A Field Researcher Is Gathering Data On Trunk Diameters

A Field Researcher Is Gathering Data On The Trunk Diameters Of Mature

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 Mean trunk diameter (cm) Sample variance What is the test value for this hypothesis test? Test value: Round your answer to three decimal places. What is the critical value? Critical value: Round your answer to three decimal places. The following table(which is the attached file) shows a break down of the 113th U.S Congress by party affiliation. Question 1: A member of congress is selected at random. what is the probability of selecting a republican? Question 2: Given that the person selected is a member of the house of representatives, what is the probability he or she is a republican? Question 3: What is the probability of selecting a member of het house of representatives or a democrat?

Paper For Above instruction

The statistical analysis of trunk diameters of pine and spruce trees aims to determine whether the mean diameter of pine trees exceeds that of spruce trees. This involves formulating and testing a hypothesis using sample data, calculating the test statistic, identifying the critical value, and interpreting the results within the context of the significance level. Additionally, a separate probability problem deals with the composition of the 113th U.S. Congress, requiring calculation of probabilities related to party affiliation and membership status.

Hypothesis Testing for Tree Trunk Diameters

The problem investigates whether there is a statistically significant difference in the average trunk diameters between pine and spruce trees. To conduct this hypothesis test, the researcher would set up the null hypothesis (H0) that the mean diameter of pine trees equals that of spruce trees (μ_pine = μ_spruce), against the alternative hypothesis (H1) that the mean diameter of pine trees is greater than that of spruce trees (μ_pine > μ_spruce). The significance level is α = 0.10, indicating that there is a 10% risk of rejecting the null hypothesis when it is actually true.

The test statistic for comparing two independent means with known or estimated variances can be calculated using the t-test formula for unequal variances (Welch's t-test). The formula is:

t = (x̄₁ - x̄₂) / √(s₁²/n₁ + s₂²/n₂)

where x̄₁ and x̄₂ are the sample means, s₁² and s₂² are the sample variances, and n₁ and n₂ are the sample sizes for pine and spruce trees, respectively. This calculation yields the test value, which is then compared to the critical value derived from the t-distribution with appropriate degrees of freedom.

The critical value depends on the significance level and the degrees of freedom, which are estimated using the Welch-Satterthwaite equation. Considering the data provided, once the test statistic is computed, it must be checked against this critical value to decide whether the null hypothesis should be rejected.

Example Calculation and Interpretation

Suppose the sample data were as follows: for pine trees, n₁ = 50, mean = 35 cm, variance = 4; for spruce trees, n₂ = 45, mean = 32 cm, variance = 5. The test statistic would be calculated as:

t = (35 - 32) / √(4/50 + 5/45) ≈ 3 / √(0.08 + 0.1111) ≈ 3 / √0.1911 ≈ 3 / 0.4369 ≈ 6.868

Using a t-distribution table or software at α = 0.10 for a one-tailed test with the appropriate degrees of freedom, the critical value can be identified. If the computed t exceeds this critical value, the null hypothesis is rejected, leading to the conclusion that pine trees indeed have a greater average trunk diameter than spruce trees.

Probabilistic Analysis of Congressional Composition

The second part of the problem involves probability calculations based on the breakdown of the 113th U.S. Congress by party affiliation. Assuming the total number of members is known, and the number of Republicans, Democrats, and other parties are listed, probabilities can be computed as follows:

• Probability of randomly selecting a Republican: P(republican) = number of republicans / total members.
• Probability a member of the House of Representatives is a Republican: P(republican | House) = number of republican house members / total house members.
• Probability of selecting either a House member or a Democrat: P(House ∪ Democrat) = P(House) + P(Democrat) - P(House ∩ Democrat). Since a member cannot be both a House member and a Democrat simultaneously, the intersection is zero, simplifying the calculation.

Using the actual counts from the table, these probabilities provide insights into the party distribution and membership structure of Congress, which can be used for political analysis or electoral predictions.

Conclusion

The hypothesis test determines whether the trunk diameter of pine trees significantly exceeds that of spruce trees, based on sample data and at a defined significance level. The probability computations shed light on the party affiliation patterns within Congress. Both analyses employ fundamental statistical methods—hypothesis testing and probability calculations—that are essential tools in scientific research and political science.

References

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