Chi-Square Goodness of Fit Test
1. The biologist Gregor Mendel wished to cross round yellow pea plants with wrinkled green pea plants,
thereby obtaining plants bearing peas in one of four categories. Mendel’s theory claimed that the expected
frequencies of these characteristics are in the proportion 9 round yellow (RY): 3 round green (RG): 3 wrinkled
yellow (WY): 1 wrinkled green (WG).
When Mendel actually crossed a random sample of round yellow pea plants and wrinkled green pea plants he
obtained the data shown in the accompanying table.
Category
Round yellow
Round green
Wrinkled yellow
Wrinkled green
Total
Frequency
315
108
101
32
556
Do the data differ significantly from what Mendel predicted?
P:
We are interested in the distribution of pea plant category of all crossbred round yellow pea plants and
wrinkled green pea plants.
H:
H0: The proportions of pea plants are as follows: round yellow 9/16 (or 0.5625), round green 3/16 (or
0.1875), wrinkled yellow 3/16 (or 0.1875), wrinkled green 1/16 (or 0.0625).
Ha: At least two of these proportions are incorrect.
A:
Random – Mendel used a random sample
Independent – We can assume there are at least 5,560 pea plants in the world
Large Counts – All the expected numbers are at least 5:
Round yellow = 556 • 9/16 = 312.75
Round green = 556 • 3/16 = 104.25
Wrinkled yellow = 556 • 3/16 = 104.25
Wrinkled green = 556 • 1/16 = 34.75
N:
We will perform a Chi-Square Goodness of Fit Test.
T:
𝜒2 =
(315  312.75) 2 (108  104.25) 2 (101  104.25) 2 (32  34.75) 2
= .4700



312.75
104.25
104.25
34.75
df = 4 – 1 = 3
O:
P-value > .25 using the table
OR
P-value = 𝜒 2 cdf(.47,100000,3) = .9254
M:
Because the P-value is not significant at the 5% level, we fail to reject the null hypothesis.
S:
There is not strong evidence to conclude that Mendel’s theory is incorrect.
2. Biologists wish to mate two random fruit flies having genetic makeup RrCc, indicating that it has one
dominant gene (R) and one recessive gene (r) for eye color, along with one dominant (C) and one recessive (c)
gene for wing type. Each offspring will receive one gene for each of the two traits from both parents. The
following table (Punnett square) shows the possible combinations of genes received by the offspring.
RC
RRCC
RRCc
RrCC
RrCc
RC
Rc
rC
rc
Rc
RRCc
RRcc
RrCc
Rrcc
rC
RrCC
RrCc
rrCC
rrCc
rc
RrCc
Rrcc
rrCc
rrcc
Any offspring receiving an R gene will have red eyes, and any offspring receiving a C gene will have straight
wings. So based on this Punnett square, the biologists predict a ratio of 9 red-eyed, straight wing: 3 redeyed, curly wing: 3 white-eyed, straight wing: 1 white-eyed, curly wing offspring.
In order to test their hypothesis about the distribution of offspring, the biologists mate the fruit flies. Of 200
offspring, 101 had red eyes and straight wings, 41 had red eyes and curly wings, 48 had white eyes and straight
wings, and 10 had white eyes and curly wings. Do these data differ significantly from what the biologists have
predicted? Follow PHANTOMS.
P:
We are interested in the distribution of genetic makeup of all offspring of two RrCc fruit flies.
H:
H0: The proportions of fruit flies are as follows: red-eyed, straight wing 9/16 (or 0.5625), red-eyed,
curly wing 3/16 (or 0.1875), white-eyed, straight wing 3/16 (or 0.1875), white-eyed, curly wing 1/16 (or
0.0625).
Ha: At least two of these proportions are incorrect.
A:
Random – Two random fruit flies were used
Independent – We can assume there are at least 2,000 fruit flies in the world
Large Counts – All the expected numbers are at least 5:
Red-eyed, straight wing = 200 • 9/16 = 112.5
Red-eyed, curly wing = 200 • 3/16 = 37.5
White-eyed, straight wing = 200 • 3/16 = 37.5
White-eyed, curly wing = 200 • 1/16 = 12.5
N:
We will perform a Chi-Square Goodness of Fit Test.
T:
𝜒2 =
(101  112.5) 2 (41  37.5) 2 (48  37.5) 2 (10  12.5) 2
= 4.94



112.5
37.5
37.5
12.5
df = 4 – 1 = 3
OR
P-value = 𝜒 2 cdf(4.94,100000,3) = .1762
O:
P-value is between .15 and .20 using table
M:
Because the P-value is not significant at the 5% level, we fail to reject the null hypothesis.
S:
There is not strong evidence to conclude that the distribution of genetic makeup is incorrect.