7: Chi Square Analysis

Students perform chi-square tests on genetic cross data to evaluate whether observed ratios fit expected Mendelian predictions.

LibreTexts reference: Chapter 7: Chi Square Analysis

Matching Chi-Square Terms to Definitions

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Match each of the following chi-square (χ²) terms with their corresponding definitions.
Note: Each choice will be used exactly once.

	Your Choice	Prompt
	Drop Your Choice Here	1. level of significance, α
	Drop Your Choice Here	2. p-value
	Drop Your Choice Here	3. chi-square (χ²) test statistic
	Drop Your Choice Here	4. alternative hypothesis, Ha
	Drop Your Choice Here	5. null hypothesis, H0

Drag one of the choices below:

• A. we attempt to find evidence against this hypothesis in our chi-square (χ²) test
• B. the probability of getting a result that is either the same or more extreme than the actual observations
• C. a measure of the discrepancy between the observed and expected data sets
• D. standard cutoff probability used to determine statistic significance
• E. for this hypothesis, the expected values are often impossible to calculate

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True/False Statements About Chi-Square Tests

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Which one of the following statements is TRUE of chi-square (χ²) tests?

• A. the null hypothesis, H₀ is NOT REJECTED if the p-value is less than the level of significance, α
• B. the alternative hypothesis, H_a is SUPPORTED if the p-value is larger than the level of significance, α
• C. the null hypothesis, H₀ is REJECTED if the chi-square (χ²) statistic is larger than the level of significance, α
• D. the alternative hypothesis, H_a is SUPPORTED if the chi-square (χ²) statistic is larger than the level of significance, α
• E. the null hypothesis, H₀ is REJECTED if the chi-square (χ²) statistic is larger than the critical value

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Chi-Square Terms from Definitions

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Which one of the following chi-square (χ²) terms correspond to the definition 'this number determines which row of the chi-square (χ²) critical value table you should use'.

• A. chi-square (χ²) test statistic
• B. critical value
• C. degrees of freedom
• D. level of significance, α
• E. p-value

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Chi-Square Values for Phenotypic Ratios

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Data Table
Phenotype	Expected	Observed	Calculation	Statistic
Yellow Round (Y–R–)	90	87	__	__
Yellow Wrinkled (Y–rr)	30	27	__	__
Green Round (yyR–)	30	31	__	__
Green Wrinkled (yyrr)	10	15	__	__
(sum) χ2 =				__

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Complete the table and calculate the chi-squared (χ²) value.
Even though not part of the question, ask yourself whether you would reject or fail to reject the null hypothesis
Note: answers need to be within 3% of the correct number to be correct.

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Hypothesis Decisions from Chi-Square Tests

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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	82	(82-90)²⁄ 82²	0.010
Yellow Wrinkled (Y–rr)	30	22	(22-30)²⁄ 22²	0.132
Green Round (yyR–)	30	40	(40-30)²⁄ 40²	0.062
Green Wrinkled (yyrr)	10	16	(16-10)²⁄ 16²	0.141
(sum) χ² =				0.345

Table 2
Phenotype	Expected	Observed	Calculation	Statistic
Yellow Round (Y–R–)	90	82	(82-90)²⁄ 90	0.711
Yellow Wrinkled (Y–rr)	30	22	(22-30)²⁄ 30	2.133
Green Round (yyR–)	30	40	(40-30)²⁄ 30	3.333
Green Wrinkled (yyrr)	10	16	(16-10)²⁄ 10	3.600
(sum) χ² =				9.778

Table 3
Phenotype	Expected	Observed	Calculation	Statistic
Yellow Round (Y–R–)	90	82	(82-90)²⁄ 82	0.780
Yellow Wrinkled (Y–rr)	30	22	(22-30)²⁄ 22	2.909
Green Round (yyR–)	30	40	(40-30)²⁄ 40	2.500
Green Wrinkled (yyrr)	10	16	(16-10)²⁄ 16	2.250
(sum) χ² =				8.440

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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 reject or fail to reject the null hypothesis using the information provided.

