use the chi-squared test to test the significance of differences between observed and expected results (the formula for the chi-squared test will be provided, as shown in the Mathematical requirements)

The roles of genes in determining the phenotype

Objective

Use the chi‑squared test to assess whether the differences between observed and expected phenotypic ratios are statistically significant.

Why a statistical test is needed

• Biological experiments involve random variation (e.g., segregation of alleles).
• Observed numbers rarely match theoretical ratios exactly.
• The chi‑squared test quantifies how likely any discrepancy is due to chance.

Mathematical requirement

The chi‑squared statistic is calculated using the formula

\$\chi^{2} = \sum{i=1}^{k} \frac{(O{i}-E{i})^{2}}{E{i}}\$

where:

• \$O_{i}\$ = observed number in class \$i\$
• \$E_{i}\$ = expected number in class \$i\$
• \$k\$ = total number of phenotypic classes

Steps to perform a chi‑squared test

1. State the null hypothesis (\$H_{0}\$): the observed ratio fits the expected Mendelian ratio.
2. Calculate the expected numbers (\$E_{i}\$) for each class using the total sample size and the theoretical proportion.
3. Compute the chi‑squared value using the formula above.
4. Determine the degrees of freedom: \$df = k - 1\$ (or \$df = k - 1 - c\$, where \$c\$ is the number of parameters estimated from the data; for a simple Mendelian test \$c = 0\$).
5. Compare the calculated \$\chi^{2}\$ with the critical value from a chi‑squared distribution table at the chosen significance level (usually \$p = 0.05\$).
6. Make a decision:

   • If \$\chi^{2} \leq \chi^{2}{\text{critical}}\$, fail to reject \$H{0}\$ – the data are consistent with the expected ratio.
   • If \$\chi^{2} > \chi^{2}{\text{critical}}\$, reject \$H{0}\$ – the data differ significantly from the expected ratio.

Worked example: Monohybrid cross (F2 generation)

Consider a cross between heterozygous pea plants (Rr × Rr) where the dominant allele \$R\$ gives round seeds and the recessive allele \$r\$ gives wrinkled seeds. The expected phenotypic ratio in the F2 generation is 3 round : 1 wrinkled.

Calculation details:

1. Total seeds counted = 215 + 85 = 300.
2. Expected numbers:

   • Round = \$300 \times \frac{3}{4} = 225\$ (rounded to 240 for illustration; in practice keep the exact value).
   • Wrinkled = \$300 \times \frac{1}{4} = 75\$ (rounded to 80).

3. Compute each component of the sum and add them to obtain \$\chi^{2}=2.91\$.
4. Degrees of freedom \$df = k-1 = 2-1 = 1\$.
5. Critical value for \$df=1\$ at \$p=0.05\$ is \$\chi^{2}_{0.05}=3.84\$.
6. Since \$2.91 < 3.84\$, we fail to reject \$H_{0}\$ – the observed data are consistent with the 3:1 ratio.

Interpreting the result in a biological context

• A non‑significant chi‑squared result supports the hypothesis that a single gene with two alleles follows Mendelian segregation.
• A significant result may indicate:

  • Linkage to another gene,
  • Incomplete dominance or codominance,
  • Environmental effects on phenotype expression,
  • Sampling error or mis‑counting.

• Further experiments (larger sample size, reciprocal crosses) can help clarify the cause.