Question

The weights of turtles caught in two different lakes were measured to compare the effects of the two lake environments on turtle growth. All the turtles were the same age and were tagged before being released into the lakes. The weights for \(n_{1}=10\) tagged turtles caught in lake 1 and \(n_{2}=8\) caught in lake 2 are listed here: $$ \begin{array}{lllllllllll} \hline \text { Lake } & \multicolumn{7}{c} {\text { Weight (grams) }} \\ \hline 1 & 399 & 430 & 394 & 411 & 416 & 391 & 396 & 456 & 360 & 433 \\ 2 & 345 & 368 & 399 & 385 & 351 & 337 & 354 & 391 & & \\ \hline \end{array} $$ Do the data provide sufficient evidence to indicate a difference in the distributions of weights for the tagged turtles exposed to the two lake environments? Use the Wilcoxon rank sum test with \(\alpha=.05 .\)

Short answer

In this problem, we used the Wilcoxon rank sum test to determine if there was enough evidence to suggest a difference in the distributions of weights of tagged turtles from two different lakes. The test statistic, W, was calculated to be 89, and the critical value was found to be approximately 1.96. As W is much larger than the critical value, there is insufficient evidence to reject the null hypothesis that the distributions of weights are the same for turtles exposed to the two lake environments at a significance level α = 0.05.

Step-by-step solution

Combine and rank the data

Merge the two datasets and then rank the combined data from smallest to largest. Assign the smallest value a rank of 1, the next smallest value a rank of 2, and so on. In case of tied values, assign the average of the ranks they span. Combined data: 345, 337, 354, 360, 368, 385, 399, 391, 399, 394, 411, 416, 430, 433, 456 Ranked data: $$ \begin{array}{lllllllllll} \hline \text {Lake} & \multicolumn{7}{c} {\text {Weight (grams)}} & \text{Rank} \\ \hline 1 & 399 & 430 & 394 & 411 & 416 & 391 & 396 & 456 & 360 & 433 \\ & 8.5 & 13 & 7 & 11 & 12 & 6 & 5 & 15 & 3 & 14 \\ 2 & 345 & 368 & 399 & 385 & 351 & 337 & 354 & 391 \\ & 1 & 4 & 8.5 & 10 & 2 & 0 & 9 & 6 \\ \hline \end{array} $$

Calculate the test statistic, W

Sum the ranks for lake 1: $$ W = 3 + 13 + 7 + 11 + 12 + 6 + 5 + 15 + 3 + 14 $$ $$ W = 89 $$

Find the critical value

Using either a Wilcoxon rank sum table or more generally using the z-test approximation, find the critical value for \(\alpha = 0.05\). For this scenario, with \(n_1 = 10\) and \(n_2 = 8\), the critical value using the z-test approximation is approximately 1.96.

Determine if there is enough evidence to reject the null hypothesis

To determine if there is enough evidence to reject the null hypothesis, compare the test statistic W to the critical value. Here, \(W = 89\) and the critical value is approximately 1.96. Since W is much larger than the critical value, there is not enough evidence to reject the null hypothesis. Hence, there is insufficient evidence to indicate a difference in the distributions of weights for the tagged turtles exposed to the two lake environments at \(\alpha = 0.05\).

Key concepts

Non-Parametric Statistics

When working with statistical data analysis, it's quite common to encounter situations where standard assumptions for parametric tests do not hold. This is where non-parametric statistics come into play. These methods allow you to perform analyses without making assumptions about the distribution of your data, such as assuming the data is normally distributed. The Wilcoxon rank sum test, which is the focus of the exercise we're discussing, is a prime example of a non-parametric test.

This test is especially useful when dealing with small sample sizes or when the data is skewed or has outliers that might influence the results of parametric tests. It is applied to compare the median of two independent samples, assessing whether their populations differ significantly.

Rather than working with the data values themselves, the Wilcoxon rank sum test uses the ranks of the data. It is robust against non-normality and can be a powerful alternative to the t-test when its assumptions cannot be met. Understanding how to apply this test can provide a great tool for situations with data that challenges the norms of standard parametric analyses.

Hypothesis Testing

In hypothesis testing, we're essentially playing a game of proof or disproof, trying to determine whether there's enough evidence to support a particular claim about a population parameter. The goal is to make an inference about the population, based on sample data. This inference comes in the form of a decision - whether to reject our initial assumption, namely the null hypothesis, in favor of an alternative hypothesis.

The process involves setting up two contradictory hypotheses: the null hypothesis (\(H_0\)) and the alternative hypothesis (\(H_a\) or \(H_1\)). The null hypothesis generally states that there is no effect or no difference, while the alternative suggests there is.

To decide which hypothesis is more likely, we calculate a test statistic and compare it to a critical value. This value is determined by the significance level (\( \text{denoted as } \beta \text { or sometimes } \text {as } \text {alpha} \text { and typically set at } 0.05\text { or } 5\text{%} \text{ probability of making a Type I error}\text {.}\text { The smaller it is, the more stringent the test is}\text {.}\text { If this statistic exceeds the critical value, the null hypothesis is rejected}\text {.}\text { The Wilcoxon rank sum test is a part of this broader process of hypothesis testing, tailored for specific situations where other tests like the t-test may not be applicable}\text {.

Probability and Statistics

The realm of probability and statistics provides the foundation for making sense of quantitative data, drawing conclusions and making predictions about real-world phenomena. Probability offers a framework for quantifying uncertainty, assigning numerical values to the likelihood of different outcomes of random events. Statistics builds upon this by creating methodologies for collecting, analyzing, interpreting, presenting, and organizing data.

These methodologies comprise both descriptive and inferential statistics. Descriptive statistics summarize the features of a dataset, such as the mean, median, and standard deviation, providing a quick glance at its shape and characteristics. Inferential statistics, on the other hand, use this data to make inferences about a population, often through hypothesis testing.

Understanding probability and statistics is pivotal for making informed decisions in various fields. It empowers researchers to differentiate between mere chance and statistically significant effects. It's also the reason why, in the context of our exercise, the Wilcoxon rank sum test is used to assess the significance of the observed differences in turtle weights across two lake environments – it incorporates concepts from probability to draw statistical inferences.