Homework Help for Probability

32. Please choose the statement that is false about random number generation:

In random number generation using linear congruent generator, we don't need to worry about cycles.

We want to explore the impact of drinking on brain size. From 12 randomly selected adult participants, their weekly alcohol consumption (X) and their brain volumes (Y) measured from their MRI scans were recorded. A simple linear regression mode is built as Y = A + BX.

From the sample data and the regression model, we computed the residual as SSR  =  65 ml^2. Please estimate the variance of the responses Yi  (i=1,..,12) with the sample data.

Please round your answer to one decimal place correctly.

Our department seminar room is lit by 7 light bulbs, controlled by a single switch. A minimum of 3 lights are needed for the room to be usable. Suppose the lifetime of the brand of light bulbs fitted is exponentially distributed with a mean of 400 hours. Assume a seminar lasts exactly 1 hour, and that the lights are switched on only during the seminar. What is the probability that after 500 seminars, exactly 3 bulbs do not need to be replaced? You may need to use a calculator. 

Give your answer rounded to 4 decimal places.

Payoff Table for Bidding and Hosting Sporting Events

                                     Event A                         Event B

Cost of Bidding.           $200,000.                 $10,000

Probability Being Awarded 0.1.                      0.9

Cost of Hosting           $1,000,000.                $60,000

Probability of Hosting

Successfully                0.7                                0 .9

Financial Benefit Expected by an Excellent Event

(including revenue for the convention center and

economic impact for the city)

For event A $50,000,000 and for event B $3,750,000

Financial Benefit Expected by a Mediocre Event

(including revenue for the convention center and

economic impact for the city)

For Event A $38,000,000 for event B$1,600,000

Construct a decision tree as the sport marketing manager of the local sports commission

and convention and visitors bureau (CVB) for Jeffersonville.

Compare the three alternatives (Event A, Event B, or not bidding for any event) for their

expected values and net gains.

Prepare your recommendation to the executives from the sports commission and

convention center and be prepared to explain your decision tree model and the preferred

choice.

Calculate the expected value and the net gain for Event A with the new probability of .3

for being awarded the event, and compare the new outcome for Event A with the

outcome of Event B from previous assignment.

2. What would your recommendation to the executives be now?

3. What are three main take-aways from this exercise?

Assume that 15% of the people in your expedition trip
own a mobile phone, and you are randomly assigned to
two partners in your group. Find the probability that:
a) The first partner will have a mobile phone.
b) The first partner will not have a mobile phone.
c) That both partners will have mobile phones.
d) That at most one will have a mobile phone.

5. The ventricle is the cavity in the brain. The mean volume of ventricles in the healthy elderly is 38.3 cm3. We know that the ventricular volume is normally distributed. To inspect the volume of ventricle as a biomarker for Alzheimer’s disease, we collected the ventricular volumes of 25 Alzheimer’s disease patients. If their sample mean was 49.9 cm3 with a sample
standard deviation of s=25.3 cm3. Can we conclude that the mean ventricular volume of Alzheimer’s disease patients is not equal to 38.3 cm3? Use the significance level of 5%.

You may use the online t-distribution calculator to solve the question:
 

In examining borrower characteristics versus loan
delinquency, a bank has collected the following information:
(1) 15% of the borrowers who have been employed
at their present job for less than 3 years are behind in
their payments, (2) 5% of the borrowers who have been
employed at their present job for at least 3 years are
behind in their payments, and (3) 80% of the borrowers
have been employed at their present job for at least
3 years. Given this information:
a. What is the probability that a randomly selected
loan account will be for a person in the same job
for at least 3 years who is behind in making
payments?
b. What is the probability that a randomly selected loan
account will be for a person in the same job for less
than 3 years or who is behind in making payments?
c. If a loan account is behind, what is the probability
that the loan is for a person who has been in the
same job for less than 3 years?

4. In proof testing of circuit boards, the probability that any particular diode will fail is 0.01. Suppose a circuit board contains 300 diodes.
(a) How many diodes would you expect to fail on a circuit board, and what is the standard deviation of the number that are expected to fail?
(b) With Poisson distribution approximation, what is the (approximate) probability that at least one diode will fail on a randomly selected board? Round your answer to 2 decimal places.
(c) If ten boards are shipped to a particular customer, how likely is it that at least two of them will work properly? (A board works properly only if all its diodes work)

6. Let X represent the amount of gasoline (gallons) purchased by a randomly selected customer at a gas station. Suppose that the mean and standard deviation of X are 11 gallons and 5 gallons, respectively.
(a) In a sample of 50 randomly selected customers, what is the approximate probability that the sample mean of the amount purchased X is at least 12 gallons? [Hint: use the central limit theorem]
(b) In a sample of 50 randomly selected customers, what is the approximate probability that the total amount of gasoline purchased is at least 600 gallons?

Question 10 The weekly demand for a product approximately has a normal distribution with mean 1,000 and standard deviation 200. The current on hand inventory is 2,200 and no deliveries will be occurring in the next two weeks. Assuming that the demands in different weeks are independent,
a) What is the probability that the demand in each of the next 2 weeks is less than 1,100?
b) What is the probability that the total of the demands in the next 2 weeks exceeds 2,200?

Question 7 A consulting engineer receives, on average, 0.7 requests per week. If the number of requests follows a Poisson process, find the probability that
a) in a given week, there will be at least one request;
b) in a given 4-week period, there will be at least 3 requests.

Question 5 Consider the diameters of cylindrical parts manufactured on a computer-controlled lathe.
a) Suppose that the mean of the diameters is known to be 0.1 millimetres. Give an upper bound to the probability that the diameter of new part will exceed 0.11 mm.
b) Suppose that the mean is unknown but the standard deviation is known to equal 0.002 mm. What can be said about the probability that the diameter of a new part will be within 0.006 mm of the unknown mean for that run?
c) How many new parts should be made so as to ensure, with probability at least 0.75, that the average diameter would be within 0.001 mm of the unknown mean? Assume each part is made independently.

Question 7 Suppose that 5 independent trials, each of which results in any of the outcomes 0, 1, or 2, with respective probabilities 0.3, 0.5, and 0.2, are performed. Find the probability that both outcome 1 and outcome 2 occur at least once. (Hint: Consider the complementary probability.)

Three unbiased coins are tossed. What is the probability of getting at most two heads?

Find the probability that a leap year has 52 Sundays.