Hypothesis Test About the Difference

12.2 Hypothesis Test about the Difference

Large Sample Case ():

Motivating Example:

Objective:

we want to test if the mean scores in two training center are different

: the mean score in the first training center.

: the mean score in the second training center.

We want to test

vs.

( vs.)

with . In addition,

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 .

As is true,

Then,

Thus,

is a sensible statistical procedure. Furthermore, denote

Thus, by dividing on the both sides, the above sensible statistical procedure can be simplified to

In addition,

Therefore, in this example,

Thus, we reject. Also,

we reject based on p-value.

General Case: as and level of significance

As are known, let

(I): vs.

Then,

In addition,

(II): vs.

Then,

In addition,

(III): vs.

Then,

In addition,

As are unknown, let

(I): vs.

Then,

In addition,

(II): vs.

Then,

In addition,

(III): vs.

Then,

In addition,

Example:

Consider the following results for two samples randomly taken from two populations.

Sample 1 / Sample 2
Sample size / 64 / 49
Mean / 1150 / 921
Standard deviation / 90 / 65

Let and be the population means.

(a)For , test using the classical hypothesis test.

(b)For , please use p-value to test .

(c)For , please use the confidence interval method to test the hypothesis .

[solution:]

(a)

Then,

Therefore, we reject .

(b)

Therefore, we do not reject .

(c)

A 95% confidence interval for is

Since , we reject .

Small Sample Case ():

Similar to 11.1, two assumptions are made:

1. Both populations have normal distribution.

2. The variance of the populations are equal ()

Motivating Example:

Objective:

we want to test if the mean project-completion time using the new software package is shorter than using current technology

: the mean project-completion time using the current technology

: the mean project-completion time using the new software package.

We want to test

vs.

( vs.)

with . In addition,

Thus,

A sensible statistical procedure would be

where is some constant. The above statistical procedure is equivalent to the following statistical test:

As is true,

where is the sample statistic with possible values .

Then,

Thus,

is a sensible statistical procedure. In addition,

Therefore, in this example,

Thus, we reject.

Also,

we reject based on p-value.

General Case: as and level of significance

(I):

vs.

Then,

In addition,

(II):

vs.

Then,

In addition,

(III):

vs.

Then,

In addition,

Example:

Consider the following results for two samples randomly taken from two normal populations with equal variance

Sample 1 / Sample 2
Sample size / 10 / 12
Mean / 48 / 44
Standard deviation / 9 / 8

(a) Test vs. at using the classical hypothesis test.

(b) Test vs. at using p-value.

(d) At 95% confidence, how many data would have to be taken to provide an interval with length 6 given equal sample sizes in two populations?

[solution:]

(a) .

Then,

Thus,

Therefore, we reject .

(b)

Therefore, we reject .

Since , we do not reject .

(d)

As sample sizes are large and equal sample sizes () in two populations, the confidence interval for is

The length of the confidence interval is . Therefore,

Therefore, and total 124 data need to be taken.

Online Exercise:

Exercise 12.2.1

Exercise 12.2.2