10.1. Estimation of the Difference between the Means

Let

: the mean of population 1, : the mean of population 2,

: the variance of population 1, : the variance of population 2,

the random sample from population 1

the random sample from population 2.

Then,

the sample mean of population 1.

the sample mean of population 2

and

: the standard deviation of population 1.

: the standard deviation of population 2.

The point estimate of :

Important Properties of :

the sample statistic with possible value

the sample statistic with possible value

Then,

and

Large Sample Case ():

Sampling Distribution of ():

.

confidence interval ():

As are known,

is a confidence interval estimate of the population difference .

As are unknown,

is a confidence interval estimate of the population difference .

Small Sample Case ():

Two assumptions are made:

Both populations have normal distribution.

The variance of the populations are equal ()

Pooled estimate of and :

The pooled estimate of , denoted by ,

is a weighted average of the two sample variance and .

The estimate of

is

.

confidence interval ():

is a confidence interval estimate of the population difference .

Example 1:

Let

: the mean balance of checking account in Chekry Grove bank

: the mean balance of checking account in Beechmont bank.

Please find a 90% confidence interval for .

[solution:]

.

Thus,

.

Then, a 90% confidence interval for is

JavaStatSoft:

Z interval:

Statistics -> Estimation -> Two Samples -> Means -> Z Interval

T interval:

Statistics -> Estimation -> Two Samples -> Means -> T Interval