A recent survey of 2,000 randomly selected U.S. residents asked whether or not they had used the Internet for making travel reservations. There were 1,100 people who answered “yes,” and the remainder answered “no.” Let X be the number of people who answered “yes." Estimate the proportion p from this information. Then calculate the probability that X is less than 1,050.

a. 0.0014

b. 0.5500

c. 0.0131

d. 0.0117

e. 0.9869

Charlene takes a quiz with 10 multiple-choice questions, each with four answer choices. If she just guesses independently at each question, she has probability 0.25 of guessing right on each. Use simulation to estimate Charlene's expected number of correct answers. (Simulate 20 repetitions using line 122 of this table of random digits. Let 00–24 be a correct guess and 25–99 be an incorrect guess.)

12 cases of a disease are found over an area of 50 square miles.

(what type of distribution is this, i.e. Poisson, Gaussian, etc.)

1. Within the study area, if the cases are distributed at random places with the same chance to occur anywhere, what is the probability of finding 4 cases within an 8 square mile area? Calculate by hand.
2. What is the average number of cases within the 8 square mile area? Calculate by hand.

A recent study has shown that 28% of 18-34 year olds check their Facebook/Instagram feeds before getting out of bed in the morning,

If we sampled a group of 150 18-34 year olds, what is the probability that the number of them who checked their social media before getting out of bed is:

a.) At least 30?  

b.) No more than 55?  

c.) between 40 and 49 (including 40 and 49)?

Two fair dice are rolled at once. Let x denote the difference in the number of dots that appear on the top faces of the two dice. For example, if a 1 and a 5 are rolled, the difference is 5−1=4, so x=4. If two sixes are rolled, 6−6=0, so x=0. Construct the probability distribution for x. Arrange x in increasing order and write the probabilities P(x) as simplified fractions.

(A universal random number generator.)Let X have a continuous, strictly increasing cdf F. Let Y = F(X). Find the density of Y. This is called the probability integral transform. Now let U ∼ Uniform(0,1) and let X = F−1(U). Show that X ∼ F. Now write a program that takes Uniform (0,1) random variables and generates random variables from an Exponential (β) distribution

A lab rat is given 6 randomly selected nuts from a bag of 20 nuts, of which 8 have poisonous chemicals.

a. What is the probability that none of the 6 nuts the rat eats are poisonous?

b. If the rat eats 3 poisonous nuts, it passes out. Let X be the number of poisoned nuts (where X cant be greater than 3). If f(x) is p.m.f of X, what is f(3)?

In a population of students, 60% study for every homework assignment. A sample of 40 students from this population is taken, and the number who study for every homework assignment from this sample is recorded as the random variable X.

a) (4 pts) Verify that X has a binomial distribution.

b) (4 pts) Find the probability that exactly 25 students from the sample study for the exam.

c) (4 pts) Find the mean for this binomial experiment