To calculate the chi-square test statistic ($\chi^2$), we will follow these steps:

1. Expected Frequency Calculation: Each expected frequency is calculated by multiplying the total number of observations by the expected proportion.

2. Chi-Square Formula: $\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}$ Where $O_i$ is the observed frequency and $E_i$ is the expected frequency.

Data from the Table

• Observed Frequencies ($O$):

  • Red Eye/Normal Wing: 66
  • Sepia Eye/Normal Wing: 20
  • Red Eye/Vestigial Wing: 18
  • Sepia Eye/Vestigial Wing: 8

• Expected Proportions:

  • Red Eye/Normal Wing: $\frac{9}{16}$
  • Sepia Eye/Normal Wing: $\frac{3}{16}$
  • Red Eye/Vestigial Wing: $\frac{3}{16}$
  • Sepia Eye/Vestigial Wing: $\frac{1}{16}$

Total Observations:

$\text{Total } = 66 + 20 + 18 + 8 = 112$

Expected Frequencies ($E$):

• Red Eye/Normal Wing: $112 \times \frac{9}{16} = 63$
• Sepia Eye/Normal Wing: $112 \times \frac{3}{16} = 21$
• Red Eye/Vestigial Wing: $112 \times \frac{3}{16} = 21$
• Sepia Eye/Vestigial Wing: $112 \times \frac{1}{16} = 7$

Chi-Square Calculation:

For each category: $\frac{(O - E)^2}{E}$

• Red Eye/Normal Wing: $\frac{(66 - 63)^2}{63} = \frac{9}{63} = 0.143$
• Sepia Eye/Normal Wing: $\frac{(20 - 21)^2}{21} = \frac{1}{21} = 0.048$
• Red Eye/Vestigial Wing: $\frac{(18 - 21)^2}{21} = \frac{9}{21} = 0.429$
• Sepia Eye/Vestigial Wing: $\frac{(8 - 7)^2}{7} = \frac{1}{7} = 0.143$

Total $\chi^2$:

$\chi^2 = 0.143 + 0.048 + 0.429 + 0.143 = 0.763$

Final Answer:

The chi-square test statistic is: $\chi^2 = 0.763$

Let me know if you need further details or assistance with hypothesis testing steps.

Questions for Further Understanding:

1. How do you interpret the chi-square value in the context of the significance level?
2. What is the next step after calculating $\chi^2$? (e.g., comparison with the critical value or p-value determination)
3. How would the result change if the sample size was larger or smaller?
4. What assumptions are made when using the chi-square test in this context?
5. Can you explain how the null hypothesis aligns with the genetic principles?

Tip:

Always double-check your expected frequencies, as they are crucial for an accurate chi-square calculation.