To complete the tables in your image, let's solve for the missing values A (observed count) and B (expected count). I’ll walk you through the process step-by-step.

Step 1: Solve for A (Observed Count)

• A is the missing observed count for sassafras in the second column (16–21 age group).

• We know that the sum for that column is 35, and we already have:

  • Blackwood: 14
  • Sassafras: A
  • Total sum: 35

  Therefore: $14 + A = 35 \quad \Rightarrow \quad A = 35 - 14 = 21$

  So, A = 21.

Step 2: Solve for B (Expected Count)

• B is the missing expected count for blackwood in the fourth column (more than 26 age group).

• We use the expected value formula: $E_{ij} = \frac{(\text{Row Total} \times \text{Column Total})}{\text{Grand Total}}$

• For blackwood in the 4th column:

  • Row total for blackwood: 58
  • Column total for the 4th column: 34
  • Grand total: 127

  Therefore: $B = \frac{58 \times 34}{127} = \frac{1972}{127} \approx 15.528$

  So, B ≈ 15.528.

Step 3: Fill the Chi-Squared Table

The chi-squared value for each cell is: $\chi^2 = \frac{(O - E)^2}{E}$ where $O$ is the observed value and $E$ is the expected value.

Let’s calculate the missing chi-squared values:

1. Blackwood, 4th column (more than 26):

   • $O = 16$, $E = 15.528$ $\chi^2 = \frac{(16 - 15.528)^2}{15.528} = \frac{0.222}{15.528} \approx 0.014$

2. Sassafras, 2nd column (16–21):

   • $O = 21$, $E = 19.016$ $\chi^2 = \frac{(21 - 19.016)^2}{19.016} = \frac{3.944}{19.016} \approx 0.207$

Summary of Solutions:

• A = 21 (observed count for sassafras in the 16–21 group)
• B ≈ 15.528 (expected count for blackwood in the more than 26 group)

Would you like further details or calculations? Here are some follow-up questions:

1. How is the chi-squared value interpreted in hypothesis testing?
2. What is the significance level (alpha) typically used for chi-squared tests?
3. How do degrees of freedom affect the chi-squared distribution?
4. What are the assumptions of the chi-squared test for independence?
5. How would the results change if the observed and expected counts were closer?

Tip: For chi-squared tests, always ensure all expected values are sufficiently large (at least 5) to ensure validity.