https://file.aiquickdraw.com/mathbot/file/172983998831021gvf0lr.jpg

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.

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### **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):**
   - \(O = 21\), \(E = 19.016\)
   \[
   \chi^2 = \frac{(21 - 19.016)^2}{19.016} = \frac{3.944}{19.016} \approx 0.207
   \]

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### **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)

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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.

We are interested in studying the association between the age of the felled tree and wood species in the population under consideration. Please complete the following tables, calculating the values of A and B.

Solve the chi-squared test problem by calculating the observed and expected counts for blackwood and sassafras trees. This problem involves calculating missing values A and B using statistical methods and evaluating chi-squared values. Perfect for university-level statistics students.

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