a. State the definition of sampling distribution?
b. Let’s assume that we have a Bernoulli random variable X,
X = a with probability 0.78,
X = b with probability 0.22,
Where a and b are 5 and 9 respectively.
Develop a sampling distribution for sample means of the Bernoulli distribution when the sample size is 6
c. What is the expected value of sample means in question 3.b? What is the variance of the sample means?
d. If the sample size increases to a large number n (n is greater than 30), how are the sample means distributed? ( 3 marks)

3. Only using the runif function with default settings, generate:

(a) n = 1e4 iid realizations from a Bernoulli distribution with probability of success parameter, θ = 0.7.

(b) n = 1e4 iid realizations from a Binomial distribution with probability of success parameter, θ = 0.7 and number of trials= 20.

(c) n = 1e4 iid realizations from an Exponential distribution with mean µ = 7 (hint: if X ~ Exp(µ) then Var(X) = µ² ).

For each case comment on the sample mean and sample variance of these realizations and if match what you expect.

Consider the machine-repair model involving two repairment and for machines. Assume that the breakdown rate is λ = 1 machine per hour, and that the service rate is µ = 2 machines per hour per serviceman. Let X(t) be the number of machines broken at time t.

Suppose that you observe at 11A.M. that two of the machines are operating and two of the machines are being repaired.

a) What is the probability that, in the next ¼ hour there will be no change in the state of the system?

b) What is the probability that both of the machines being repaired at 11:00 A.M. will be fixed before any more breakdowns occur?

Criteria for a Binomial Distribution

A. The number of observation or trials is fixed.

B. Each observation or trial is independent.

C. For each trial, there are only two possible outcomes.

A quality control expert at a large factory estimates that 10% of all batteries produced are defective. If he takes a random sample of fifteen batteries, what is the probability that exactly two are defective?

A quality control expert at a large factory estimates that 10% of all the batteries produced at the factory are defective. If he takes a random sample of fifteen batteries, what is the probability that no more than two are defective?

Customers arrive at a service facility according to a Poisson process of rate 5 /hour Let N(t) be the number of customers that have arrived up to time t hours).

a.

What is the probability that there is at least 2 customer walked in 30 mins?

b.If there was no customer in the first30 minutes, what is the probability that you have to wait in total of more than 1 hours for the 1 st customer to show up?

c.For an y random customer, if there is 50% chance the customer is female, what is the expected waiting time until the

5th female customer comes in?

1.) Years of scores have indicated that the mean on a stats exam is 30 and the standard deviation is 7. The number of people in the sample class 47.

a- What is the population mean of the sampling distribution?

b-What is the population variance and standard error of this sampling distribution?

c-What minimum sample mean would this class need to obtain to be in the top 5% of the sample mean ?

d) What is the probability of obtaining a sample mean of 28.2 or bellow for a class of 49 students?

e) What is the probability that an individual student in class would obtain a score of 28.2 or below?

In a certain population of mussels (mytilus edulis) 80% of the individuals are infected with an intestinal parasite. A marine biologist plans to examine 100 randomly chosen mussels from the population. Let ? represent the number of mussels in this sample with the intestinal parasite.

9) Clearly state the distribution that ? follows, Explain why you picked this distribution? State the distribution we may use to approximate it.

10) Approximate the probability that between 75% and 90% (inclusive) of the mussels in the sample are infected. Note: show your R code for calculating the probability under the normal curve or z-value.

In a certain population of mussels (mytilus edulis) 80% of the individuals are infected with an intestinal parasite. A marine biologist plans to examine 100 randomly chosen mussels from the population. Let ? represent the number of mussels in this sample with the intestinal parasite.

9) Clearly state the distribution that ? follows, Explain why you picked this distribution? State the distribution we may use to approximate it.

10) Approximate the probability that between 75% and 90% (inclusive) of the mussels in the sample are infected. Note: show your R code for calculating the probability under the normal curve or z-value.

A manufacturing process produces semiconductor chips with a known failure rate of 6.3%. Assume that the chip failures are independent of one another. You will be producing 2,000 chips tomorrow.

e. Find the probability that you will produce more than 120 defects.

f. You just learned that you will need to ship 1,860 working chips out of tomorrow’s production of 2,000. What are the chances that you will succeed? Will you need to increase the scheduled number produced?

g. If you schedule 2,100 chips for production, what is the probability that you will be able to ship 1,860 working ones?

After preliminary testing, it is determined that .01 of the residents of Selma have a certain virus. In Sangrian, a suburb of Selma (incidence of the virus is the same), the authorities test the 10,000 residents of the suburb. The test has the following characteristics. If you have the virus, the probability that the test is positive is .95, and if you don’t have the virus, the probability that the test is positive is .10. Suppose that 1,100 of the 10,000 residents test positive. Let X = the number among those 1,100 who really do have the virus.

Find E(X) and Variance(X). thank you :) !!