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STAT 3000
Spring 2001
MIDTERM EXAM II
NAME: __________________________
The total number of points for this exam is 200. Your grade on this exam will account for approximately 14% of your total course grade.
The exam consists of 4 problems. The maximum number of points for each problem and for each partial question is given in parentheses.
Write in the space provided below each question. If necessary, write also on the back. Write neatly.
Show your work in order to get the full number of points. Write down each step you are taking. You may use your calculator if applicable. If you use a calculator, always indicate that you are using a calculator and write down the function you are working with. Just the end result, even if it is entirely correct, is not enough.
You have 50 minutes to complete this exam. Read the questions carefully. Start with the question which is easiest for you and then move on to more difficult problems.
Most of the questions are multi-part questions, in which an answer to part a) is needed to answer the following parts. If you cannot answer part a) and you feel that you cannot proceed without knowing this answer, make up a (plausible) answer and proceed to part b) etc.
Problem I (60 pts)
A continuous random variable X takes values between 0 and ∞ with a cumulative distribution function
F ( x )= A + Be −^ x
for 0 ≤ x ≤∞.
a) (25 pts) Find the values of A and B and write down the equation of the cdf.
b) (10 pts) What is P ( 2 ≤ X ≤ 3 )?
c) (15 pts) What is the probability density function of X?
d) (10 pts) Looking at cdf and pdf of X you can see that X follows one of the theoretical probability distributions we discussed in class. What is this theoretical probability distribution? What is the parameter of this distribution? What kind of data is usually modeled using this distribution?
Problem III (50 pts)
A machine-filled milk carton is labeled as containing 2 liters. However, the actual amount of milk deposited into the carton varies according to a normal distribution with mean μ = 2.06 liters and standard deviation σ = 0.07 liters.
a) (15 pts) What is the probability that a randomly chosen milk carton is underweight (i.e., the actual amount of milk in the carton is smaller than the advertised amount of 2.00 liters)?
b) (15 pts) The milk cartons are shipped in boxes of 6 cartons. What is the probability that the average milk content of 6 milk cartons in a box is smaller than the advertised amount of 2.00 liters?
c) (20 pts) What is the probability that a box of 6 milk cartons contains exactly one milk carton with content under 2.00 liters (you need the result from part a) to answer this question).
Problem IV (30 pts)
Many models of parasitic infections use the Poisson distribution. Suppose that the number of parasites on a host has a Poisson distribution with a parameter λ = 5.
a) (10 pts) What is the expected number of parasites on a host? What is the standard deviation of the number of parasites on a host?
b) (10 pts) What is the probability that there are no parasites on a host?
c) (10 pts) What is the probability that there are more than 3 parasites on a host?
