7: Chi Square Analysis
Statistical evaluation of genetic data, chi-square test statistic, and hypothesis testing.
LibreTexts reference: Chi Square Analysis
Matching Chi-Square Terms to Definitions
Click to show Matching Chi-Square Terms to Definitions example problem
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. alternative hypothesis, Ha | ||
| 3. chi-square (χ²) test statistic | ||
| 4. null hypothesis, H0 |
Drag one of the choices below:
- A. represents how many independent values can vary in the calculation after constraints are applie
- B. we want to accept this hypothesis in our chi-square (χ²) test
- C. this hypothesis makes it easier to calculate the expected values
- D. the bigger this number, the smaller the p-value
Identifying Chi-Square Terms
Click to show Identifying Chi-Square Terms example problem
Which one of the following chi-square (χ²) terms correspond to the defintion 'the smaller this number, the bigger the chi-square (χ²) test statistic'.
Determining True/False Statements About Chi-Square Tests
Click to show Determining True/False Statements About Chi-Square Tests example problem
Which one of the following statements is TRUE concerning chi-square (χ²) tests?
Identifying Chi-Square Tests for Phenotypic Ratios
Click to show Identifying Chi-Square Tests for Phenotypic Ratios 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 |
| Table 1 | ||||
|---|---|---|---|---|
| Phenotype | Expected | Observed | Calculation | Statistic |
| Yellow Round (Y–R–) | 90 | 94 | (94-90)⁄ 90 | 0.044 |
| Yellow Wrinkled (Y–rr) | 30 | 28 | (28-30)⁄ 30 | -0.067 |
| Green Round (yyR–) | 30 | 31 | (31-30)⁄ 30 | 0.033 |
| Green Wrinkled (yyrr) | 10 | 7 | (7-10)⁄ 10 | -0.300 |
| (sum) χ² = | -0.289 | |||
| Table 2 | ||||
|---|---|---|---|---|
| Phenotype | Expected | Observed | Calculation | Statistic |
| Yellow Round (Y–R–) | 90 | 94 | (94-90)²⁄ 90 | 0.178 |
| Yellow Wrinkled (Y–rr) | 30 | 28 | (28-30)²⁄ 30 | 0.133 |
| Green Round (yyR–) | 30 | 31 | (31-30)²⁄ 30 | 0.033 |
| Green Wrinkled (yyrr) | 10 | 7 | (7-10)²⁄ 10 | 0.900 |
| (sum) χ² = | 1.244 | |||
| Table 3 | ||||
|---|---|---|---|---|
| Phenotype | Expected | Observed | Calculation | Statistic |
| Yellow Round (Y–R–) | 90 | 94 | (94-90)²⁄ 94 | 0.170 |
| Yellow Wrinkled (Y–rr) | 30 | 28 | (28-30)²⁄ 28 | 0.143 |
| Green Round (yyR–) | 30 | 31 | (31-30)²⁄ 31 | 0.032 |
| Green Wrinkled (yyrr) | 10 | 7 | (7-10)²⁄ 7 | 1.286 |
| (sum) χ² = | 1.631 | |||
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.
Determining Chi-Square Tests for Goodness of Fit
Click to show Determining Chi-Square Tests 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 | 77 | (77-90)²⁄ 90 | 1.878 |
| Yellow Wrinkled (Y–rr) | 30 | 26 | (26-30)²⁄ 30 | 0.533 |
| Green Round (yyR–) | 30 | 45 | (45-30)²⁄ 30 | 7.500 |
| Green Wrinkled (yyrr) | 10 | 12 | (12-10)²⁄ 10 | 0.400 |
| (sum) χ² = | 10.311 | |||
The final result gives the chi-squared (χ²) test value of 10.31 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 10.31 is greater than the critical value of 7.81, the null hypothesis is ACCEPTED.
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?