Question:

Community ecologists draw "food webs" to describe the predator and prey relationships between all organisms living in an area. A theoretical model predicts that a measure of the structure of food webs called "diet discontinuity" should be zero. Diet discontinuity is a measure of the relative numbers of predators whose prey are not ordered contiguously. Researchers have measured discontinuity scores for seven different food webs in nature. The values are: 0.35, 0.08, 0.004, 0.08, 0.32, 0.28, 0.17. Assume discontinuity in natural food webs has a normal distribution. Are the results significantly different from the theoretical expectation? Answer TRUE for 'yes' and FALSE for 'no'.

True

False

2.As the world warms, geographic ranges of species might shift toward cooler areas. In the file 'RangeShiftsWithClimateChange.csv' are data from a study on the highest elevations at which species occur. Typically, higher elevations are cooler than lower elevations. The aforementioned file contains data in the changes in highest elevation for 31 taxa in meters between the late 1990's and the early 2000's. Positive numbers indicate upward shifts in elevation and negative numbers indicate shifts to lower elevations. Based on this information, the mean altitude shift is , the lower 95% confidence limit is , and the upper 95% confidence limit is . Please round your answers to zero decimal places (i.e., only provide rounded numbers to the left of where the decimal point would go).

3.Assume that Z is a number randomly chosen from a standard normal distribution. Use R to calculate P[Z

4.The babies born in singleton births in the United States have birth weights that are approximately normally distributed with mean 3.296 kg and standard deviation 0.560 kg. What is the probability of a newborn weighing between 3 and 4 kg? Round your answer to two decimal places.

5.It was rumored that Britain's domestic intelligence service, MI5, has an upper limit on the height of its spies, on the assumption that tall people stand out. The rumor also said that to be a spy, you could be no taller than 5 feet 11 inches (180.3 cm) if you are a man. If the true mean height of men in Britain is 177 cm and the true standard deviation is 7.1 cm, what fraction of men would be precluded from applying to MI5? Please round your answer to two decimal places.

6.Can a human swim faster in water or in syrup? It is unknown whether the increase in the friction of the body moving through syrup is compensated by the increased power of each stroke. The results of an experiment in which researchers filled one pool with water mixed with syrup and another with normal water are in the file 'SyrupSwimming.csv'. The data are presented as the relative speed of each swimmer in syrup (i.e., speed in syrup / speed in water). If syrup has no effect on swimming, the mean should be one. Assuming these data are normally distributed, use this information to test the null hypothesis that syrup has no effect on swim peed. Answer TRUE if syrup does have a statistically significant effect and FALSE if the test fails to

A Markov chain with state space {1, 2, 3) has transition probability matrix 0.6 0.3 0.1\\ P. = 0.3 0.3 0.4 0.4 0.1 0.5 (a) Is this Markov chain irreducible? Is the Markov chain recurrent or transient? Explain your answers. (b) What is the period of state 1? Hence deduce the period of the remaining states. Does this Markov chain have a limiting distribution? (c) Consider a general three-state Markov chain with transition matrix Pit P12 P13 P = P21 P22 P23 P31 P32 P33 Give an example of a specific set of probabilities paj for which the Markov chain is not irreducible (there is no single right answer to this, of course !).Problem 3. Consider the Markov chain shown in Figure 2. Figure 2: Problem 3 Markov chain 1' Let the initial distribution be Pr(A) = Pr(B) = 0.5. What is the probability distribution after one step? 2. What is the stationary distribution of the Markov chain? State-space equation of a linear autonomous system are given below -5/3 4/3 a x ( t ) = - 1/ 3 0 7/3 O - 2 a ) find the eat state transition matrix of system by the Lapalace transform b ) find the At state transition matrix of system with the Cayley - Hamilton Theorem ( ) find the efit state transition matrix of system using eigenvectors