Research question: We wish to investigate whether the age of tree felled at a specialty timber plantation is distributed according to the table below.
the age of tree | 1 (less than 15) | 2 (16-21) | 3 (22-26) | 4 (more than 26)
Percentage      |               10 |        60 |        20 |               10
Please complete the following table calculating the values of A, B and C.

               | 1 (less than 15) | 2 (16-21) | 3 (22-26) | 4 (more than 26) | Total  
Observed Count | 44               | A=***     | 31        | 26               | 137    
Expected Count | B=***            | 82.2      | 27.4      | 13.7             | 137    
chi squared    | 67.014           | 25.966    | 0.473     | C=***            | 104.496
A [integer]:	

B [use at least 3dp]

To complete this problem, we need to:

1. Calculate \( A \), the observed count for the age group 16-21.
2. Calculate \( B \), the expected count for the age group 1 (less than 15).
3. Calculate \( C \), the chi-squared value for the age group 4 (more than 26).

Let's go step by step:

### 1. **Calculate \( A \) (Observed Count for Age Group 2)**

The total observed count is given as 137. Since we know the observed counts for the other three age groups (44, 31, and 26), we can find \( A \) by subtracting the sum of these values from the total.

\[
A = 137 - (44 + 31 + 26)
\]
\[
A = 137 - 101 = 36
\]

### 2. **Calculate \( B \) (Expected Count for Age Group 1)**

The total expected count should also add up to 137. We are given percentages for each age group, and the expected counts are calculated based on these percentages. The percentage for age group 1 is 10%.

\[
B = 10\% \times 137 = 0.10 \times 137 = 13.7
\]

### 3. **Calculate \( C \) (Chi-Squared Value for Age Group 4)**

The chi-squared value for each group is calculated using the formula:

\[
\chi^2 = \frac{(O - E)^2}{E}
\]
Where \( O \) is the observed count and \( E \) is the expected count.

For age group 4:
- Observed count \( O = 26 \)
- Expected count \( E = 13.7 \)

\[
C = \frac{(26 - 13.7)^2}{13.7}
\]
\[
C = \frac{(12.3)^2}{13.7} = \frac{151.29}{13.7} \approx 11.04
\]

### Final Table

|                       | 1 (less than 15) | 2 (16-21) | 3 (22-26) | 4 (more than 26) | Total |
|----------------------- |----------------- |---------- |---------- |----------------- |-------|
| **Observed Count**     | 44               | 36        | 31        | 26               | 137   |
| **Expected Count**     | 13.7             | 82.2      | 27.4      | 13.7             | 137   |
| **Chi-squared**        | 67.014           | 25.966    | 0.473     | 11.04            | 104.496 |

Thus:
- \( A = 36 \)
- \( B = 13.7 \)
- \( C \approx 11.04 \)

Would you like more details on these calculations, or have any questions?

### 5 Related Questions:
1. What is the chi-squared test used for in statistical analysis?
2. How are expected counts calculated based on percentages in a chi-squared test?
3. Why is it important to ensure the total observed and expected counts are the same?
4. Can a chi-squared test be used for any type of data distribution?
5. What is the significance level in a chi-squared test, and how is it determined?

**Tip:** In chi-squared tests, if the observed count is significantly different from the expected count, the chi-squared value will be larger, indicating that the observed data may not fit the expected distribution.

Research question: We wish to investigate whether the age of tree felled at a specialty timber plantation is distributed according to the table below.

The age of tree | 1 (less than 15) | 2 (16-21) | 3 (22-26) | 4 (more than 26)
Percentage      |               10 |        60 |        20 |               10
Please complete the following table calculating the values of A, B and C.

|               | 1 (less than 15) | 2 (16-21) | 3 (22-26) | 4 (more than 26) | Total  |
|---------------|------------------|-----------|-----------|------------------|--------|
| Observed Count| 44               | A=***     | 31        | 26               | 137    |
| Expected Count| B=***            | 82.2      | 27.4      | 13.7             | 137    |
| Chi-squared   | 67.014           | 25.966    | 0.473     | C=***            | 104.496|

Investigate the distribution of tree age at a specialty timber plantation using chi-squared tests. This problem focuses on calculating the observed and expected counts for different age groups of trees, with solutions for values A, B, and C. Suitable for Grades 11-12 students studying statistics.

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