The Chi Squared Test

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

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

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

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

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 5474 +1850= 7324 7324 x .75= 5493 7324 x .25= 1831 Is the difference between the observed and the expected significant?

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

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

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

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.

Answer

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