7: Chi Square Analysis
Statistical evaluation of genetic data, chi-square test statistic, and hypothesis testing.
LibreTexts reference: Chi Square Analysis
Chi-Square Terms and Definitions Matching
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Match each of the following chi-square (χ²) terms with their corresponding defintions.
Note: Each choice will be used exactly once.
| Your Choice | Prompt | |
|---|---|---|
| 1. degrees of freedom | ||
| 2. level of significance, α | ||
| 3. chi-square (χ²) test statistic | ||
| 4. critical value |
Drag one of the choices below:
- A. in a chi-square (χ²) test, it is usually the number of categories minus one
- B. if this value is large, then there is stronger evidence in favor of the alternative hypothesis, Ha
- C. the boundary of how extreme a test statistic we need to reject the null hypothesis, H0
- D. a constant probability that provides a cutoff for falsification of the null hypothesis, H0
Chi-Square Test Terms and Definitions
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Which one of the following chi-square (χ²) terms correspond to the defintion 'the bigger this number, the smaller the p-value'.
True/False Statements on Chi-Square Tests
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Which one of the following statements is TRUE regarding chi-square (χ²) tests?
Chi-Squared Test for Phenotypic Ratios
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| Table of Chi-Squared (χ²) Critical Values | ||||||||
|---|---|---|---|---|---|---|---|---|
| Degrees of Freedom | Probability | |||||||
| 0.95 | 0.90 | 0.75 | 0.50 | 0.25 | 0.10 | 0.05 | 0.01 | |
| 1 | 0.00 | 0.02 | 0.10 | 0.45 | 1.32 | 2.71 | 3.84 | 6.63 |
| 2 | 0.10 | 0.21 | 0.58 | 1.39 | 2.77 | 4.61 | 5.99 | 9.21 |
| 3 | 0.35 | 0.58 | 1.21 | 2.37 | 4.11 | 6.25 | 7.81 | 11.34 |
| 4 | 0.71 | 1.06 | 1.92 | 3.36 | 5.39 | 7.78 | 9.49 | 13.28 |
| Table 1 | ||||
|---|---|---|---|---|
| Phenotype | Expected | Observed | Calculation | Statistic |
| Yellow Round (Y–R–) | 90 | 96 | (96-90)²⁄ 96 | 0.375 |
| Yellow Wrinkled (Y–rr) | 30 | 28 | (28-30)²⁄ 28 | 0.143 |
| Green Round (yyR–) | 30 | 23 | (23-30)²⁄ 23 | 2.130 |
| Green Wrinkled (yyrr) | 10 | 13 | (13-10)²⁄ 13 | 0.692 |
| (sum) χ² = | 3.341 | |||
| Table 2 | ||||
|---|---|---|---|---|
| Phenotype | Expected | Observed | Calculation | Statistic |
| Yellow Round (Y–R–) | 90 | 96 | (96-90)²⁄ 96² | 0.004 |
| Yellow Wrinkled (Y–rr) | 30 | 28 | (28-30)²⁄ 28² | 0.005 |
| Green Round (yyR–) | 30 | 23 | (23-30)²⁄ 23² | 0.093 |
| Green Wrinkled (yyrr) | 10 | 13 | (13-10)²⁄ 13² | 0.053 |
| (sum) χ² = | 0.155 | |||
| Table 3 | ||||
|---|---|---|---|---|
| Phenotype | Expected | Observed | Calculation | Statistic |
| Yellow Round (Y–R–) | 90 | 96 | (96-90)²⁄ 90 | 0.400 |
| Yellow Wrinkled (Y–rr) | 30 | 28 | (28-30)²⁄ 30 | 0.133 |
| Green Round (yyR–) | 30 | 23 | (23-30)²⁄ 30 | 1.633 |
| Green Wrinkled (yyrr) | 10 | 13 | (13-10)²⁄ 10 | 0.900 |
| (sum) χ² = | 3.067 | |||
Your lab partner is trying again (eye roll) and did another a chi-squared (χ²) test on the F2 generation in a dihybid cross based on your lab data (above). They wanted to know if the results confirm the expected phenotype ratios.
You helped them set up the null hypothesis, so you know that part is correct, but they got confused and were unsure about how to calculate the chi-squared (χ²) value. So much so that they did it three (3) different ways.
Before you ask your instructor for a new lab partner, tell them which table is correct AND whether they can accept or reject the null hypothesis using the information provided.
Chi-Squared Test for Goodness of Fit
Click to show Chi-Squared Test for Goodness of Fit example problem
| Table of Chi-Squared (χ²) Critical Values | ||||||||
|---|---|---|---|---|---|---|---|---|
| Degrees of Freedom | Probability | |||||||
| 0.95 | 0.90 | 0.75 | 0.50 | 0.25 | 0.10 | 0.05 | 0.01 | |
| 1 | 0.00 | 0.02 | 0.10 | 0.45 | 1.32 | 2.71 | 3.84 | 6.63 |
| 2 | 0.10 | 0.21 | 0.58 | 1.39 | 2.77 | 4.61 | 5.99 | 9.21 |
| 3 | 0.35 | 0.58 | 1.21 | 2.37 | 4.11 | 6.25 | 7.81 | 11.34 |
| 4 | 0.71 | 1.06 | 1.92 | 3.36 | 5.39 | 7.78 | 9.49 | 13.28 |
| Phenotype | Expected | Observed | Calculation | Statistic |
|---|---|---|---|---|
| Yellow Round (Y–R–) | 90 | 86 | (86-90)²⁄ 86 | 0.186 |
| Yellow Wrinkled (Y–rr) | 30 | 19 | (19-30)²⁄ 19 | 6.368 |
| Green Round (yyR–) | 30 | 47 | (47-30)²⁄ 47 | 6.149 |
| Green Wrinkled (yyrr) | 10 | 8 | (8-10)²⁄ 8 | 0.500 |
| (sum) χ² = | 13.203 | |||
The final result gives the chi-squared (χ²) test value of 13.20 with 3 degrees of freedom. Consulting the Table of χ² Critical Values and a level of significance α=0.05, we obtain a critical value of 7.81.
Since the chi-squared value of 13.20 is greater than the critical value of 7.81, the null hypothesis is REJECTED.
Your lab partner completed a chi-squared (χ²) test on your lab data (above) for the F2 generation in a standard dihybrid cross. The goal was to verify if the observed results matched the expected phenotype ratios.
However, it appears they made an error. What did they do wrong?