1. The Chi Squared Test

2. Chi-Squared Test
Used to compare data in the field with data in an experiment
Is the difference between what you observed and what you expected due to random chance or some other factor?
Ex. If you toss a coin 10 times, you expect 5 heads and 5 tails, but that might not be what you observe to happen.
There is variability in the real world
Chi-squared enables us to
distinguish the natural variability between what is observed and what is expected
the possibility of something else significantly playing a role

3. Hypothesis Testing
Hypothesis- a causal relationship exists between an underlying factor (variable) and an observable phenomenon.
Ex. Do English ivy leaves grow bigger in the shade?
Relationship- sunlight amount is the variable causing bigger leaves.
Hypothesis testing- Are the effects real? Is there something in this data?
trying to reject a null hypothesis
A statement explaining that there is no causal relationship
Ex. Leaf widths are the same in sunny and shady environments
Alternative to the null hypothesis- the causal relationship exists
Ex. Leaves grow larger in the shade to capture more sunlight
In most cases, experiments do not prove the alternate to the null hypothesis but reject the null hypothesis

4. Critical Value
How much variation should be tolerated before rejecting the null hypothesis?
If the observed deviate from the predictions, how much do we allow to chance?
In biology we usually have a 5% critical value
Known as a probability value, or p-value
Ex. Data is collected on leaf width in shade and sun
A statistical analysis is used
A p-value is generated- if it is less than 5%, we should reject the null
The leaves of shady plants are significantly larger than leaves in the sun

5. Chi-Squared Test Used in introductory biology to
Test observed outcomes of genetic test compare with Mendel’s predicted outcomes
See how gene frequencies match up to Hardy-Weinberg equilibrium
Usually trying to prove that any variation away from Mendel or Hardy-Weinberg was due to chance
You want to fail to reject the null, p-value is greater than .05
Proving Mendel and Hardy-Weinberg correct
Can also be used to reject the null (p ≤.05)
Pill bugs choosing one environment over another
Want to show that one environment is desired significantly
In medicine- comparing a drug to a placebo
Null- effects are the same
Chi-squared p-value greater than .05- drug and placebo are equally effective
Chi-squared p-value less than or equal to .05- there is a significant different
Null is rejected

6. Calculating Chi-Squared
O: observed
E: expected
Example- Mendel’s monohybrid cross
Round is dominant over wrinkled
F2 generation should show a 3:1 ratio
Data- 5,474 round, 1,850 wrinkled
This is the observed
Expected data- 5,493 round, 1,831 wrinkled
= 7324
7324 x .75= 5493
7324 x .25= 1831
Is the difference between the observed and the expected significant?

7. Calculating Chi-Squared
Degrees of Freedom- one less than the number of results
Two results here, round or wrinkled, so the degrees of freedom is one

8. Calculating Chi-Squared
Once you have determined the value of chi-squared, use a Chi-squared table to calculate the p-value:

9. Calculating Chi-Squared
Results:
Our p-value is well above .05
do not reject the null hypothesis
The observed vs. the expected values are not significantly different
The differences are due to chance
Summary of Steps:
Calculate the observed and the expected.
Plug into the Chi-squared formula
Get a number answer for χ2 by adding up the last column
Calculate degrees of freedom
Use a chi-squared table to calculate the p-value
If p≤.05, then reject the null hypothesis

10. Your Turn Stream snails were marked and recaptured
Question asked: Do snails tend to move upstream or downstream after initial capture?
If no preference, half would go upstream, half would go down stream
Completed in two stream beds:
Sandy stream: 43 of 50 snails were recovered upstream
Rocky stream: 22 of 38 snails were recovered upstream
Do a Chi-squared analysis for each bed type.

11. Answer

12. Answer Sandy bed- definitely reject the null hypothesis
Rocky bed- do not reject the null hypothesis