:
(therefore a
one
‐
tailed
test)
2)
ࢻ ൌ
3)
ݔ ൌ
For this one
‐
tailed test
ܺ
counts the number of ‘overs’
because
suggests there should be more of these
Denote each age by a plus or
minus sign, depending on
whether it’s above or below the
hypothesised median value
23.1; 27.4; 29.2; 25.8; 30.0;
41.2; 31.3; 28.4; 29.5; 19.4
1)
Form two opposing
hypotheses
and
2)
Decide on a
significance level
ࢻ
for the
test
3)
Calculate a
test statistic
x
4)
Find the
p
‐
value
using the test statistic
5)
Either
reject
or
fail to reject
the
null
hypothesis
6)
Make a
conclusion
about the
population
44
4)
For the
p
‐
value we require the probability that
X
is at least as
extreme as 6 out of 10 mothers being over the age of 27.5
ܺ~ܤ݅݊ሺ10,0.5ሻ
ܲ
ܺ 6
ൌ ?
ܲ
ܺ 6
ൌ 0.377 ሺ3݀ሻ
from Excel
This is a
one
‐
tailed test
so we should
not
double
ܲሺܺ 6ሻ
p
‐
value
ൌ 0.377 ሺ3݀ሻ
_
_
+
_
+
23.1; 27.4; 29.2; 25.8; 30.0;
+
+
+
+
_
41.2; 31.3; 28.4; 29.5; 19.4
45

21/11/2018
16
Bar graph showing the probabilities for
ܺ~ܤ݅݊ሺ10,0.5ሻ
46
1
)
ܪ
: ݉݁݀݅ܽ݊ ൌ 27.5
ܪ
ଵ
: ݉݁݀݅ܽ݊ 27.5
(so one
‐
tailed test)
2)
ࢻ ൌ 0.05
3)
ݔ ൌ 6
overs
ܺ
counts the number of ‘overs’ because
ܪ
ଵ
suggests there should be
more of these
4)
For
p
‐
value we require prob. that
X
is at least as extreme as 6 out of 10
ܺ~ܤ݅݊ሺ10,0.5ሻ
so
ܲ
ܺ 6
ൌ 0.377 ሺ3݀ሻ
from Excel
(one
‐
tailed test so do not double)
p
‐
value
ൌ 0.377 ሺ3݀ሻ
5)
p
‐
value
ࢻ ሺ0.377 0.05ሻ
so do not reject
H
6)
Conclude that we do not have sufficient evidence of an increase in
the median age above the 1990 figure of 27.5 years
_
_
+
_
+
23.1; 27.4; 29.2; 25.8; 30.0;
+
+
+
+
_
41.2; 31.3; 28.4; 29.5; 19.4
47
Clearly show all six
steps:
Summary of how to calculate both test statistic
and
p
‐
value for a Sign Test
H
1
:
Test statistic
p
‐
value
median
m
0
ݔ
= no. of +
(overs)
ܲሺܺ ݔሻ
median
൏
m
0
ݔ
= no. of
‐
(unders)
ܲሺܺ ݔሻ
median
്
m
0
ݔ
= maximum of (no. of +, no. of
‐
)
2 ൈ ܲሺܺ ݔሻ
H
0
:
median
ൌ
m
0
if it’s a two
‐
sided test, need to find the probabilities in
both
tails
48

21/11/2018
17
Useful mnemonic for remembering what conclusion the
p
‐
value leads us to:
‘if
p
‐
value is low, the null must go!’
(lower than alpha)
49
Hypothesis testing in one sentence!
50
•
Scores have been collected in a psychological test as
follows. Use a Sign Test to determine if there is
evidence at the 10% level of significance that the
median has changed from 29 (the median value when
the test was last used)
36;
43;
9;
37;
48;
29;
31;
24;
40;
37
1)
ܪ
:
ܪ
ଵ
:
2)
ߙ ൌ
3)
ݔ ൌ
4)
ܲ
ܺ
ൌ
so
p
‐
value
ൌ
5) Compare
p
‐
value to
ߙ
and then do or do not reject
ܪ
6) Conclusion:
51
1)
Form two opposing
hypotheses
and
2)
Decide on a
significance
level
ࢻ
for the test
3)
Calculate a
test statistic
x
4)
Find the
p
‐
value
using the
test statistic
5)
Either
reject
or
fail to reject
the
null
hypothesis
6)
Make a
conclusion
about
the population
Sign Test Example 3

21/11/2018
18
ܺ~ܤ݅݊
݊′,
ܺ~ܤ݅݊
9,0.5
ܲ
ܺ 7
ൌ 0.0898 ሺ4݀ሻ
from Excel
p
‐
value
ൌ 2 ൈ 0.0898 ൎ 0.18
(two
‐
tailed test)
See ‘Help with Excel #12’ for
more detail
52
•
Scores have been collected in a psychological test as follows.