6)  You have a group of 500 students.  On a particular test, μ = 72 and σ = 10.

      a)  How many students scored above 88?

      b)  What is the number of students scoring below 60?

7)  100 9-year old boys take turns throwing a baseball as far as they can.  For the group,

      average distance thrown is 80 feet and σ = 20 feet.      

      a) What percentage threw 100 feet or more?

      b) How many threw 45 feet or less?

      c) What distance would be the top 10%?

      d) What is the probability that a child picked at random threw between 59-99 feet?

      e) What distances are so extreme that only 1% did it?

      f) What distances are so extreme that only 5% did it?

Do Americans trust advertisements? A survey asked Americans who view advertisements at least once a month how honest the advertisements that they see, read, and hear are. The results were:

1. Develop and populate a frequency and percentage table to show the number and percentage differences between areas of the country on their perspective of honesty in commercials.
2. Is there evidence of a significant difference among the areas of the country on their perspective of the honesty in commercials. Use the probability of 0.05.
3. Determine the p-value and interpret its meaning.
4. If appropriate, use the Marascuilo procedure and p=0.05 to determine which regions of the country differ.

An office employs several clerks who create documents, and one operator who enters the document information in a computer system. The group creates documents at a rate of 14 per hour. The operator can enter the information with an average exponentially distributed time of 3 minutes. Assume the population is infinite, arrivals are Poisson, and queue length is infinite with the FCFS discipline.

a. Calculate the percentage utilization of the operator. (Round your answer to 2 decimal places.)

b. Calculate the average number of documents in the system. (Round your answer to 1 decimal place.)

c. Calculate the average time in the system. (Round your answer to 1 decimal place.)

d. Calculate the probability of four or more documents being in the system. (Round your intermediate calculations to 3 decimal places and final answer to 1 decimal place.

Consider tossing a coin and rolling a four-sided die (with the numbers 1 through 4 printed on the sides).

(a) Describe the sample space.

(b) Whatistheprobabilityofrollingaheadsandanevennumber?

(c) What are the odds of rolling a heads and an even number?

(d) Whatistheprobabilityofrollingaheads?

(e) What are the odds against rolling a heads?

Determine whether or not the following statements are correct or incorrect and explain why (Be thorough and clear in your explanation!):

(a) A person says “the odds of rolling a 1 on a standard six-sided die is 1/6.”

(b) Inthepastfiveseasons,crosstownfootballrivalstheQuakersandtheCometshaveplayed

each other with the Comets winning twice and the Quakers winning three times. Someone

says “the odds in favor of the Quakers winning are 3:5.”

(c) If the odds in favor of an event occurring are A to B, then the probability of the event

occurring is A/(A+B).

(d) Aneventwithprobability75%meansthattheeventisthreetimesmorelikelytooccurthan

not occur.

A system consists of three machines and two repairmen. At most two machines can operate at any time. The amount of time that an operating machine works before breaking down is exponentially distributed with mean 5 hours. The amount of time that it takes a single repairman to x a machine is exponentially distributed with mean 4 hours. Only one repairman can work on a failed machine at any given time. Let X(t) be the number of machines in working order at time t.

a) Calculate the long-run probability distribution of X(t).

b) In the long run, what fraction of time are both repairmen busy?

c) If an operating machine produces 100 units of output per hour, what is the long run output per hour of the system?

1. Suppose X follows the normal distribution with mean μ and variance σ^2 . Then:

(Circle all that apply.)

A. X is symmetric with respect to the y-axis.

B. P(X=2)=P(X=-2).

C. Y=aX follows the same distribution as X, where a is a constant.

D. None of the above statements is correct.

2.

Given a random variable X having a normal distribution with μ=50, and σ=10. The probability that Z assumes a value between 45 and 62 is: ___________.

3. Which of the following statements is true?

A. The Gamma distribution is a special case of the exponential distribution.

B. The Exponential distribution is used to describe the number of arrivals

within one unit.

C. The expectation of the exponential distribution does not equal to that of

the Poisson distribution in general.

D. All of the above is correct.

Consider a 4-particle system. Each particle may be in one of three levels called "A", "B", and "C". Answer the questions below, using the process of finding micro- and macrostates described in the January 11th lecture. Assume the particles are distinguishable from one another.(a)  How many microstates does this system have?(b)  For this system, define a macrostate as a set of microstates with the same number of particles in each named state, e.g., (AAAB) and (AABA) are in the same macrostate, but in a different macrostate from (BBBA). How many macrostates are there?(c)  Determine the multiplicity of each macrostate using the multiplicity equation from class and confirm that you have accounted for all of the microstates for the system. (d)  What is the probability of getting the particles in the levels AABB in any order? (e)  What is the most likely macrostate?

Company XYZ know that replacement times for the DVD players it produces are normally distributed with a mean of 4.7 years and a standard deviation of 2.3 years.

Find the probability that a randomly selected DVD player will have a replacement time less than -0.8 years?
P(X < -0.8 years) =

Enter your answer accurate to 4 decimal places. Answers obtained using exact z-scores or z-scores rounded to 3 decimal places are accepted.

If the company wants to provide a warranty so that only 1.8% of the DVD players will be replaced before the warranty expires, what is the time length of the warranty?
warranty =  years

Enter your answer as a number accurate to 1 decimal place. Answers obtained using exact z-scores or z-scores rounded to 3 decimal places are accepted.

The Federal Deposit Insurance Corporation is a United States government corporation providing deposit insurance to depositors in U.S. commercial banks and savings institutions. The FDIC was created by the 1933 Banking Act, enacted during the Great Depression to restore trust in the American banking system. The FDIC recently conducted a survey and found that 45% of all financial consumers were very satisfied with their primary financial institution. If this figure still holds true today, suppose 28 financial consumers are sampled randomly. If X is a binomial random variable representing the number of financial consumers who were very satisfied with their primary financial institution, what is the probability that exactly 20 of the 28 are very satisfied with the primary financial institution? (Report your answer to 4 decimal places, using conventional rounding rules.)

1. Suppose a $50,000 annual loss occurs randomly with a probability of one in one hundred.

a. What is the actuarially fair premium? If the insurance company charges $750 per person per year, what is the loading factor?

b. Assume the insurance company has 100 people in the pool and the insurance company is charging $750 per person per year. If one person in the pool suffers from the loss, will the insurance company be able to pay for the loss?

c. Assume the insurance company has 100 people in the pool and the insurance company is charging $750 per person per year. If two people in the pool suffer from the loss, will the insurance company be able to pay the losses?

d. What happens to the variability of the expected value of a loss as the number of people in the group increases?