The Chi-square (χ²) test is a crucial statistical tool in AP Biology, used to determine if observed data differs significantly from expected data. Mastering this test is vital for success in the course and the AP exam. This guide provides practice problems, explanations, and tips to help you confidently tackle Chi-square analysis.
Understanding the Chi-Square Test
The Chi-square goodness-of-fit test assesses how well observed results align with expected results based on a specific hypothesis. A small χ² value suggests a good fit (observed data closely matches expected data), while a large χ² value indicates a poor fit (significant difference between observed and expected). We use degrees of freedom (df) and a critical χ² value (from a Chi-square distribution table) to determine statistical significance.
Key Components:
- Observed (O): The actual data you collect from your experiment.
- Expected (E): The data you anticipate based on your hypothesis (often based on ratios like Mendelian genetics).
- Degrees of Freedom (df): The number of categories minus 1 (df = n - 1).
- Critical χ² Value: Found in a Chi-square distribution table, based on your df and chosen significance level (usually 0.05, representing a 5% chance of observing the results if the null hypothesis is true).
The Formula:
χ² = Σ [(O - E)² / E]
AP Biology Chi-Square Practice Problems
Let's dive into some practice problems to solidify your understanding:
Problem 1: Mendelian Genetics
A genetics experiment involving pea plants predicted a 3:1 phenotypic ratio (purple:white flowers) in the F2 generation. The observed data were 730 purple and 270 white flowers. Test the hypothesis that the observed data fits the expected 3:1 ratio.
Solution:
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Calculate Expected Values (E): Total plants = 730 + 270 = 1000. Expected purple: 1000 * (3/4) = 750. Expected white: 1000 * (1/4) = 250.
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Calculate χ²:
Phenotype Observed (O) Expected (E) (O - E)² (O - E)² / E Purple 730 750 400 0.533 White 270 250 400 1.6 Total χ² = 2.133 -
Determine Degrees of Freedom (df): df = n - 1 = 2 - 1 = 1
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Find Critical χ² Value: At a significance level of 0.05 and df = 1, the critical χ² value is approximately 3.84.
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Conclusion: Our calculated χ² (2.133) is less than the critical χ² value (3.84). Therefore, we fail to reject the null hypothesis. The observed data is consistent with a 3:1 ratio.
Problem 2: Population Genetics
A population of butterflies has three color morphs: red, blue, and yellow. You hypothesize that the morphs are in Hardy-Weinberg equilibrium with the following allele frequencies: p (red) = 0.6, q (blue) = 0.3, and r (yellow) = 0.1. You observe the following numbers of butterflies: 360 red, 180 blue, and 60 yellow. Does the data support your hypothesis?
Solution: (Similar steps as Problem 1, but calculating expected values using the Hardy-Weinberg equation: p² + 2pq + 2pr + q² + 2qr + r² = 1) This problem requires more complex calculations, applying the Hardy-Weinberg principle to determine expected genotype frequencies and then converting those into expected numbers of butterflies based on the total population size (600). Remember to adjust your degrees of freedom accordingly (df = number of phenotypes -1).
Problem 3: Enzyme Activity
An experiment on enzyme activity yields the following observed data: 10, 12, 15, 13, and 10. The expected value for each trial is 12. Perform a chi-square analysis.
Solution: (Follow the same steps as problem 1, but note that you're comparing five observed values to a single expected value. This impacts the calculation of degrees of freedom.)
Tips for Success
- Practice Regularly: The more Chi-square problems you solve, the more comfortable you'll become with the process.
- Organize Your Work: Create a table to organize your observed, expected, and calculated values. This minimizes errors.
- Understand the Concepts: Don't just memorize the formula; understand the underlying statistical principles.
- Use a Chi-Square Table: Keep a Chi-square distribution table handy to look up critical values.
- Interpret Your Results: Clearly state your conclusions based on whether you reject or fail to reject the null hypothesis.
By working through these practice problems and understanding the underlying principles, you'll be well-prepared to tackle Chi-square analysis on the AP Biology exam. Remember to consult your textbook and teacher for further assistance if needed.
