Explain your answers including formulas. (Do not rely on built-in functions in Excel or other software.)

1. We are told by someone who claims to be an authority on the subject that the lifespan of a certain species of horse parasites stated in days is distributed according to the gamma distribution with parameters a = 7 and b = 2.5.

a.) Given that we accept this statement, what is the average lifespan of this type of parasite?

b.) What is the standard deviation in the lifespan of these parasites?

c.) What is the median lifespan of these parasites?

d.) What is the probability that any given parasite of this type will live more than 20 days?

e.) What is the probability that any given parasite of this type will live somewhere between 10 and 30 days?

2. Someone else claims that the lifespan of these horse parasites is distributed according to the Weibull distribution with parameters a = 2 and b = 20. What would be the answers to parts a through e of question 4 assuming that this second party is correct? (You really do have enough information to answer parts a and b if you look through the class notes carefully.)

3. Now along comes a third authority who claims that both of these first two guys are all wrong. This third guy claims that the lifespan of these horse parasites are distributed according to the lognormal distribution with parameters m = 2.8 and s = 0.4. What would be the answers to parts a through e of question 4 assuming now that this third person is correct?

4. There is a species of goat parasite, whose larval stage lasts for 6 days and who never live longer than 16 days total. Thus, considering a parasite of this species that reaches adulthood, the lifespan is always somewhere between 6 and 16 days. A certain authority (it happens to be the same as the 2^nd guy above) claims that the lifespan of these goat parasites is distributed according to the beta distribution with parameters a = 3, b = 3, A = 6, and B = 16. Assuming that this guy is correct, answer the following. (This problem is going to take a lot of work. Be sure to give yourself plenty of time.)

a.) State (in simplified form as much as possible) the formula for the probability density function of this distribution.

b.) State (again in as simplified a form as you can) the formula for the cumulative distribution function of this distribution.

c.) What is the average lifespan of these goat parasites?

d.) What is the standard deviation in lifespan of these?

e.) What is the median lifespan of these?

f.) What is the probability that a given goat parasite will live for somewhere between 8 and 12 days?

5. You are teaching a lab section of a class in which you need to teach the students laboratory methods including collecting data on horse parasites. You know that the weights of these parasites are distributed in a way that is very close to a normal distribution. You require each student to collect 10 horse parasites, weigh them carefully to the nearest hundredth of a gram, and report these data to you. However, you know that although some of the students take the lab seriously, others are either too lazy or too squeamish to collect real data and try to make up their own data instead. You wish to test how plausible it is that each student collected real data as opposed to making up data. Here are examples of data sets submitted by two students (in weights measured to the nearest hundredth of a gram).

Student A: Student B:

For which of these two students is it more plausible that the data are real? Justify your answer in detail.

6. There is a process whose output X is a random variable that you know (never mind how you know this) to be normally distributed with m = 50 and s = 30. You are going to collect a sample of size 20 from this process and calculate a sample mean from those data. What is the probability that the value you calculate will fall somewhere between 45 and 55?