9.4. Modern Approach: P-value

Example 1 (continue):

In the coffee example,

vs. , , .

Suppose , the p-value in this case is

(the error we might make < (the allowable error as reject at )

The error we might make by rejecting at is smaller than the allowable error.

Therefore, we reject .

Suppose ,

Since the error we might make by rejecting at is larger than

the allowable error. Therefore, we can not reject .

Note: The conclusion based on the p-value is the same as the one based on the previous

hypothesis testing procedure.

Note: In the above example, let be the sample mean and . Then,

.

For example, as

General Case: as and level of significance

As is known, let

.

As is unknown,

.

(I): vs.

Then,

(II): vs.

Then,

In (a) and (b), we then compare p-value and the predetermined level of significance .

If , then reject . Otherwise, we do not reject .

Example 3 (continue):

A sample with provides a sample mean of 36 and sample standard deviation of 12.

Consider the following hypothesis test

.

Find the p-value. What is your conclusion as ?

[solution:]

.

Since

,

we do not reject .

JavaStatSoft:

Z test:

Statistics -> Tests -> One Sample -> Mean -> Z Test