9.3. One-Tailed Tests about a Population Mean: Large Sample Case

Example 1 (continue):

A sensible statistical procedure would be

where is some constant.

Next Question: how to determine the value of ?

Answer: control (the probability of making type I error) to determine the value of .

Suppose is the predetermined Type I error (the level of significance).

As is true, for ease of explanation, suppose

.

Then,

.

Thus, a sensible statistical procedure would be

with

.

By dividing both sides by , the resulting statistical procedure

Therefore, if then

not reject .

General Case: as and level of significance

As is known, let

.

As is unknown,

.

(I): vs.

Then,

(II): vs.

Then,

Note:

in (I) or in (II) are called critical value.

Example 3:

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

Consider the following hypothesis test

.

Using a critical value, test the hypothesis at the 10% level of significance (classical approach).

[solution:]

Therefore, we reject .

JavaStatSoft:

Z test:

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