Download Chi squared test

Chi squared test
In an examination, this is likely to be presented to you in one of three different
ways:
1. Results of an investigation that have to be worked through using the
formula and a table set out for completion.
2. Judging the significance of results given the figure for 2.
3. Selecting the chi squared test as the appropriate way to assess results
from an investigation.
Here is an example to show how the chi squared test is used.
Pure bred tomato plants with cut leaves and purple stems were crossed with
pure bred plants with potato leaves and green stems. All the F1 generation
had cut leaves and purple stems. These F1 plants were testcrossed against
tomato plants showing the recessive phenotype – potato leaves and green
stems. The F2 offspring showed the following numbers of plants in each of
four phenotypes
purple, cut
70
purple, potato green, cut
91
86
green, potato
77
The ratio of phenotypes expected in the testcross offspring of a dihybrid cross
such as this is 1:1:1:1.
Use the chi squared test (2), and the test of probabilities shown in the table,
to find out if these results are significantly different from the expected results.
2 = E
E 
categories


sum of….
observed value
E = expected value
O
E
O-E
(O-E)2
(O-E)2 / E
purple cut
purple potato
green cut
green potato
2 =
totals
degrees of
Distribution of 2
1
freedom
1
2
3
4
0.10
2.71
4.61
6.25
7.78
0.05
3.84
5.99
7.82
9.49
probability, p
0.02
5.41
7.82
9.84
11.67
0.01
6.64
9.21
11.35
13.28
0.001
10.83
13.82
16.27
18.47
What is the probability for the value of 2 that you have calculated?
(Hint: better to give the range of probabilities for 2)
State what conclusions can be drawn from the probability.
That was a typical question.
It would be good to be given the null hypothesis in these questions. But the
null hypothesis can be derived from the information given.
What is the null hypothesis for this investigation involving inheritance in
tomatoes?
What is the underlying reason for this expectation?
Here is a more detailed table of probabilities that you can use while revising
the chi squared test.
distribution of 2
degrees
of
freedom
increasing values of p
decreasing values of p
probability, p
1
2
3
4
0.99
0.90
0.00016 0.016
0.02
0.10
0.12
0.58
0.30
1.06
p > 0.90
result is
‘dodgy’
= too good!
0.50
0.46
1.39
2.37
3.36
0.10
2.71
4.61
6.25
7.78
p > 0.05
result is not
significantly
different from
expected
outcome
0.05
3.84
5.99
7.82
9.49
0.02
5.41
7.82
9.84
11.67
p < 0.05
result is
significantly
different from
expected
outcome
0.01
6.64
9.21
11.35
13.28
p < 0.01
highly
significant
0.001
10.83
13.82
16.27
18.47
p < 0.001
very
highly
significant
Ruling lines at p = 0.05 helps to find out if the 2 value is significant or not.
2
Some results have been analysed using the chi squared test. What can be
concluded from the results?
1. The F1 generation of tomato plants described earlier were interbred. The
total number of plants in each class was as follows:
purple,
cut
972
purple,
potato
325
green,
cut
334
green,
potato
93
2 = 2.39
what are the degrees of freedom? tick the appropriate box
1
2
3
4
where does the 2 value fit in the range of probabilities? tick the appropriate
box
p > 0.90
p < 0.90,
but
> 0.50
p < 0.50, but
> 0.10
p < 0.10, but
> 0.05
p < 0.05, but
> 0.01
p < 0.01, but
> 0.001
p < 0.001
What can you conclude from the result? Tick and/or complete the appropriate
box.
either
The results are significant at the …………….. level
The null hypothesis can be rejected
or
The results are not significant
The null hypothesis can be accepted
2. A choice chamber was divided into two compartments – light and dark. 50
woodlice were put into the choice chamber and left for 15 minutes. After 15
minutes the number in each compartment were counted. There were 34 in the
dark side and 16 in the light side. If the woodlice showed no preference for
light or dark, there should be equal numbers in each compartment.
The 2 value for these results is 6.48
what are the degrees of freedom? Tick the appropriate box
1
2
3
4
where does the 2 value fit in the range of probabilities? tick the appropriate
box
3
p > 0.90
p < 0.90,
but
> 0.50
p < 0.50, but
> 0.10
p < 0.10, but
> 0.05
p < 0.05, but
> 0.01
p < 0.01, but
> 0.001
p < 0.001
What can you conclude from the result? Tick and/or complete the appropriate
box.
either
The results are significant at the …………….. level
The null hypothesis can be rejected
or
The results are not significant
The null hypothesis can be accepted
3. A student crossed some vestigial winged fruit flies with some wild type
(long winged) fruit flies. All the F1 fruit flies were long winged. The results of
the F2 generation were as follows:
long winged
vestigial winged
65
21
2 = 0.015504
what are the degrees of freedom? tick the appropriate box
1
2
3
4
where does the 2 value fit in the range of probabilities? tick the appropriate
box
p > 0.90
p < 0.90,
but
> 0.50
p < 0.50, but
> 0.10
p < 0.10, but
> 0.05
p < 0.05, but
> 0.01
p < 0.01, but
> 0.001
p < 0.001
4
What can you conclude from the result? Tick and/or complete the appropriate
box.
either
The results are significant at the …………….. level
The null hypothesis can be rejected
or
The results are not significant
The null hypothesis can be accepted
5
Reinforcement
1
The results of certain types of experiments can be analysed with the 2
test – what do these experiments have in common?
2
What type of data can be analysed using the 2 test?
3
Given the formula:
2 = E
E
what are the row and column headings in the table
for calculating 2?
4
How do you calculate degrees of freedom?
5
What does the 2 value mean?
6
What is the importance of the 5% level (or 0.05)?
7
What does p stand for?
8
What do you conclude if the value for p is greater than (>) 0.05 (5%)?
9
What do you conclude if the value of p is less than (<) 0.05 (5%)?
10
What do you conclude if the value of p is greater than (>) 0.9 (90%)?
6