Which of the following is an example of the feel-good, do-good phenomenon? a. Anagha tries to keep her 2-year-old son from becoming upset for any reason. b. Niko is praised by his teacher and later volunteers to help his mother clean the garage. c. Carrsan felt satisfied after helping the ecology club pick up trash around the school. d. Maggie’s parents doubled her allowance when she was nice to her sister all day. e. Rowen was glad to finally be over the flu so he could return to his volunteer work.

Which of the following has shown effectiveness in improving sleep, reducing depression, and increasing heart and lung fitness? a. Meditation. b. Aerobic exercise. c. Religious involvement. d. Subjective well-being. e. Social support.

Isle Royale, an island in Lake Superior, has provided an important study site of wolves and their prey. In the National Park Service Scientific Monograph Series 11, Wolf Ecology and Prey Relationships on Isle Royale, Peterson gives results of many wolf–moose studies. Of special interest is the study of the number of moose killed by wolves. In the period from 1958 to 1974, there were 296 moose deaths identified as wolf kills. The age distribution of the kills is as follows.

$$\begin{matrix} \text{Age of Moose in Years} & \text{Number Killed by Wolves}\\ \text{Calf (0.5 yr)} & \text{112}\\ \text{1-5} & \text{53}\\ \text{6-10} & \text{73}\\ \text{11-15} & \text{56}\\ \text{16-20} & \text{2}\\ \end{matrix}$$

For each age group, compute the probability that a moose in that age group is killed by a wolf.

Elk dwell in a certain forest. There are $N$ elk, of which a simple random sample of size $n$ are captured and tagged (“simple random sample” means that all

$$\left( \begin{array} { c } { N } \\ { n } \end{array} \right)$$

sets of $n$ elk are equally likely). The captured elk are returned to the population, and then a new sample is drawn, this time with size $m$. This is an important method that is widely used in ecology, known as capture-recapture. What is the probability that exactly $k$ of the $m$ elk in the new sample were previously tagged? (Assume that an elk that was captured before doesn’t become more or less likely to be captured again.)

According to ecology writer Jacqueline Killeen, phosphates contained in household detergents pass right through our sewer systems, causing lakes to turn into swamps that eventually dry up into deserts. The following data show the amount of phosphates per load of laundry, in grams, for a random sample of various types of detergents used according to the prescribed directions: $$ \begin{matrix} \textbf{Laundry Detergent} & \textbf{Phosphates per Load (grams)} \\ \hline \\ \text{A & P Blue Sail} & 48 \\ \text{Dash} & 47 \\ \text{Concentrated All} & 42 \\ \text{Cold Water All} & 42 \\ \text{Breeze} & 41 \\ \text{Oxydol} & 34 \\ \text{Ajax} & 31 \\ \text{Sears} & 30 \\ \text{Fab} & 29 \\ \text{Cold Power} & 29 \\ \text{Bold} & 29 \\ \text{Rinso} & 26 \end{matrix} $$ For the given phosphate data, find (a) the mean; (b) the median; (c) the mode.

An ecology center wants to set up an experimental garden using 300 meters of fencing to enclose a rectangular area of 5000 square meters.

(a) Let x meters represent the width of the garden. Why must the length be 150-x meters?

(b) Define an expression A(x) that represents the area of the garden.

(c) What are the restrictions on x?

(d) Use the given values for area and length of fencing to find the dimensions of the garden.

Ecologists believe that organic material decays over time according to an exponential decay model. This is the case 0 < $\beta$ < 1 in the exponential regression model, for which $\mu_y$ decreases over time. The rate of decay is determined by a number of factors, including composition of material, temperature, and humidity. In an experiment carried out by researchers at the University of Georgia Ecology Institute, leaf litter was allowed to sit for a 20-week period in a bag in a moderately forested area. Initially, the total weight of the organic mass in the bag was 75.0 kg. Each week, the remaining amount (y) was measured. The table shows the weighty by x = number of weeks of time that have passed.

x	y	x	y	x	y	x	y	x	y	x	y
0	75.0	1	60.9	2	51.8	3	45.2	4	34.7	5	34.6
6	26.2	7	20.4	8	14.0	9	12.3	11	8.2	15	3.1
20	1.4

Plot the log of y against x. Does a straight-line model now seem appropriate?

