Analyzing Timber Log Age Distribution Using Chi-Squared Test

Math Problem Statement

A random sample of 118 logs felled at a specialty timber plantation has been selected to study diversity of undergrowth and timber quality. Some of the variables recorded for each log are listed below.

Variable Description subid Subject ID timber wood species numSpecies number of different species of plant observed in the undergrowth within 30m of the tree usable the percentage of the log producing usable timber ageTreeCat the age of tree

Part 1

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 | A=*** | 23 | 37 | 19 | 118 Expected Count | 11.8 | 70.8 | B=*** | 11.8 | 118 chi squared | 62.698 | C=*** | 7.608 | 4.393 | 106.971 A [integer]: Answer 1 Question 1 B [use at least 3dp]: Answer 2 Question 1

Solution

To complete the table, we need to calculate the observed counts for age category 1 (A) and the expected count for age category 3 (B).

Given Information:

Total sample size: 118
Age distribution percentages:
- 1 (less than 15): 10%
- 2 (16-21): 60%
- 3 (22-26): 20%
- 4 (more than 26): 10%

Step 1: Calculate the Expected Counts

The expected counts are calculated based on the total sample size and the given percentages.

Age Category 1 (less than 15): $E_1 = 118 \times 0.10 = 11.8$
Age Category 2 (16-21): $E_2 = 118 \times 0.60 = 70.8$
Age Category 3 (22-26): $E_3 = 118 \times 0.20 = 23.6 \quad \text{(this is B)}$
Age Category 4 (more than 26): $E_4 = 118 \times 0.10 = 11.8$

Step 2: Calculate the Observed Counts

We already have observed counts for age categories 2, 3, and 4. We need to find the count for age category 1 (A).

The total observed count must equal 118: $A + 23 + 37 + 19 = 118$ $A + 79 = 118$ $A = 118 - 79 = 39$

Summary of Findings:

A (Observed Count for Age Category 1): 39
B (Expected Count for Age Category 3): 23.600

Completed Table:

Age Category	1 (less than 15)	2 (16-21)	3 (22-26)	4 (more than 26)	Total
Observed Count	A = 39	23	37	19	118
Expected Count	11.8	70.8	B = 23.600	11.8	118
Chi Squared	62.698	C = 1.500	7.608	4.393	106.971

Now, for part C, we can calculate the Chi-squared for Age Category 2 using the formula:

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

C (Chi-Squared for Age Category 2): $C = \frac{(23 - 70.8)^2}{70.8} = \frac{(-47.8)^2}{70.8} \approx 34.211$

Final Values:

A = 39
B = 23.600
C = 34.211

Would you like any further details or explanations? Here are some questions you might consider:

How do we interpret the results of the chi-squared test?
What assumptions are made when performing a chi-squared test?
How does the observed distribution compare to the expected distribution?
What implications does this analysis have for timber management practices?
How might additional variables impact the study's findings?

Tip: Always check the assumptions of the chi-squared test, especially the expected frequency in each category, which should ideally be at least 5.

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Math Problem Analysis

Mathematical Concepts

Statistics
Chi-Squared Test
Data Analysis

Formulas

Expected Count = Total Sample Size × Percentage
Chi-Squared = Σ((O - E)² / E)

Theorems

Chi-Squared Goodness of Fit

Suitable Grade Level

Grades 11-12

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