’t believe the “fail-to-reject-results” of a
study based on a small sample.
If sample is really large ...
• … the power can be extremely high for
identifying even meaningless differences
between the null and alternative values.
– In addition to performing hypothesis tests, use a
confidence interval to estimate the actual
population value.
– If a study reports a “reject result,” ask how
much different?
The moral of the story
as researcher
• Always determine how many measurements
you need to take in order to have high
enough power to achieve your study goals.
• If you don’t know how to determine sample
size, ask a statistical consultant to help you.
The moral of the story
as reviewer
• When interpreting the results of a study,
always take into account the sample size.