Question:

You are a research psychologist at ASU and are interested in recommending stress management strategies for students. You know that a requirement of a good strategy is that it is something students are likely to use (i.e., they enjoy it or, at the very least, don't hate it). You have scoured the literature and identified four strategies that you have been shown to be both effective and have a high usage rate in past studies. These are Strategies A, B, C, and D. You know that Strategy B is most commonly used by students but you suspect that is because it is often the only strategy that is taught and, therefore is often the only strategy of which students are aware. You design an experiment to find out which strategy is actually the most used by students when they know about all four strategies. You deliver training that teaches all four strategies to a group of students, and then you measure which strategy they tend to use over a period of time. You do not measure other variables.

Below I have listed five different research hypotheses based on the scenario above. For each, briefly discuss why the hypothesis is, or is not, appropriate paying particular attention to (a) whether the hypothesis makes a prediction, and (b) whether the hypothesis is testable given the research design and measures.

2. When all four strategies are available to students, they will choose Strategy B because it requires the least amount of time/effort.

3. When all four strategies are available to students, they will most often choose Strategy B.

4. Students who use Strategy B will experience less stress than students who use Strategy A, C, or D.

5. When all four strategies are available to students, which strategy will they prefer?

6. Which of the five hypotheses above do you think is most appropriate given the research scenario described?

7. If you were interested in testing one of the other research hypotheses (i.e., hypotheses other than the hypothesis you listed for #6), how might you modify the research scenario? Be sure to list the number of the hypothesis, as well as the change(s) to the design of the research.

6.3 The waiting time If until the rst error in a binary sequence is described by the (type 1} geometric PDF. Let the probability of an error be p = US. [a] Compute the mean and Variance of K. (b) Apply the Markov inequality to bound the probability of the event K 23 16. [e]: Use the Cbebyshev inequality to bound the probability of this event. (cl) Use the oue~sidetl {Zlhebyshetr inequality to bound the probability. {e} Compute the actual probability of the event K E 16. {Child} la=3 4. An inequality developed by Russian mathematician Chebyshev gives the minimum percentage of values in ANY sample that can be found within some number (k 1} standard deviations from the mean. Let P be the percentage of values within .14: standard deviations of the mean value. Chebyshevs inequality states that for ANY distribution, P 100* (i 1/k:)2. (a) For any distribution, what does Chebyshevs inequality say about the percentage of values that are within 2, 3. 5. 10 standard deviations of the mean? (b) For each of the distributions below, determine the percentage of observations within 2 and 3 standard deviations of the mean value. Comment on how these percentages compare to the percentage found using Chebyshevs inequality. i. A standard normal distribution (X N N(0,1)) ii. An exponential distribution with A = 2 (X N exp(2)) iii. A Poisson distribution with A = 2 X ~Pois(2) iv. A binomial distribution with n, = 10 and 'p = 0.2. X m binom(10..45) 4. An inequality developed by Russian mathematician Chebyshev gives the minimum percentage of values in ANY sample that can be found within some number ( 1) standard deviations from the mean. Let P be the percentage of values within k standard deviations of the mean value. Chebyshevs inequality states that for ANY distribution, P 100 * (1 -1/k)2. (a) For any distribution, what does Chebyshevs inequality say about the percentage of values that are within 2, 3, 5, 10 standard deviations of the mean? (b) For each of the distributions below, determine the percentage of observations within 2 and 3 standard deviations of the mean value. Comment on how these percentages compare to the percentage found using Chebyshevs inequality. i. A standard normal distribution (X ~ N(0,1)) ii. An exponential distribution with A = 2 (X ~ exp(2)) iii. A Poisson distribution with A = 2 X ~Pois(2) iv. A binomial distribution with n = 10 and p = 0.2. X ~ binom(10,0.45)Question 2. (31] points) Consider a 3state Markov chain as shown in the state diagram in Figure 2, in which the reward of each state are 112(51): 1, R(52} : 3, and R{s3} : 1. 0.0 Figure 2: The 3state Markov chain diagram for lQuestion 2. {a} [15 points) Compute the values of each state, assmniug a discount factor of '7 = 0.9. {b} {15 points) Compute the values of each state with a discount factor of T : {1.1. Give your comments on the difference between the outcomes of Questions 2(a) and 2|[b]? Hint: For any state 5 anal policy 71' of a Markov chain with rewards, the state value function at a state 3 satises the following Bellman equation was) = Hts) + szs'lmcsn ms')