Chapter 13
Nonparametric Statistics
132
Sign Test
1. The sign test is “nonparametric” or “distributionfree” because it does not require the data to
come from a particular distribution defined in terms of certain parameters.
2. The procedure in this section is called the “sign” test because it reduces the data to plus and
minus signs in order to perform the analysis.
3. The alternative hypothesis is that the proportion of girls is greater than 0.5, but the sample
proportion of 39/(172+39) = 39/211 is less than 0.5.
Without doing any mathematics, it is
apparent that the conclusion must be that there is not enough evidence to support the claim that
the method increases the likelihood of girl births.
In order for the data to support the claim the
number of girl births would have to be significantly greater than (½)·(211) = 105.5.
4. The efficiency of the sign test is 0.63.
This means that for normal populations, the sign test
requires a sample of size n=100 to identify departures from the null hypothesis that the
appropriate corresponding
parametric test could identify with a sample of size n=63.
5. 13 +’s and 1 –’s
n = 14
≤
25; use C.V. = 2 from Table A7
Since min(13,1) = 1
≤
2 = C.V., reject the hypothesis of no difference.
Since there were more +’s, conclude that the first variable has the larger scores.
6. 5 +’s and 7 –’s
n = 12
≤
25; use C.V. = 2 from Table A7
Since min(5,7) = 5 > 2 = C.V., do not reject the hypothesis of no difference.
There is not sufficient evidence to reject the hypothesis of no difference.
7. 360 +’s and 374 –’s
n = 734 > 25; use C.V. = 1.96 from the z table
z = [(x+0.5) – n/2]/
[
n/2]
= [360.5 – 367]/
[
734/2]
= 6.5/13.546 = 0.480
Since 0.480 > 1.96, do not reject the hypothesis of no difference.
There is not sufficient evidence to reject the hypothesis of no difference.
8. 512 +’s and 327 –’s
n = 839 > 25; use C.V. = 1.96 from the z table
z = [(x+0.5) – n/2]/
[
n/2]
= [327.5 – 419.5]/
[
839/2]
= 92/14.483 = 6.35
Since 6.35 < 1.96, reject the hypothesis of no difference.
Since there were more +’s, conclude that the first variable has the larger scores.
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442
CHAPTER 13
Nonparametric Statistics
NOTES FOR THE REMAINING EXERCISES IN THIS SECTION.
(1)
FOR
n
≤
25.
Table A7 gives only x
L
, the lower
critical value for the sign test.
And so the text
lets x be the smaller
of the number of +’s or the number of –’s, and warns the reader to use
common sense to avoid concluding the reverse of what the data indicates.
But the problem’s
symmetry means that the upper critical value is x
U
= n – x
L
and that
μ
x
= n/2, the natural expected
value of x when H
o
is true.
For completeness, this manual indicates those values whenever using
the sign test – and uses a normal curve for illustration, even though the distribution of x is discrete.
Letting x always be the number of +’s is an alternative approach that maintains the natural
agreement between the alternative hypothesis and the critical region – and is consistent with the
logic and notation of parametric tests.
Many sign test software programs use this approach.
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 Spring '11
 Dr.Kalluri
 Statistics, Nonparametric statistics, sufficient evidence

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