Transcript Understanding Power

By Jessica Jorge
Definitions
Type I Error
Incorrectly rejecting a true null hypothesis
Type II Error
Failing to reject the null hypothesis when it
should be rejected
Power
Rejecting the null hypothesis when Ha is true
THIS IS WHAT YOU WANT
Helpful Chart for remembering definitions
Drawing this chart
before doing any power
presentation will be
very helpful!!!
Decision
From
Sample
Reject
Ho:
Fail to
Reject
Ho:
Truth About The Population
Ho: is true
Type I Error
Ha: is true
POWER

1-
Correct
Type II Error

Let’s start at the beginning...
The  level is the predetermined place where Ho
is rejected. That is, if a
observation falls above the
 level, one would consider
it significant, and would
therefore reject Ho.
Ho
The level is very
important for
understanding
Power
The most frequently
used  level is .05.
In this presentation,
 will always equal
.05.

 = .05
Type One Error is denoted by the symbol…. 
Power is found when…
the null hypothesis is rejected and the alternative
hypothesis is true. THIS IS WHAT YOU WANT!!
The most frequently used  level is... … .05
Type Two Error is found when…
we fail to reject Ho and Ha is true
The Ha curve is the second curve used when
looking at power.
Ha
Similarly, the Ha curve is a mound
shaped symmetrical curve, but will have
a different mean than the Ho curve
and sometimes a different standard
deviation.
Note: Unless told otherwise,
assume  is the same for the Ho
and Ha curve.

Also note that the Ha curve
does not have an  level
marked. ONLY the Ho curve
will have an  level.
Let’s put the two curves together…
Ho
Ha
Notice the  level is
marked clearly with
a vertical line.
Drawing this line
clearly is helpful to
visualize what is
happening.
Always place the Ha curve
in accordance to the Ho
curve.
=.05
Another little quiz….
What curve is this? Ho
If the mean of
the Ho curve is
=10, and the
mean of the Ha
curve is =0,
where would
the Ha curve be
placed?
=10
Another little quiz….
What curve is this? Ho
If the mean of
the Ho curve
is =10, and
the mean of
the Ha curve
is =0, where
would the Ha
curve be
placed?
Ha
=0
Ho
=10
You will now see the importance of the  level
Fail to reject Ho
Reject Ho
As previously mentioned,
the  level plays an
important role in Power.
In this illustration we have
added where to reject
and accept Ho
according to the
 level. If an
observation falls to the
left of the  level, we fail
to reject Ho (accept Ho),
and if it falls to the right,
we reject.
Power
The power of this test is the shaded region. Once again, power is found
when we reject Ho and Ha is correct.
Fail to reject Ho
Reject Ho
To find the probability of getting
power, one can find the z-score of
the  level. Because power says
that Ha is true, use the mean and
standard deviation of the Ha
curve to find the z-score. A quick
way to find the z-score is to use
the invNorm function on the TI83.
Ex. invNorm(.05, ,,)
Once the z-score is calculated,
simply use normalcdf to calculate
the probability of finding power.
=0
=10
Type I Error
Fail to accept Ho
Reject Ho
Type one error is the shaded
region and is found when we
fail to accept Ho but Ho is true.
The probability of getting Type I Error will always
be the  level.
To find the probability of getting Type I Error, first
find the z-score of the  level. You can use the
invNorm function of your calculator to do this.
Next, you can use the z-table to find the probability
of Type II Error.
An easier and more accurate way is to use the
normalcdf function on you calculator.
Ex. Normalcdf(lower bound, upper bound, , )
=0
=10
Type II Error
Fail to reject Ho
Reject Ho
Type II Error is found when we fail
to reject Ho based on the sample,
but Ha is true.
To find the probability of getting Type
II Error, find the z-score of the 
level. (Use invNorm, the same function
used in calculating Power and Type I
error.)
Once the z-score is found, simply use
either the z-table to find the p-value
(same as the probability) OR use
normalcdf. If you use normalcdf, be
sure to use the mean and standard
deviation of the Ha curve.
=0
=10
An Example:
If the null hypothesis is Ho: =0 and the alternative hypothesis is Ha:
=1.1
Find the power of the test using =.05 and =.316
1.) Draw the Ho
curve
=.316
2.) Label the appropriate
 level on the Ho curve.
Also label the mean and
standard deviation.
=0
=.05
An Example Continued:
If the null hypothesis is Ho: =0 and the alternative hypothesis is Ha:
=1.1
Find the power of the test using =.05 and =.316
Ho:
Ha:
3.) Draw the Ha curve
4.) In this problem we
want to find Power, so
shade the desired
region
=.316
=0
=.05
=1.1
An Example Continued:
If the null hypothesis is Ho: =0 and the alternative hypothesis is Ha: =1.1
Find the power of the test using =.05 and =.316
Ho:
Ha:
5.) Find the z-score of the 
level.
invNorm(.95,0,.316) = .52
6.) Use normalcdf to
find the probability of
getting Power.
Normalcdf(.52,e99,1.1,.316)=.966
=.316
Conclusion:
=0
=.05
z=.52
=1.1
The probability of
correctly rejecting Ho and
accepting Ha is 96.6%
A couple problems to try on your
own…
From Moore’s Basic Practice of Statistics
1.)You have a SRS of size n=9 from a normal distribution with =1. You wish to test
Ho:  = 0 and Ha:  >0
You decide to reject Ho if x-bar > 0 and to accept Ho otherwise.
a.
Find the probability of Type I error. That is, find the probability that the test rejects Ho when in
fact  = 0
•
Find the probability of Type II error when  = .3.
•
Find the probability of Type II error when  = 1.
The hypothesis are:  = 300 and Ha:  < 300. The sample size is n=6, and the population is assumed to
have a normal distribution with =3. A 5% significance test rejects Ho if z< -1.645, where the test
statistic z is:
Z= x-bar – 300/(3/sqr6)
a.
Find the power of this test against the alternative hypothesis  = 299.
•
Find the power against the alternative  = 295.
•
Is the power against  = 290 higher or lower than the value you found in b? (Don’t actually
calculate that power.) Explain your answer.