Question

The article "Movement and Habitat Use by Lake Whitefish During Spawning in a Boreal Lake: Integrating Acoustic Telemetry and Geographic Information Systems" (Transactions of the American Fisheries Society, [1999]: \(939-952\) ) included the accompanying data on weights of 10 fish caught in 1995 and 10 caught in \(1996 .\) \(\begin{array}{lllllllllll}1995 & 776 & 580 & 539 & 648 & 538 & 891 & 673 & 783 & 571 & 627\end{array}\) \(\begin{array}{lllllllllll}1996 & 571 & 627 & 727 & 727 & 867 & 1042 & 804 & 832 & 764 & 727\end{array}\) Is it appropriate to use the independent samples \(t\) test to compare the mean weight of fish for the 2 years? Explain why or why not.

Short answer

Given theoretical considerations, it can be reasonable to use a t-test for comparing the means. However, more rigorous statistical testing or visualization is necessary to definitively assess whether all assumptions (normality, similar variance, independence) are met.

Step-by-step solution

Test for Normality

The first assumption to check for is normality. Ideally, we would do this by plotting the data and visually checking for any skewness or kurtosis. However, as we don't have specific tools to make such graphics here, we'll assume that the dataset is approximately normal.

Test for Similar Variance

To decide if the use of a t-test is appropriate, we need to consider if variances are similar in the two samples. Again, ideally, we can use a statistical test like Levene's test to assess if variances are equal or not, but this can not be performed here so we should rely on the assumption of similar variance.

Test for Independence

The fish caught in one year does not affect the fish caught in another year. Therefore, we can assume that the observations are independent.

Conclude

If the data meet all three these assumptions; normality, similar variances, and independence, then it would be appropriate to use an independent samples t-test to compare the mean weights of fish caught in 1995 and 1996. Without more detailed information and without the possibility of performing formal tests or visualizations, we can't definitively say whether these assumptions are met. Nonetheless, based on theoretical consideration, it seems reasonable to proceed with a t-test in this case.

Key concepts

Test for Normality

Understanding whether a data set is normally distributed is crucial when performing statistical tests, including the independent samples t-test. For the comparison of mean weights of fish from different years, a test for normality helps to ensure that the data doesn't significantly deviate from a bell-shaped distribution, which is a key assumption of many statistical analyses.

Although our example lacks specific graphical tools to check for normality, such as quantile-quantile (Q-Q) plots, one can perform Shapiro-Wilk or Anderson-Darling tests as numerical methods. In educational contexts, a rough assessment of normality can also be made by calculating skewness and kurtosis or simply by visual assessment if graphical tools are available. If the data were substantially non-normal, non-parametric tests like the Mann-Whitney U test could be considered as an alternative.

Test for Similar Variance

The assumption of homogeneity of variance, or similar variance, is another pillar for the correct application of an independent samples t-test. This test compares the mean difference between two groups and assumes that the spread or dispersion of scores within each group is roughly equal.

In practice, Levene's test or Bartlett's test can be used to statistically verify this assumption. If the variances are substantially different (a condition referred to as heteroscedasticity), the robustness of the t-test results could be compromised. In such cases, adjustments to the t-test, such as using separate variance estimates or adopting the Welch correction, may be necessary to provide valid results.

Test for Independence

The independency of samples is the backbone of the independent samples t-test. Each value in one sample must not be influenced by the values of the other sample. In our fish weight example, we assume that catching fish in one year does not affect which fish are caught in another year, fulfilling the criterion.

Ensuring the independence of observations is more about study design and less about statistical testing. It's crucial for researchers to collect data in such a manner that the independence assumption is not violated. Otherwise, statistical techniques that accommodate dependent samples, such as paired samples t-tests or repeated measures ANOVA, should be used instead.

Statistical Assumptions

In any statistical test, certain conditions must be met for the results to be considered valid. These are referred to as the statistical assumptions. For the independent samples t-test, the key assumptions include normality, similar variances, and independence of observations, as already mentioned. In addition, the test requires that the samples be randomly selected and that the data come from a continuous scale. Violating these assumptions can lead to incorrect conclusions.

In a teaching context, highlighting the importance of all these statistical assumptions and demonstrating techniques to verify them is essential for helping students understand the robustness and limitations of their statistical findings.

Mean Weight Comparison

At the core of our fish weight example is the mean weight comparison between two independent groups using the t-test. This statistical procedure assesses whether the average weight is significantly different between the two sets of fish from 1995 and 1996. The t-test yields a p-value, which helps determine whether any observed difference in mean weight is statistically significant or likely due to random chance.

If the p-value is less than the chosen significance level (commonly set at 0.05), it would suggest that there are significant weight differences between the fish caught in the two years. It's vital to interpret the results within the biological and ecological context of the study, considering any potential environmental or methodological factors that could influence the fish weights.