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

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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
	Drop Your Choice Here	1. level of significance, α
	Drop Your Choice Here	2. null hypothesis, H0
	Drop Your Choice Here	3. degrees of freedom
	Drop Your Choice Here	4. p-value

Drag one of the choices below:

• A. this value determines which row of the chi-square (χ²) critical value table you should use
• B. this hypothesis makes it easier to calculate the expected values
• C. the smaller this number, the bigger the chi-square (χ²) test statistic
• D. the statistical cutoff of the result for the null hypothesis, H₀ to be TRUE or false

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Chi-Square Test Cutoff Terms

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Which one of the following chi-square (χ²) terms correspond to the defintion 'for this hypothesis, the expected values may be impossible to calculate'.

• A. alternative hypothesis, H_a
• B. chi-square (χ²) test statistic
• C. critical value
• D. degrees of freedom

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True/False Statements About Chi-Square Tests

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

• A. the null hypothesis, H₀ is REJECTED if the chi-square (χ²) statistic is less than the critical value
• B. the null hypothesis, H₀ is ACCEPTED if the p-value is larger than the level of significance, α
• C. the alternative hypothesis, H_a is ACCEPTED if the p-value is larger than the critical value
• D. the alternative hypothesis, H_a is ACCEPTED 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 level of significance, α

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Chi-Squared Test for Genetic Variation

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Data Table
Phenotype	Expected	Observed	Calculation	Statistic
Yellow Round (Y–R–)	90	93	__	__
Yellow Wrinkled (Y–rr)	30	26	__	__
Green Round (yyR–)	30	35	__	__
Green Wrinkled (yyrr)	10	6	__	__
(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 2% of the correct number to be correct.

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Chi-Square Tests for Genetic 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	87	(87-90)²⁄ 87	0.103
Yellow Wrinkled (Y–rr)	30	35	(35-30)²⁄ 35	0.714
Green Round (yyR–)	30	30	(30-30)²⁄ 30	0.000
Green Wrinkled (yyrr)	10	8	(8-10)²⁄ 8	0.500
(sum) χ² =				1.318

Table 2
Phenotype	Expected	Observed	Calculation	Statistic
Yellow Round (Y–R–)	90	87	(87-90)²⁄ 87²	0.001
Yellow Wrinkled (Y–rr)	30	35	(35-30)²⁄ 35²	0.020
Green Round (yyR–)	30	30	(30-30)²⁄ 30²	0.000
Green Wrinkled (yyrr)	10	8	(8-10)²⁄ 8²	0.062
(sum) χ² =				0.084

Table 3
Phenotype	Expected	Observed	Calculation	Statistic
Yellow Round (Y–R–)	90	87	(87-90)²⁄ 90	0.100
Yellow Wrinkled (Y–rr)	30	35	(35-30)²⁄ 30	0.833
Green Round (yyR–)	30	30	(30-30)²⁄ 30	0.000
Green Wrinkled (yyrr)	10	8	(8-10)²⁄ 10	0.400
(sum) χ² =				1.333

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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 = 1.318 is correct and we REJECT the null hypothesis
• B. Table 2 with χ² = 0.084 is correct and we REJECT the null hypothesis
• C. Table 3 with χ² = 1.333 is correct and we REJECT the null hypothesis
• D. Table 1 with χ² = 1.318 is correct and we FAIL to REJECT the null hypothesis
• E. Table 2 with χ² = 0.084 is correct and we FAIL to REJECT the null hypothesis
• F. Table 3 with χ² = 1.333 is correct and we FAIL to REJECT the null hypothesis

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Chi-Square Calculations and Hypothesis Evaluations

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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	98	(98-90)²⁄ 90	0.711
Yellow Wrinkled (Y–rr)	30	23	(23-30)²⁄ 30	1.633
Green Round (yyR–)	30	31	(31-30)²⁄ 30	0.033
Green Wrinkled (yyrr)	10	8	(8-10)²⁄ 10	0.400
(sum) χ² =				2.778

The final result gives the chi-squared (χ²) test value of 2.78 with 4 degrees of freedom. Consulting the Table of χ² Critical Values and a level of significance α=0.05, we obtain a critical value of 9.49.
Since the chi-squared value of 2.78 is less than the critical value of 9.49, 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 numbers in the calculation have to be squared and they are not squared.
• B. The wrong numbers in the calculation were used for division, you need to divide by a different number.
• C. The degrees of freedom is wrong, it should be a different value.
• D. The wrong rejection criteria was used. the significance level, α is wrong.
• E. The expected progeny for the null hypothesis is incorrect. They did the calculation wrong.

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Chi-Squared Test 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	9263	9205.2	(9263-9205.2)2⁄ 9205.2	0.363
Pink Flowers	6738	6854.7	(6738-6854.7)2⁄ 6854.7	1.987
White Flowers	1334	1276.1	(1334-1276.1)2⁄ 1276.1	2.627
(sum) χ2 =				4.977

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You finally have a new competent lab partner that you trust.
This lab partner calculated the allele frequencies of p=0.73 and q=0.27. 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 LESS than the critical value and we REJECT the null hypothesis
• B. the chi-squared (χ²) sum is GREATER than the critical value and we ACCEPT 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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Chi-Squared Goodness-of-Fit Test for Mendelian Inheritance

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You perform a dihybrid testcross (AaBb × aabb) and count the offspring phenotypes.
Total offspring scored: 248

Observed data
Category	Ratio	Expected	Observed
A–B–	1	62	67
A–bb	1	62	67
aaB–	1	62	62
aabb	1	62	52

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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 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).

• B.

  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.

• C.

  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).

• D.

  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).

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Chi-Squared Goodness-of-Fit Test for Mendelian Inheritance

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Your lab partner is trying again (eye roll).
You perform a dihybrid testcross (AaBb × aabb) and count the offspring phenotypes.
Total offspring scored: 244

Observed data
Category	Ratio	Expected	Observed
A–B–	1	61	71
A–bb	1	61	58
aaB–	1	61	54
aabb	1	61	61

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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 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).
What is the main problem with their hypotheses?

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

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