Comparing Population Means and Variances.
Testing for the difference between means in two populations.
Given two independent random samples X
1i
, i = 1,.
.,N
1
and X
2j
, j=1,.
.,N
2
(the
independence is both between and within the samples) from two populations which are
respectively N(!
1
,"
1
2
) and N(!
2
,"
2
2
) the sample means, which will be independent of
each other,
will be distributed:
22
12
11
2 2
,
,;
XN
NN
±±
²²
³´
³
µ
¶·
¶
¸¹
¸
´
·
¹
(Note that if the distributions are not normal the sample means will still be
approximately normal by the Central Limit Theorem). Since it is a property of the normal
distribution that sums and differences of normal random variables are also normal with a
variance equal to the sum of the variances in both cases (we shall demonstrate this in a
later chapter) it follows that:
1
2
,
XX
N
µº
º
»
±
which, when the variances are known, can be used to test hypotheses about !
1
 !
2
in a
fashion similar to the tests for a single population mean which were outlined in course
notes 5. This follows since transforming to a Z variable we have:
¼½
1
2
(0,1)
ZN
µº
º
º
¾
»
±
which for hypothesized values of !
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '11
 Kambourov
 Normal Distribution, Standard Deviation, Variance, Probability theory, probability density function, n1

Click to edit the document details