AMS 102.7 Spring 2006
Jingyu Zou
Elements of Statistics
Lecture Notes # 3
Today
• How to form a decision rule.
• Direction of extreme. Most extreme value. Cutoff value/critical value. Rejection region.
Frequency plot.
• Calculation of α and β based on decision rules.
1
How to form a decision rule
Definition 1.1 A Decision rule is a formal rule that states, based on the data obtained, when to
reject the null hypothesis H0 . Generally, it specifies a set of values based on the data to be collected,
which are contradictory to the null H0 and which favor the alternative hypothesis H1 .
In order to propose a decision rule, we need several more definitions.
Definition 1.2 The Direction of extreme corresponds to the position of the values that are more
likely under the alternative hypothesis H1 than under the null hypothesis H0 . If the larger values
are more likely under H1 than under H0 , then the direction of extreme is said to be one-sided to
the right. Similarly, we can define direction of extreme one-sided to the left or two-sided.
Note that it may be possible that we are unable to identify a direction of extreme.
1.1
Examples
Consider two bags each containing vouchers of the same size and shape. The only things that differ
between the two bags are the face values and their frequencies of vouchers.
Bag A : -$1000,1; $10,7; $20,6; $30,2; $40,2; $50,1; $60,1
Bag B : $10,1; $20,1; $30,2; $40,2; $50,6; $60,7; $1000,1
(1) Could you calculate the total values of Bag A and B? (-$560 and $1890)
You will be shown only one bag. Pick one voucher from it. And decide whether you want to
keep this bag or the other one. You are required to receive the sum of the face value of the voucher
in your bag. Of course you want to take the other one if you convince yourself the current one is
Bag A.
(2) H0 : The shown bag is A. H1 : The shown bag is B.
(3) What are the Type I and II errors and what’re their consequences? (Both consequences are
that you have to pay $560.)
1
Now we want to develop a more formal decision rule based on the possible values that you could
select from the shown bag.
Face Value Chance under H0 Chance under H1
-$1000
1/20
0
$10
7/20
1/20
$20
6/20
1/20
$30
2/20
2/20
$40
2/20
2/20
$50
1/20
6/20
$60
1/20
7/20
$1000
0
1/20
It is obvious that it is the larger vouchers values that are extreme under the null hypothesis,
we say the direction of extreme for this scenario is to the right.
A similar example with direction of extreme to the left can be obtained by reversing the ordering
of chances in the above table. Be sure to make changes to both two columns.
2
Rejection Region
Definition 2.1 Definition: The value under the null hypothesis H0 that is least likely, but at the
same time is very likely under H1 , is called the most extreme value.
Note: it’s possible that it may not be possible to find the most extreme value. Or say there is
no such most extreme value.
Example : Draw two frequency plots of the first example, one for Bag A, and one for Bag B.
Questions: What are the most extreme values under H0 ? Which one is very likely to happen under
H1 ? That is the most extreme value. ( $60)
Definition 2.2 A rejection region is a the set of values for which you would reject the null
hypothesis H0 .
A cutoff value, or critical value, is a value that marks the starting point of a set of values
that comprise the rejection region.
After a rejection region is determined. The decision rule would be: if the observed data is in
the rejection region, then reject H0 . If not, we fail to reject H0 .
In the above example, we choose the most extreme value as the cutoff value and formulate our
rejection region and corresponding decision rule : Reject H0 if your selected voucher is ≥ $60.
(reject H0 if you select a $60 or a $1000 voucher)
Note that it is not always the best way that we choose most extreme value as the cutoff value.
Or else the world is simple. It is just acting as a reasonable approach to get started.
2
3
Calculating the level of significance and power of the test
α = chance of rejecting H0 when H0 is true = chance of selecting a $60 or $1000 voucher from Bag
1
A = 20
.
β = chance of selecting a -$1000, $10, $20, $30, $40, or $50 voucher from Bag B = 12
20
So the power of the test is 0.4
This is a very large value of β. So our chance of committing a Type II error is very high if
following this decision rule.
Question: Why do we have such large value of β and could you suggest an alternative way of
determining rejection region with smaller β.
Answer: Rejection region is too small, only containing two values. One way is to enlarge then
rejection region.
3