Download A brief summary of the statistics used when testing for the mean of a distribution

Summary Of Test Statistis
For brevity, we onsider only tests involving the mean(s) of normal random
variables, assuming we are always dealing with simple random samples.
1
Testing for the mean of a population
In general, the variane will be unknown, hene we use the following statisti
√ x−µ
n
s
where
and
the
µ
x
is the sample mean,
s
is the sample standard deviation,
n
is the ount,
E = √sn ,
is the true mean, as speied in the null hypothesis. Dening
standard error, this an also be written as
x−µ
E
This value (the
t-sore ) is (under the Null Hypothesis) distributed aording
It will be heked against the ritial
to a (Student) tn−1 distribution.
value(s)
as determined by the signiane level
tailed or two-tailed (one tail:
1.1
tn−1,α ,
two tails:
α and whether
tn−1,α/2 ).
the test is one-
Testing two paired samples
Two paired samples are handled as before, with the adjustment that the data
we will be alulating with are the
2
dierenes
between the paired data.
Testing Proportions
The null hypothesis speies an assumed true proportion p, implying a true
p(1−p)
. Hene the statisti (assuming the sample is large enough to
n
justify a normal approximation) is
variane of
pb − p √
p
n
p(1 − p)
(p
b is, as usual, the sample mean, i.e., the proportion of suesses in the sample)
whih is (essentially) distributed as a
standard normal.
Hene its value will
be ompared with the ritial value(s) of he standard normal distribution as
α,
zα/2 )
determined by the signiane level
two-tailed (one tail:
zα ,
two tails:
1
and whether the test is one-tailed or
3 Testing two independent samples
3
2
Testing two independent samples
Again, the varianes are generally unknown.
3.1
Varianes unknown but assumed to be equal
The test statisti,
E=
(x1 and
anes )
x2
s
x1 − x2 − (µ1 − µ2 )
E
(n1 − 1) s21 + (n2 − 1) s22
n1 + n2 − 2
are the two sample means, and
s21
r
and
1
1
+
n1
n2
s22
are the two sample
vari-
is, under our usual assumptions, and under the null hypothesis that
µ1 − µ2 is the dierene of the true means, distributed aording to a
tn1 +n2 −2 Student distribution. From this point on we proeed as in setion
1.
3.2
Varianes unknown and assumed to be unequal
This is an approximate test, as the test statisti
x1 − x2 − (µ1 − µ2 )
E
with
E=
s
s2
s21
+ 2
n1
n2
has a distribution unt for our usual methods, sine its spei form atually
depends on the (unknown) values of the varianes. However, it has been heked
that treating it as if it was distributed aording to a tmin(n1 −1,n2 −1)
Student distribution produes aeptable results. A more rened hoie for
the number of degrees of freedom (but the test is still only approximate, as the
exat distribution is not of STudent type) is usually implemented in software,
but is impratial for table-based work:
1
n1 −1
s22
s21
n1 + n2
2 2
s
1
n1
+
2
1
n2 −1
s22
n2
2
Again, with these hoies we an now proeed as in setion 1.