Solution to: QSCI 482: LAB
for Topic 4
Here’s another set of
data regarding color-shape distribution of green-yellow, round-wrinkled peas.
Under Mendelian genetics, the distribution of the four categories
(in the order listed below) should be: 9/16, 3/16, 3/16, 1/16; or equivalently
9:3:3:1.
Having alread rejected this null hypothesis at the 0.05 level of
significance (observed chi-square test statistic = 8.01, use partitioning to
show why we rejected. (Are all categories “off” the null hypothesis? Or only some?) Part of the table is not significantly
different from Mendelian predictions; the other part
of the table is. The chi-square statistics from the parts of the table should
add up to a number close to 8.01, the original chi-square (and the df should add up also). Remember,
in subdividing chi-square tables, for df=1 do NOT use
the continuity correction.
Here's the data that
you have:
COLOR OBSERVED COUNT EXPECTED COUNT RESIDUAL
STD. RESIDUAL
Yellowround 135 140.625 -5.625 -0.474
Yellowwrinkled 36 46.875 -10.875 -1.588
Greenround 56 46.875 9.125 1.333
Greenwrinkled 23 15.625 7.375 1.866
Total 250 250 0 NA
Note that
there are two negative and two positive standardized residual—a first sign of
what should be grouped together.
Furthermore, the two positive ones are relatively close
together and larger than zero. This
indicates we should try the following two subgroups:
YR,Yw
gR.gW
The
subtests should preserve the relative proportions under Ho.
Subtest 1. Ho: YR:Yw
= 9:3 = 3:1; P(YR) = 3/4, P(Yw) = ¼
Color Obs Exp (O-E)2/E
YR 135 171*3/4=128.25 0.355
Yw 36 171*1/4=42.75 1.066
Total 171 171 1.421<3.841, fail to reject
Subtest
2. Ho: gR:gw = 3:1; P(gR) = 3/4, P(gw=1/4)
Color Obs Exp (O-E)2/E
gR 56 79*3/4=59.25 0.178
gw 23 79*1/4=19.75 0.535
Total 79 79 0.713<3.841, fail to reject
Accept
these groupings, now perform the “summary” test
Ho: YR,Yw:gR,gw = (9+3):(3+1) = 12:4 = 3:1; P(YR,Yw) =
3/4, P(:gR,gw)=1/4
Color Obs Exp (O-E)2/E
YR,Yw 135+36=171 250*3/4=187.5 1.452
gR.gW 56+23=79 250*1/4=62.5 4.356
Total 250 250 5.808>3.841,
Reject
Conclusion: It would seem that given the color of the pea
the wrinkled or round trait follows the proportions expected under Mendelian theory (e.g. within yellow or within green). It may be that it is the color trait that is
not sorting in a Mendelian way.