AMS 102.7 Spring 2006 Jingyu ZouElements 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 toreject 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 morelikely under the alternative hypothesis H1 than under the null hypothesis H0. If the larger valuesare more likely under H1 than under H0, then the direction of extreme is said to be one-sided tothe 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 differbetween 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 voucherin your bag. Of course you want to take the other one if you convince yourself the current one isBag 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.)

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Now we want to develop a more formal decision rule based on the possible values that you couldselect 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 1/20It 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 thesame 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 isno 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 underH1? 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 nullhypothesis H0.

A cutoff value, or critical value, is a value that marks the starting point of a set of valuesthat comprise the rejection region.

After a rejection region is determined. The decision rule would be: if the observed data is inthe 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 ourrejection 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.

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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 BagA = 1

20 .

β = chance of selecting a -$1000, $10, $20, $30, $40, or $50 voucher from Bag B = 1220

So the power of the test is 0.4This 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 ofdetermining rejection region with smaller β.

Answer: Rejection region is too small, only containing two values. One way is to enlarge thenrejection region.

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