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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

Expert verified
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

01

Calculate the Pooled Standard Deviation

First, use the provided formula to calculate the pooled standard deviation (\(s_p\)): \(s_p=\sqrt{((n1 - 1) * s1^2 + (n2 - 1) * s2^2) / (n1 + n2 - 2)} = \sqrt{((11 - 1) * 0.0066^2 + (15 - 1) * 0.0071^2) / (11 + 15 - 2)}\).
02

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.
03

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

After the 2010 earthquake in Haiti, many charitable organizations conducted fundraising campaigns to raise money for emergency relief. Some of these campaigns allowed people to donate by sending a text message using a cell phone to have the donated amount added to their cell-phone bill. The report "Early Signals on Mobile Philanthropy: Is Haiti the Tipping Point?" (Edge Research, 2010 ) describes the results of a national survey of 1526 people that investigated the ways in which people made donations to the Haiti relief effort. The report states that \(17 \%\) of Gen \(Y\) respondents (those born between 1980 and 1988 ) and \(14 \%\) of Gen \(X\) respondents (those born between 1968 and 1979 ) said that they had made a donation to the Haiti relief effort via text message. The percentage making a donation via text message was much lower for older respondents. The report did not say how many respondents were in the Gen \(\mathrm{Y}\) and Gen \(\mathrm{X}\) samples, but for purposes of this exercise, suppose that both sample sizes were 400 and that it is reasonable to regard the samples as representative of the Gen \(\mathrm{Y}\) and Gen \(\mathrm{X}\) populations. a. Is there convincing evidence that the proportion of those in Gen Y who donated to Haiti relief via text message is greater than the proportion for Gen X? Use \(\alpha=.01\). b. Estimate the difference between the proportion of Gen \(\mathrm{Y}\) and the proportion of Gen \(\mathrm{X}\) that made a donation via text message using a \(99 \%\) confidence interval. Provide an interpretation of both the interval and the associated confidence level.

The paper "If It's Hard to Read, It's Hard to Do" (Psychological Science [2008]\(: 986-988)\) described an interesting study of how people perceive the effort required to do certain tasks. Each of 20 students was randomly assigned to one of two groups. One group was given instructions for an exercise routine that were printed in an easy-to-read font (Arial). The other group received the same set of instructions, but printed in a font that is considered difficult to read (Brush). After reading the instructions, subjects estimated the time (in minutes) they thought it would take to complete the exercise routine. Summary statistics are given below. $$ \begin{array}{ccc} & \text { Easy font } & \text { Difficult font } \\ \hline n & 10 & 10 \\ \bar{x} & 8.23 & 15.10 \\ s & 5.61 & 9.28 \\ \hline \end{array} $$ The authors of the paper used these data to carry out a two-sample \(t\) test, and concluded that at the .10 significance level, there was convincing evidence that the mean estimated time to complete the exercise routine was less when the instructions were printed in an easy-to-read font than when printed in a difficult-to-read font. Discuss the appropriateness of using a two-sample \(t\) test in this situation.

In a study of malpractice claims where a settlement had been reached, two random samples were selected: a random sample of 515 closed malpractice claims that were found not to involve medical errors and a random sample of 889 claims that were found to involve errors (New England Journal of Medicine [2006]: \(2024-2033\) ). The following statement appeared in the referenced paper: "When claims not involving errors were compensated, payments were significantly lower on average than were payments for claims involving errors \((\$ 313,205\) vs. \(\$ 521,560, P=0.004) . "\) a. What hypotheses must the researchers have tested in order to reach the stated conclusion? b. Which of the following could have been the value of the test statistic for the hypothesis test? Explain your reasoning. i. \(t=5.00\) ii. \(t=2.65\) iii. \(t=2.33\) iv. \(t=1.47\)

The report "Audience Insights: Communicating to Teens (Aged 12-17)" (www.cdc.gov, 2009) described teens' attitudes about traditional media, such as TV, movies, and newspapers. In a representative sample of American teenage girls, \(41 \%\) said newspapers were boring. In a representative sample of American teenage boys, \(44 \%\) said newspapers were boring. Sample sizes were not given in the report. a. Suppose that the percentages reported had been based on a sample of 58 girls and 41 boys. Is there convincing evidence that the proportion of those who think that newspapers are boring is different for teenage girls and boys? Carry out a hypothesis test using \(\alpha=.05\) b. Suppose that the percentages reported had been based on a sample of 2000 girls and 2500 boys. Is there convincing evidence that the proportion of those who think that newspapers are boring is different for teenage girls and boys? Carry out a hypothesis test using \(\alpha=.05\). c. Explain why the hypothesis tests in Parts (a) and (b) resulted in different conclusions.

The paper "The Observed Effects of Teenage Passengers on the Risky Driving Behavior of Teenage Drivers" (Accident Analysis and Prevention [2005]: 973-982) investigated the driving behavior of teenagers by observing their vehicles as they left a high school parking lot and then again at a site approximately \(\frac{1}{2}\) mile from the school. Assume that it is reasonable to regard the teen drivers in this study as representative of the population of teen drivers. Use a .01 level of significance for any hypothesis tests. a. Data consistent with summary quantities appearing in the paper are given in the accompanying table. The measurements represent the difference between the observed vehicle speed and the posted speed limit (in miles per hour) for a sample of male teenage drivers and a sample of female teenage drivers. Do these data provide convincing support for the claim that, on average, male teenage drivers exceed the speed limit by more than do female teenage drivers? $$ \begin{array}{cc} \hline \text { Male Driver } & \text { Female Driver } \\ \hline 1.3 & -0.2 \\ 1.3 & 0.5 \\ 0.9 & 1.1 \\ 2.1 & 0.7 \\ 0.7 & 1.1 \\ 1.3 & 1.2 \\ 3 & 0.1 \\ 1.3 & 0.9 \\ 0.6 & 0.5 \\ 2.1 & 0.5 \\ \hline \end{array} $$ b. Consider the average miles per hour over the speed limit for teenage drivers with passengers shown in the table at the top of the following page. For purposes of this exercise, suppose that each driver-passenger combination mean is based on a sample of size \(n=40\) and that all sample standard deviations are equal to .8 . $$ \begin{array}{l|cc} & \text { Male Passenger } & \text { Female Passenger } \\ \hline \text { Male Driver } & 5.2 & .3 \\ \text { Female Driver } & 2.3 & .6 \\ \hline \end{array} $$ i. Is there sufficient evidence to conclude that the average number of miles per hour over the speed limit is greater for male drivers with male passengers than it is for male drivers with female passengers? ii. Is there sufficient evidence to conclude that the average number of miles per hour over the speed limit is greater for female drivers with male passengers than it is for female drivers with female passengers? iii. Is there sufficient evidence to conclude that the average number of miles per hour over the speed limit is smaller for male drivers with female passengers than it is for female drivers with male passengers? c. Write a few sentences commenting on the effects of gender on teenagers driving with passengers.

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