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 essential for success on the exam. This post provides practice problems of varying difficulty to help you hone your skills and build confidence. Remember, understanding the underlying concepts is just as important as the calculations themselves.
Understanding the Chi-Square Test
Before diving into the problems, let's briefly review the core concepts:
- Purpose: The Chi-square test assesses the significance of the difference between observed and expected frequencies. A significant difference suggests a relationship between the variables being studied.
- Null Hypothesis (H₀): This hypothesis assumes there's no significant difference between observed and expected values. The Chi-square test helps determine whether we can reject this null hypothesis.
- Degrees of Freedom (df): Calculated as (number of categories - 1). This value is crucial for determining the critical χ² value from a Chi-square distribution table.
- Critical Value: The value from the Chi-square distribution table that determines the significance level (typically 0.05 or 5%). If your calculated χ² value exceeds the critical value, you reject the null hypothesis.
- P-value: The probability of obtaining the observed results if the null hypothesis is true. A p-value less than the significance level (e.g., 0.05) indicates a significant difference.
Practice Problems
Let's work through some examples, gradually increasing in complexity:
Problem 1: Simple Mendelian Genetics
A monohybrid cross between two heterozygous pea plants (Rr x Rr) is expected to produce a 3:1 phenotypic ratio (3 round, 1 wrinkled). You perform the cross and observe the following results:
- Round peas: 72
- Wrinkled peas: 28
Calculate the χ² value and determine if the observed results significantly deviate from the expected 3:1 ratio. Use a significance level of 0.05.
Solution:
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Calculate expected values: Total offspring = 100. Expected round peas = (3/4) * 100 = 75. Expected wrinkled peas = (1/4) * 100 = 25.
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Calculate χ²: χ² = Σ [(Observed - Expected)² / Expected] = [(72-75)²/75] + [(28-25)²/25] = 0.12 + 0.36 = 0.48
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Determine degrees of freedom: df = (number of categories - 1) = 2 - 1 = 1
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Find the critical value: Consult a Chi-square distribution table with df = 1 and α = 0.05. The critical value is approximately 3.84.
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Compare calculated χ² to the critical value: 0.48 < 3.84. Therefore, we fail to reject the null hypothesis. The observed results are not significantly different from the expected 3:1 ratio.
Problem 2: Dihybrid Cross
A dihybrid cross involving flower color (purple, P, is dominant to white, p) and plant height (tall, T, is dominant to short, t) is performed. The expected phenotypic ratio is 9:3:3:1. You observe the following:
- Purple tall: 78
- Purple short: 21
- White tall: 27
- White short: 8
Calculate the χ² value and determine if the observed results significantly deviate from the expected ratio. Use a significance level of 0.05.
Solution (Outline):
Follow the same steps as Problem 1, but with four categories. Remember to calculate the expected values for each phenotype based on the total number of offspring and the expected 9:3:3:1 ratio. The degrees of freedom will be 3 (4 categories - 1). Compare your calculated χ² value to the critical value from the Chi-square table for df = 3 and α = 0.05.
Problem 3: Population Genetics
In a population of 100 butterflies, you observe the following wing color phenotypes:
- Red: 60
- Blue: 30
- White: 10
You hypothesize that the population is in Hardy-Weinberg equilibrium, with allele frequencies of p = 0.7 (red) and q = 0.3 (blue). White is a rare recessive phenotype.
Calculate the χ² value and determine if the population is in Hardy-Weinberg equilibrium. Use a significance level of 0.05.
Solution (Outline):
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Calculate expected genotype frequencies: Using the Hardy-Weinberg equation (p² + 2pq + q² = 1), determine the expected frequencies of each genotype (RR, RB, BB).
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Calculate expected phenotype frequencies: Based on the genotype frequencies, determine the expected number of butterflies with each phenotype (red, blue, white).
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Calculate χ²: Use the formula as in previous problems.
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Determine degrees of freedom: df = (number of categories - 1) = 3 - 1 = 2
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Find the critical value: Use the Chi-square table for df = 2 and α = 0.05.
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Compare calculated χ² to the critical value: Interpret your results in terms of Hardy-Weinberg equilibrium.
These practice problems will help you solidify your understanding of the Chi-square test. Remember to practice regularly, and consult your textbook or teacher if you need additional help. Good luck with your AP Biology studies!
