**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 | 33               | 26        | A=***     | 25               | 105
Expected Count | B=***            | 63        | 21        | 10.5             | 105
`chi squared    | 48.214           | 21.73     | 0         | C=***            | 89.968`

A [integer]:

Answer 6 Question 1

B [use at least 3dp]:

Answer 7 Question 1

To solve this, we'll need to use the given percentages and observed counts to compute the missing values \(A\), \(B\), and \(C\). The steps involve calculating the expected count, observed count, and chi-squared statistic for each category.

### Step 1: Calculate A (Observed Count for category 3: 22-26)
We know the total observed count is 105. Therefore, using the observed counts for other categories, we can calculate \(A\).

\[
A = 105 - (33 + 26 + 25) = 105 - 84 = 21
\]

### Step 2: Calculate B (Expected Count for category 1: less than 15)
The expected count for each category is based on the percentage provided in the table. For category 1, the percentage is 10%. So, the expected count for category 1 is:

\[
B = 10\% \text{ of } 105 = 0.10 \times 105 = 10.5
\]

### Step 3: Calculate C (Chi-squared value for category 4: more than 26)
The chi-squared statistic for each category 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 category 4 (more than 26), we know:
- \(O = 25\)
- \(E = 10.5\)

Substituting these values into the formula:

\[
C = \frac{(25 - 10.5)^2}{10.5} = \frac{(14.5)^2}{10.5} = \frac{210.25}{10.5} = 20.024
\]

### Final Table

| Age of tree    | 1 (less than 15) | 2 (16-21) | 3 (22-26) | 4 (more than 26) | Total |
|----------------|------------------|-----------|-----------|------------------|-------|
| Observed Count | 33               | 26        | 21        | 25               | 105   |
| Expected Count | 10.5             | 63        | 21        | 10.5             | 105   |
| Chi-squared    | 48.214           | 21.73     | 0         | 20.024           | 89.968|

### Answers:
- \( A = 21 \)
- \( B = 10.5 \) (3 decimal places)
- \( C = 20.024 \)

Would you like further details or clarifications?

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**Related Questions:**
1. How is the chi-squared statistic used in hypothesis testing?
2. What assumptions are made when using the chi-squared test?
3. How do we determine if the chi-squared value is significant?
4. What is the role of degrees of freedom in the chi-squared test?
5. How would different expected percentages affect the chi-squared statistic?

**Tip:** When calculating expected counts, always ensure the total expected counts sum to the total observed count.

We wish to investigate whether the age of tree felled at a specialty timber plantation is distributed according to the given table. Calculate the values of A, B, and C in the table for observed count, expected count, and chi-squared statistic.

Explore the chi-squared goodness-of-fit test applied to tree age distribution at a specialty timber plantation. The solution calculates observed count (A = 21), expected count (B = 10.5), and chi-squared statistic (C = 20.024), with step-by-step guidance. Ideal for college-level statistics students.

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