The school Ecology Club wants to fence in an area along the riverbank to protect endangered wildflowers that grow there. The club has enough money to buy. 200 feet of fencing. It decides to enclose a rectangular space. The fence will form three sides of the rectangle, and the riverbank will form the fourth side. a. If the width of the enclosure is 30 feet, how much fencing material is available for the length? Sketch this situation. What is the area? b. If the width is w, how much fencing material remains for the length, l? c. Use your answer from 11b to write an equation of the area of the rectangle in factored form. Check your equation with your width and area from 11a. d. Which two different widths would give an area equal to 0? e. Which width will give the maximum area? What is that area?

Ecology Labs Inc. will pay a dividend of $\$ 6.40$ per share in the next 12 months $\left(D_1\right)$. The required rate of return $\left(K_e\right)$ is 14 percent and the constant growth rate is 5 percent. (For parts $b, c$, and $d$ in this problem, all variables remain the same except the one specifically changed. Each question is independent of the others.) Assume $D_1$ is $\$ 7.00$. What will be the new value of $P_0$ ? Assume $K_e$ is at its original value of 14 percent and $g$ goes back to its original value of 5 percent.

(Ecology-Pollution) A manufacturer makes two kinds of products, $P$ and $Q$, at each of two plants, $X$ and $Y$. In making these products, the pollutants sulfur dioxide, nitric oxide, and particulate matter are produced. The amounts of pollutants produced are given (in kilograms) by the matrix

State and federal ordinances require that these pollutants be removed. The daily cost of removing each kilogram of pollutant is given (in dollars) by the matrix

$$B=\left[\begin{array}{cc}\text { Plant } X & \text { Plant } Y \\ 8 & 12 \\ 7 & 9 \\ 15 & 10\end{array}\right]$$

Sulfur dioxide What do the entries in the matrix product $A B$ tell the manufacturer?

In the haploid model of natural selection, there is one copy of every gene within the genome. Consider two groups of individuals, one carrying the allele $A$ and the other the allele a. Then the proportion that carries the allele $A$ can be written as

$$p=\frac{n_A}{n_A+n_\mu} .$$

Source: A Biologist's Guide to Mathematical Modeling in Ecology and Evolution. Show that if both $n_A$ and $n_a$ are changing with time. then the rate of change of the proportion that carry the allele $A$ is

$$\frac{d p}{d t}=\frac{n_a \frac{d n_A}{d t}-n_A \frac{d n_a}{d t}}{\left(n_A+n_d\right)^2} .$$

Ecologists believe that organic material decays over time according to an exponential decay model. This is the case 0 < $\beta$ < 1 in the exponential regression model, for which $\mu_y$ decreases over time. The rate of decay is determined by a number of factors, including composition of material, temperature, and humidity. In an experiment carried out by researchers at the University of Georgia Ecology Institute, leaf litter was allowed to sit for a 20-week period in a bag in a moderately forested area. Initially, the total weight of the organic mass in the bag was 75.0 kg. Each week, the remaining amount (y) was measured. The table shows the weighty by x = number of weeks of time that have passed.

x	y	x	y	x	y	x	y	x	y	x	y
0	75.0	1	60.9	2	51.8	3	45.2	4	34.7	5	34.6
6	26.2	7	20.4	8	14.0	9	12.3	11	8.2	15	3.1
20	1.4

Construct a scatterplot. Why is a straight-line model inappropriate?

The convict surgeonfish is a common tropical reef fish that has been found to delay metamorphosis into adult by extending its larval phase. This delay often leads to enhanced survivorship in the species by increasing the chances of finding suitable habitat. In the paper “Delayed Metamorphosis of a Tropical Reef Fish (Acanthurus triostegus): A Field Experiment” (Marine Ecology Progress Series, Vol. 176, pp. 25-38), M. McCormick published data that he obtained on the larval duration, in days, of 90 convict surgeonfish. The data are given on the WeissStats CD. At the 5% significance level, do the data provide sufficient evidence to conclude that the mean larval duration of convict surgeonfish exceeds 52 days? a. Employ the Wilcoxon signed-rank test. b. Employ the t-test. c. Compare your results from parts (a) and (b).

Opisthotrochopodus n. sp. is a polychaete worm that inhabits deep-sea hydrothermal vents along the Mid-Atlantic Ridge. According to the article “Reproductive Biology of Free-Living and Commensal Polynoid Polychaetes at the Lucky Strike Hydrothermal Vent Field (Mid-Atlantic Ridge)” (Marine Ecology Progress Series, Vol. 181, pp. 201-214) by C. Van Dover et al., the lengths of female polychaete worms are normally distributed with mean 6.1 mm and standard deviation 1.3 mm. Let X denote the length of a randomly selected female polychaete worm. Determine and interpret a. $P(X \leq 3)$. b. P(5<X<7).