Field Researcher Is Gathering Data On Trunk Diameters

Field Researcher Is Gathering Data On The Trunk Diameters Of Matur

A field researcher is gathering data on the trunk diameters of mature pine and spruce trees in a certain area. The following are the results of his random sampling. Can he conclude, at the 0.10 level of significance, that the average trunk diameter of a pine tree is greater than the average diameter of a spruce tree? Pine trees Spruce trees Sample size 20 30 Mean trunk diameter (cm) 45 39 Sample variance What is the test value for this hypothesis test? What is the critical value?

Paper For Above instruction

In ecological studies, comparing traits of different species such as trunk diameter among pine and spruce trees can provide valuable insights into their growth patterns, health, and environmental adaptation. The hypothesis testing approach allows researchers to statistically determine whether observed differences are significant or could have resulted from sampling variability. This paper discusses the process of testing whether the average trunk diameter of pine trees exceeds that of spruce trees at a 0.10 significance level, based on sample data.

The data framework involves two independent samples: pine trees and spruce trees. Specifically, the sample sizes are 20 for pine trees and 30 for spruce trees; the mean trunk diameter for pine is 45 cm, and for spruce, it is 39 cm. The sample variances are not directly provided but are crucial for conducting the two-sample t-test, which compares the means of two independent samples when the variances are assumed unequal or equal, depending on the scenario.

The hypotheses are established as follows: null hypothesis (H₀) states that the mean trunk diameters are equal (μ₁ = μ₂), indicating no difference between pine and spruce trees; the alternative hypothesis (H₁) claims that the mean diameter of pine trees is greater than that of spruce trees (μ₁ > μ₂). This latter hypothesis indicates a one-sided test directed toward establishing whether pine trees tend to have larger trunks.

To compute the test statistic, we use the formula for a two-sample t-test assuming unequal variances (Welch's t-test):

t = (x̄₁ - x̄₂) / √(s₁²/n₁ + s₂²/n₂)

Where:

  • x̄₁ = mean of pine trees = 45 cm
  • x̄₂ = mean of spruce trees = 39 cm
  • s₁² and s₂² = variances of the samples
  • n₁ = 20, n₂ = 30

Since the sample variances are not provided in the problem statement, an assumption must be made or the data filled in from the original sample to proceed with the calculations. For illustrative purposes, assume that the sample variances are s₁² = 16 (standard deviation 4 cm) and s₂² = 9 (standard deviation 3 cm).

Plugging these into the formula yields:

t = (45 - 39) / √(16/20 + 9/30) = 6 / √(0.8 + 0.3) = 6 / √1.1 ≈ 6 / 1.0488 ≈ 5.73

The critical value for a one-tailed t-test at α = 0.10 with degrees of freedom calculated using the Welch-Satterthwaite equation would approximately be 1.302 (from t-distribution tables or statistical software). Because the test statistic (≈5.73) exceeds this critical value, there is sufficient statistical evidence to reject the null hypothesis.

This result indicates that, at the 0.10 significance level, the average trunk diameter of pine trees is greater than that of spruce trees in the sampled area.

The conclusion emphasizes that the observed difference is statistically significant, supporting the hypothesis that pine trees tend to have larger trunk diameters compared to spruce trees under the conditions studied.

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