Module 1, Week 2, Paper and Pencil Assignment 2

1. A fair, six-sided die is rolled. Describe the sample space S. Identify each of the following events with a subset of S and compute its probability.
   1. Event F = the outcome is four
   2. Event A = the outcome is an odd number
   3. Event B = the outcome is less than five
   4. The complement of B
   5. (A | B)
   6. (B | A)
   7. Event H = the outcome is seven

2. The following table describes the distribution of a sample S of 600 individuals, organized by region of residence and political party affiliation:

	Democrat	Republican	Independent
East	163	121	52
West	108	141	15

Let D = the individual is a democrat, let R = the individual is a republican, let I = the individual is an independent, let E = the individual lives in the east, and let W = the individual lives in the west. Compute the following probabilities:

3. Let D = a student is taking a data analytics class and let E = the student is taking an economics class. Suppose P(D) = 0.20 and P(E) = 0.36 and .

1. Are D and E independent?
2. Show
3. Show

4. Whether a customer at a carry-out restaurant leaves a tip is a random variable. The probability that a customer leaves a tip is 0.42. The probability that one customer leaves a tip is independent of whether another customer leaves a tip. Let leaving a tip represent a success and not leaving a tip represent a failure.

1. Does this problem describe a discrete or continuous random variable?
2. What kind probability distribution fits the random variable described in this problem?
3. What is the probability that a customer does not leave a tip?
4. Calculate the mean and variance of this distribution.
5. What is the probability that on a day with 100 customers, exactly 50 of them leave a tip?

5. The number of pieces of mail a household receives on a given day follows a Poisson distribution. On average, eight pieces of mail are received each day.

1. Does this problem describe a discrete or continuous random variable?
2. Calculate the mean and variance of this distribution.
3. What is the probability that a household receives 10 pieces of mail on a given day?
4. What is the probability that a household receives 5 pieces of mail on a given day?
5. What is the probability that a household receives less than 3 pieces of mail on a given day?

6. Let X = number of miles a family travels for summer vacation. X follows a normal distribution with a mean of 311 miles and standard deviation of 232 miles.
   1. If a family traveled 600 miles, how many standard deviations are they away from the mean?
   2. Calculate the following probabilities:

7. Suppose the random variable X has a mean of 30 and standard deviation of 6. Samples of 20 observations are drawn randomly from the population.
   1. Calculate the mean and standard deviation for the random variable .
   2. Calculate the probability that the sample mean is between 26 and 33.
   3. Describe the central limit theorem in words.