Question

Do certain behaviors result in a severe drain on energy resources because a great deal of energy is expended in comparison to energy intake? The article "The Energetic Cost of Courtship and Aggression in a Plethodontid Salamander" (Ecology [1983]: 979-983) reported on one of the few studies concerned with behavior and energy expenditure. The accompanying table gives oxygen consumption \((\mathrm{mL} / \mathrm{g} / \mathrm{hr})\) for male-female salamander pairs. (The determination of consumption values is rather complicated. It is partly for this reason that so few studies of this type have been carried out.) $$ \begin{array}{lccc} \text { Behavior } & \begin{array}{c} \text { Sample } \\ \text { Size } \end{array} & \begin{array}{c} \text { Sample } \\ \text { Mean } \end{array} & \begin{array}{c} \text { Sample } \\ \text { sd } \end{array} \\ \hline \text { Noncourting } & 11 & .072 & .0066 \\ \text { Courting } & 15 & .099 & .0071 \\ \hline \end{array} $$ a. The pooled \(t\) test is a test procedure for testing \(H_{0}: \mu_{1}-\mu_{2}=\) hypothesized value when it is reasonable to assume that the two population distributions are normal with equal standard deviations \(\left(\sigma_{1}=\right.\) \(\sigma_{2}\) ). The test statistic for the pooled \(t\) test is obtained by replacing both \(s_{1}\) and \(s_{2}\) in the two-sample \(t\) test statistic with \(s_{p}\) where \(s_{p}=\sqrt{\frac{\left(n_{1}-1\right) s_{1}^{2}+\left(n_{2}-1\right) s_{2}^{2}}{n_{1}+n_{2}-2}}\) When the population distributions are normal with equal standard deviations and \(H_{0}\) is true, the resulting pooled \(t\) statistic has a \(t\) distribution with \(\mathrm{df}=n_{1}+\) \(n_{2}-2 .\) For the reported data, the two sample standard deviations are similar. Use the pooled \(t\) test with \(\alpha=.05\) to determine whether the mean oxygen consumption for courting pairs is higher than the mean oxygen consumption for noncourting pairs. b. Would the conclusion in Part (a) have been different if the two-sample \(t\) test had been used rather than the pooled \(t\) test?

Short answer

The conclusion may change if using the two-sample t-test instead of the pooled t-test, since these two tests are based on different conditions, especially the assumption on the population standard deviations. It is important to choose the appropriate test based on the information available and the context of the study.

Step-by-step solution

Calculate the Pooled Standard Deviation

First, use the provided formula to calculate the pooled standard deviation (\(s_p\)): \(s_p=\sqrt{((n1 - 1) * s1^2 + (n2 - 1) * s2^2) / (n1 + n2 - 2)} = \sqrt{((11 - 1) * 0.0066^2 + (15 - 1) * 0.0071^2) / (11 + 15 - 2)}\).

Apply Pooled t-test

Next, apply the pooled t-test using the formula \(t = (x1 - x2) / (s_p * \sqrt{1/n1 + 1/n2})\) where \(x1\) and \(x2\) are the sample means, \(n1\) and \(n2\) are the sample sizes. Substitute the given and calculated values to get \(t = (0.099 - 0.072) / (s_p * \sqrt{1/11 + 1/15})\). Compare the calculated t value with the critical t value to determine whether the null hypothesis should be rejected.

Two-sample t-test

For Part (b) we use two-sample t-test without assuming that the population standard deviations are equal, the t statistic is calculated with the formula \(t = (x1 - x2) / \sqrt{(s1^2/n1) + (s2^2/n2)}\). This test would yield a different t value when compared to the pooled t-test.

Key concepts

Pooled t-test

Understanding the pooled t-test is important for comparing the means of two independent groups when their population standard deviations are assumed to be equal. This test is based on the idea that the variability of each group can be pooled into a single estimate, thereby simplifying the comparison between the two.

• The pooled standard deviation, denoted as \( s_p \), combines the variances of both groups to provide a more stable estimate when sample sizes are small.
• The formula for the pooled standard deviation is \[s_p = \sqrt{\frac{(n_1 - 1)s_1^2 + (n_2 - 1)s_2^2}{n_1 + n_2 - 2}}.\]

For our salamander study, since the standard deviations between courting and noncourting pairs are similar, this method can be applied. The pooled t-test checks if there is a significant difference between the two means by comparing the calculated t-value against a critical value from the t-distribution.

Two-sample t-test

The two-sample t-test is a method used to determine if there are significant differences between the means of two independent groups. Unlike the pooled t-test, this test does not assume equal variances between the groups.

• This flexibility makes it preferable when the equality of variances is questionable.
• The formula for the two-sample t-test is \[t = \frac{(x_1 - x_2)}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}.\]

For the salamander problem, if we chose to use this test rather than the pooled t-test, we might have obtained a slightly different t-value and therefore a different conclusion regarding our hypothesis on energy expenditure. It is important to consider which test provides the most accurate results for your data set.

Energy expenditure

Energy expenditure is a crucial aspect of studying animal behavior, as it sheds light on how much energy different activities consume and their impact on an organism’s overall energy balance. In research contexts, like the salamander study, oxygen consumption is a common measure.

• Higher oxygen consumption indicates greater energy expenditure, as oxygen is vital in the metabolic process of energy production.
• In our exercise, the mean oxygen consumption differs between courting and noncourting salamanders, suggesting that courtship behavior may demand more energy compared to non-courting behavior.

By examining energy expenditure, researchers can infer potential impacts on survival, reproduction, and overall fitness, making it a critical component in behavioral ecology studies.

Salamander behavior studies

Studying salamander behavior is essential in behavioral ecology as it provides insight into their ecological roles and adaptations. Salamanders, like many other species, exhibit behaviors that are closely tied to their energy requirements.

• In the context of the provided study, investigating behaviors such as courtship and aggression reveals their energy costs and how they affect the salamanders’ energy budgets.
• Behavioral studies in this context can help scientists understand the balance organisms need to maintain between energy intake and expenditure for successful reproduction and survival.

Such research is pivotal in understanding broader ecological interactions and can illuminate the constraints faced by organisms in their natural habitats. This particular study helps to measure the fundamental costs associated with different behavioral strategies.