Therefore a one tailed test 2 ࢻ 3 ݔ for this one

: (therefore a one ‐ tailed test)
2) ࢻ ൌ
3) ݔ ൌ
For this one ‐ tailed test ܺ counts the number of ‘overs’
because ࡴ
૚
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 ࡴ
૙
and ࡴ
૚
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 ሺ3݀݌ሻ from Excel
This is a one ‐ tailed test so we should not double ܲሺܺ ൒ 6ሻ
p ‐ value ൌ 0.377 ሺ3݀݌ሻ
_ _
+
_
+
23.1; 27.4; 29.2; 25.8; 30.0;
+ + + +
_
41.2; 31.3; 28.4; 29.5; 19.4
45

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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 ሺ3݀݌ሻ from Excel
(one ‐ tailed test so do not double)
p ‐ value ൌ 0.377 ሺ3݀݌ሻ
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

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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 ࡴ
૙
and ࡴ
૚
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

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18
ܺ~ܤ݅݊ ݊′, ݌
ܺ~ܤ݅݊ 9,0.5
ܲ ܺ ൒ 7 ൌ 0.0898 ሺ4݀݌ሻ 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.