You will roll two standard dice together 5 times. You are interested in the outcome where both dice are six. Let X be the number of times you observe

this outcome. Answer the following questions.

1. What are the possible values for X? (values the random variable X can

take)

2. Is X binomial random variable? If so, state its parameter n and p. If

not, explain why.

3. Find the probability that you will see both dice being six at least once.

Round your answer to the three decimal places.

4. Find the mean and variance of X.

Problem 3

Suppose that the distance of fly balls hit to the outfield (in baseball) is normally distributed with a mean of   270   feet and a standard deviation of   50 feet. Let   X=   distance in feet for a fly ball.

In each appropriate box you are to enter either a rational number in "p/q" format or a decimal value accurate to the nearest   0.01 .

1. (.3)   X∼    (pick one) BENU   (    ,    ) .
   
2. 
   
3. (.35) For a random fly ball, what is the probability that this ball traveled fewer than   220   feet?   P(X<220)=    .
   
4. 
   
5. (.35) The   80th   percentile of the distribution of fly balls is given by   P(X<  )=0.80 .

70% of people who are 30 years old will be alive ten years from now. In the random sample of 15 people, calculate the probability that...

a) exactly 10 of them will be alive ten years from now

b) at least 9 will be alive ten years from now

c) at most 8 will be alive ten years from now

d) how many of them would you expect to be alive 10 years from now

e) let X be the variable denoting the number of people alive ten years from now. Calculate the standard deviation of X

An average of 10 cars per hour reaches an ATM with a single server that provides service without leaving the car. Suppose the time of average service for each client is 4 minutes, and how long the times between arrival and service times are exponential. Answer the following questions:
a) What is the notation for this problem?
b) What is the probability that the cashier is idle?
c) What is the average number of cars in the cashier's queue?
d) What is the average amount of time a customer spends in the parking lot of the bank (including service time)?
e) How many customers will the cashier serve on average per hour?

The number of minutes between successive bus arrivals at NewYork's bus stop is exponentially distributed with unknown parameter lambda. John's prior information regarding lambda (real valued, i.e., a continuous random variable) can be summarized as a continuous uniform PDF in the range from 0 to 0.85. john has recorded his bus waiting times for three days. These are 13, 19, and 26 minutes. Assume that John's bus waiting times in different days are independent. Compute the MAP (maximum a posteriori probability) rule based estimate for lambda given the data collected by John.

A company produces steel rods. The lengths of the steel rods are normally distributed with a mean of 195.3 cm and a standard deviation of 1 cm. For shipment, 24 steel rods are bundled together.

Note: Even though our sample size is less than 30, we can use the z score because
1) The population is normally distributed and
2) We know the population standard deviation, sigma.

Find the probability that the average length of a randomly selected bundle of steel rods is between 194.8 cm and 194.9 cm.


Enter your answer as a number accurate to 4 decimal places.

Hogg, who is risk-neutral over votes, is running for office with 500,000 sure voters. To add voters, he wants to choose n, the number of negative campaign ads to run, where 0 n 4. The ads will backfire with probability n/5 and give him no extra votes. Otherwise, the ads will work and give him 100,000 + 40,000n extra votes. So n = 0 implies a total of 600,000 votes. He should choose n = :

a. 0.

b. 1.

c. 2.

d. 3.

e. 4.

Can someone explain how the answer is B

An individual has three umbrellas, some at his office, and some at home. If he is leaving home in the morning (or leaving work at night) and it is raining, he will take an umbrella, if there is one there. Otherwise, he gets wet. Assume that, independent of the past, it rains on each trip with probability 0.2. To formulate a Markov chain, let Xn be the number of umbrellas at his current location “before” he starts his n-th trip. Note that “current location” can either be home or office, depending on whether the trip is from home to office or vice versa. Find the transition probabilities of this Markov chain

Suppose that the manager of the MileagePlus frequent flier program is promoted and consequently another individual is hired to replace him. Also suppose that United publishes in internal documentation that the average number of Premier Qualifying Miles (PQM) earned by individuals who travel for work at least once a month is 45,000 with a standard deviation of 5,000 miles. Further suppose that the new manager desires to test the claims that United has made to see if the statistics have changed.

a) First, are these statistics given by United describing the parent population or a sample ?

b) Define appropriate null and alternative hypothesis

c) Suppose that the new manager takes a sample of 50 such United customers. What is the probability that a sample of size 50 provides a sample mean within + - 1,000 miles of the 45,000 mile figure provided by United?

d) What is the probability that a sample of 50 such United customers provides a sample mean wiithin + - 500 miles?

e) Suppose that the new manager's sample has a sample average of 47,500 miles. Compute the 95% confidence interval for the population mean.

f) Based on the confidence interval you computed in part e), does this sample provide evidence for or against United's claim that the average number of Premier Qualifying Miles (PQM) earned is 45,000 ?

g) If the new manager wants to determine a specific p-value for the likelihood that the null hypothesis is true based on the sample collected in part c), should he use Z- or t- scores for the test statistic?

h) Determine and interpret the p-value