• A. Table 1 with χ² = 0.345 is correct and we REJECT the null hypothesis
• B. Table 2 with χ² = 9.778 is correct and we REJECT the null hypothesis
• C. Table 3 with χ² = 8.440 is correct and we REJECT the null hypothesis
• D. Table 1 with χ² = 0.345 is correct and we FAIL to REJECT the null hypothesis
• E. Table 2 with χ² = 9.778 is correct and we FAIL to REJECT the null hypothesis
• F. Table 3 with χ² = 8.440 is correct and we FAIL to REJECT the null hypothesis

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Errors in Chi-Square Calculations and Hypothesis Decisions

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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

Phenotype	Expected	Observed	Calculation	Statistic
Yellow Round (Y–R–)	90	95	(95-90)²⁄ 90²	0.003
Yellow Wrinkled (Y–rr)	30	18	(18-30)²⁄ 30²	0.160
Green Round (yyR–)	30	41	(41-30)²⁄ 30²	0.134
Green Wrinkled (yyrr)	10	6	(6-10)²⁄ 10²	0.160
(sum) χ² =				0.458

The final result gives the chi-squared (χ²) test value of 0.46 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 0.46 is less than the critical value of 7.81, the null hypothesis has FAILED TO BE REJECTED.

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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?

• A. The wrong rejection criteria was used. the significance level, α is wrong.
• B. The degrees of freedom is wrong, it should be a different value.
• C. The wrong numbers in the calculation were used for division, you need to divide by a different number.
• D. The expected progeny for the null hypothesis is incorrect. They did the calculation wrong.
• E. The numbers in the calculation have to be squared and they are not squared.

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Chi-Square Tests for Hardy-Weinberg Equilibrium

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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	Observed	Expected	Calculation	Statistic
Red Flowers	125	112.9	(125-112.9)2⁄ 112.9	1.297
Pink Flowers	224	249.4	(224-249.4)2⁄ 249.4	2.587
White Flowers	150	137.7	(150-137.7)2⁄ 137.7	1.099
(sum) χ2 =				4.982

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You finally have a new competent lab partner that you trust.
This lab partner calculated the allele frequencies of p=0.48 and q=0.52. Then they did a chi-squared (χ²) test for your Hardy-Weinberg data.
They need you to decide whether you reject or accept the null hypothesis using the information provided.

• A. the chi-squared (χ²) sum is GREATER than the critical value and we ACCEPT the null hypothesis
• B. the chi-squared (χ²) sum is LESS than the critical value and we REJECT the null hypothesis
• C. the chi-squared (χ²) sum is GREATER than the critical value and we REJECT the null hypothesis
• D. the chi-squared (χ²) sum is LESS than the critical value and we ACCEPT the null hypothesis

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Null and Alternative Hypotheses for Genetic Crosses

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You perform a dihybrid testcross (AaBb × aabb) and count the offspring phenotypes.
Total offspring scored: 220

Observed data
Category	Ratio	Expected	Observed
A–B–	1	55	53
A–bb	1	55	46
aaB–	1	55	63
aabb	1	55	58

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For a chi-squared (χ²) goodness-of-fit test, which option correctly states the null hypothesis (H₀) and the alternative hypothesis (H_A)?

• A.

  H₀: The offspring proportions are consistent with the expected 9:3:3:1 ratio (any differences from the expected ratio are due to chance).
  H_A: The offspring proportions are not consistent with the expected 9:3:3:1 ratio (the differences are too large to explain by chance alone).

• B.

  H₀: The offspring proportions are consistent with the expected 1:1:1:1 ratio (any differences from the expected ratio are due to chance).
  H_A: The offspring proportions are not consistent with the expected 1:1:1:1 ratio (the differences are too large to explain by chance alone).

• C.

  H₀: The observed counts are exactly in a 1:1:1:1 ratio.
  H_A: The observed counts are not exactly in a 1:1:1:1 ratio.

• D.

  H₀: The offspring proportions are not consistent with the expected 1:1:1:1 ratio (the differences are too large to explain by chance alone).
  H_A: The offspring proportions are consistent with the expected 1:1:1:1 ratio (any differences from the expected ratio are due to chance).

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Misstated Null Hypotheses for Genetic Ratios

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Your lab partner is trying again (eye roll).
You perform a standard dihybrid cross and count the F₂ offspring phenotypes.
Total offspring scored: 256

Observed data
Category	Ratio	Expected	Observed
Yellow Round (Y–R–)	9	144	144
Yellow Wrinkled (Y–rr)	3	48	61
Green Round (yyR–)	3	48	43
Green Wrinkled (yyrr)	1	16	8

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They are setting up a chi-squared (χ²) goodness-of-fit test, but they wrote the hypotheses below:
H₀: The offspring proportions are consistent with the expected 8:2:4:2 ratio (any differences from the expected ratio are due to chance).
H_A: The offspring proportions are not consistent with the expected 8:2:4:2 ratio (the differences are too large to explain by chance alone).
What is the main problem with their hypotheses?

• A. They swapped the null and alternative hypotheses.
• B. They incorrectly wrote the null hypothesis as an exact match (no random variation).
• C. Nothing is wrong; the hypotheses are stated correctly.
• D. They used the wrong expected ratio in the hypotheses.

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