Decision Limits Based on the p Value

3.4.4 Decision Limits Based on the

p p

ValueValue

In the preceding section, the use of the z value to make a hypothesis test decision was described. The z statistic was calculated from the sample mean and then compared to critical z values indicated by ± za /2. Then, the decision to accept or reject the null

hypothesis test was based on where the z statistic fell with respect to the acceptance interval. Although this method of making a decision is correct, it requires that we keep a table of critical za /2values handy. As we consider more hypothesis tests for different

conditions and statistics, there will be many new probability distributions that will have their own tables of critical values. Instead of dealing with so many tables of critical values, there is a simple, concise, and universal way of making hypothesis test decisions that is very important to DOE. This method involves the calculation of a quantity called the p value.

The p value for a hypothesis test is calculated from the experimental test statistic under the assumption that the null hypothesis is true. The pvalue is related to the tail area under the distribution that characterizes the test statistic relative to the specific value of the test statistic obtained from the sample data. pvalues are compared directly to

a

so, to be fair, when

a

is split between two tails of the– xdistribution, the p value also gets contributions from both tails. Since the normal distribution that characterizes the distribution of – xs is symmetric, the p value is just twice the normal distribution tail area relative to the experimental test statistic– x or its corresponding z value. That is:

(3.6) 1

− = − < < +

p

Φ(

z _{p /2} z z_{p/ 2}

)

46 46 Chapter Three 0.005 0.005

Reject H H 0 Accept H H 0 Reject H H 0

0.99 80 74.4 78.2 85.6 –0.82 x x 0 –2.575 2.575 z z Figure 3.6

where

(3.7) Figure 3.7 shows the relationship between – x, its corre sponding z value, and the p value.

Once the p value is known, the decision to accept or reject the null hypothesis is made using the following rules:

• If p <

a

then reject H 0.

• If p >

a

then accept H 0or reserve judgment.

Instead of memorizing these rules, just remember that when the p value is very small, the test statistic– xmust fall far away from

m

0so we should reject H 0:

m

=

m

0, and when the

pvalue is very large, the test statistic must fall close to

m

0so we should accept H 0or

reserve judgment.

A helpful way to think about the p value is to recognize that it measures how unusual the experimental test statistic is given that H 0_{was true. The p value’s size cor-}

responds to the probability of obtaining the observed test statistic, or something even more unusual, if H 0was true. This means that when p is very small, the observed sta-

tistic would be a rare event if H 0was true, so H 0is more likely to be false. And when p

is large, a value like the observed statistic is an expected result if H 0was true, so we

should accept H 0or reserve judgment.

Example 3.6 Example 3.6

Perform the hypothesis test for Example 3.4 using the p value method.

z p x x /2 0

= −µ

σ

Inferential Statistics 4747 p p /2

Reject H H 0 Accept H H 0 Reject H H 0

1 –p p p p /2 m 0 x x x x 0 – z z _{p p /2} z z _{p /2p} z z Figure 3.7

Solution: From Example 3.5 we have z p /2= –0.82 so the p value is determined from:

and p= 1 – 0.59 = 0.41.The p value is the shaded area under the curve shown in Figure 3.8. Since we are working with

a

= 0.01 we have ( p = 0.41) > (

a

= 0.01) so we must accept H 0or reserve judgment.

Initially, p values may seem abstract; however, their use is very common, so it is essential to become completely comfortable with their use and interpretation. The reasons that they are so important are:

• The p value provides a clear and concise summary of the significance of the experimental data. There is no need to specify the conditions of the test, the type of data collected, or the statistic used in the test.

• The p value is perfectly general and can be applied to hypothesis tests of every type. Even if you don’t recognize the hypothesis test being used, its pvalue behaves like every other p value.

• In many technical journals, experimental p values are considered to be

important enough to be included prominently in abstracts. A common question after describing an experiment to a knowledgeable experimenter is, “What is the p value?”

• In DOE, especially in computer analyses of DOE problems, the p value is commonly provided for a number of relevant statistics.

1

− = −

p

Φ

( 0 82

< < +

. z

=

0 82. ) 0 59.

48

48 Chapter Three

p p = 0.41

Reject H H 0 Accept H H 0 Reject H H 0

0.59 80 78.2 –0.82 0 0.82 –2.575 2.575 Figure 3.8

There is a trap associated with the use of p values that you must be careful to avoid. Since the decision to accept or reject the null hypothesis is made by comparing the experimental p value to

a

, it is essential that the value of

a

be chosen before the data are collected, and based on economic or business requirements. Otherwise, if you pick

a

after the data are collected and analyzed, you might be tempted to influence the deci- sion to accept or reject H 0with your choice of

a

.* This practice is unethical but quite

common and gives statisticians and statistical methods a bad reputation. Example 3.7

Example 3.7

A sample drawn from a population yields a mean that gives p= 0.014 under the null hypothesis. What decision is made if

a

= 0.05? If

a

= 0.01?

Solution: For

a

= 0.05 we have ( p = 0.014) < (

a

= 0.05) so we must reject H 0. For

a

= 0.01 we have ( p = 0.014) > (

a

= 0.01) so we must accept H 0or reserve judgment.

In document Mathews Paul G. Design of Experiments With MINITAB (Page 61-64)