Strategic Placement Of Lobster Traps Is One Of The Keys For

Strategic Placement Of Lobster Traps Is One Of The Keys For A Successf

Strategic placement of lobster traps is one of the keys for a successful lobster fisherman. An observational study of teams fishing for the red spiny lobster in Baja California Sur, Mexico, was conducted and the results published in Bulletin of Marine Science (April 2010). One of the variables of interest was the average distance separating traps—called trap spacing—deployed by the same team of fishermen. Trap-spacing measurements (in meters) for a sample of seven teams of red spiny lobster fishermen are shown below. (Source: Based on Shester, G. G. “Explaining catch variation among Baja California lobster fishers through spatial analysis of trap-placement decisions,” Bulletin of Marine Science, Vol. 86, No. 2, April 2010, pp. 479–498.) Of interest is the mean trap spacing for the population of red spiny lobster fishermen fishing in Baja California Sur, Mexico. Previously, you calculated the mean and standard deviation of the seven sample measurements to be x̄ = 89.9 meters and s = 11.6 meters, respectively. Now, the researchers want to know whether σ², the variation in the population of trap-spacing measurements, is larger than 10 m². They will conduct a test of hypothesis using α = 0.05.

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In the realm of marine resource management, especially in lobster fisheries, optimal trap placement plays a crucial role in maximizing catch efficiency and ensuring sustainable harvesting. The strategic placement of lobster traps influences both the catch rates and the ecological impact of fishing activities. Analyzing the data collected from fishermen in Baja California Sur highlights the importance of understanding spatial variability in trap positioning to inform management decisions and improve fishery outcomes.

The study outlined above seeks to statistically assess whether the population variance in trap spacing exceeds a threshold of 10 m². This involves hypothesis testing with the primary goal of evaluating the variability in trap deployment among fishermen. Specifically, the null hypothesis (H₀) states that the population variance σ² is less than or equal to 10 m², while the alternative hypothesis (H₁) posits that σ² exceeds 10 m². Conducting this test helps fishery managers and scientists understand the degree of inconsistency in trap spacing, which can be related to behavioral, environmental, or operational factors.

Before executing the statistical test, it is essential to recognize potential issues stemming from the decision rule. The fisherman’s desire to reject H₀ based on the sample variance s² > 10 introduces problems, particularly because the sample variance may not accurately reflect the population variance in small samples. With a small sample size of only seven teams, the variability in the estimate and the possibility of Type I errors—incorrectly rejecting a true null hypothesis—are heightened. Relying solely on the sample variance without considering the sampling distribution's properties can lead to misleading conclusions, especially when the sample size is limited, and the data may not satisfy underlying assumptions.

Calculating the test statistic involves using the chi-square distribution, given that the test pertains to variance. The formula for the chi-square test statistic is:

χ² = (n - 1) * s² / σ₀²

where n is the sample size and σ₀² is the hypothesized population variance. Here, n = 7, s² = (11.6)² = 134.56, and σ₀² = 10. Substituting these values:

χ² = (7 - 1) 134.56 / 10 = 6 134.56 / 10 = 806.76 / 10 = 80.676

The p-value corresponds to the probability of observing a chi-square statistic as extreme or more so under the null hypothesis. Since the alternative hypothesis is that the variance is larger than 10, this is a one-tailed test on the right tail. Using chi-square distribution tables or software to find the p-value for χ² = 80.676 with degrees of freedom df = 6, we find that such a high chi-square value is extremely unlikely under H₀.

The approximate p-value is practically zero, indicating strong evidence against H₀. Therefore, the data suggest that the population variance in trap spacing exceeds 10 m². Given the significance level of α = 0.05, the decision would typically be to reject the null hypothesis, concluding that the variation in trap placement among fishermen is statistically larger than the specified threshold.

However, this conclusion hinges on certain assumptions being satisfied. The primary conditions for the validity of the chi-square test of variance include the data being independent, normally distributed, and randomly sampled. Small sample sizes, like n=7, make the normality assumption particularly critical because the chi-square distribution's shape is sensitive to deviations from normality. If these conditions are not met, the validity of the test results could be compromised, potentially leading to inaccurate conclusions about the true variance in trap spacing.

In sum, the hypothesis test indicates significant variability in trap spacing among Baja California's lobster fishermen, underscoring the need for targeted management strategies that consider spatial behavior. Ensuring the underlying assumptions are satisfied is vital for accurate inference, especially given the small sample size. Future research should aim to collect larger datasets and verify normality, to reinforce the robustness of such statistical assessments and improve sustainable fishing practices.

References

  • Shester, G. G. (2010). Explaining catch variation among Baja California lobster fishers through spatial analysis of trap-placement decisions. Bulletin of Marine Science, 86(2), 479–498.
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  • Buell, M. F. & Selig, E. R. (2014). Spatial variability and implications for lobster trapping practices. Marine Policy, 45, 181-188.
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