Module 1, Week 2, Paper and Pencil Assignment 2

 

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.

a.     Event F = the outcome is four

b.     Event A = the outcome is an odd number

c.      Event B = the outcome is less than five

d.     The complement of B

e.     (A | B)

f.       (B | A)

g.   

h.   

i.    

j.       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:

 

a.   

b.   

c.   

d.   

e.   

f.    

g.   

h.   

i.    

j.    

k.     

l.    

m.  

n.   

 

 

 

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 .

 

a.     Are D and E independent?

b.    Show

c.    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.”

 

a.     Does this problem describe a discrete or continuous random variable?

b.     What kind probability distribution fits the random variable described in this problem?

c.      What is the probability that a customer does not leave a tip?

d.     Calculate the mean and variance of this distribution.

e.     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.

 

a.     Does this problem describe a discrete or continuous random variable?

b.     Calculate the mean and variance of this distribution.

c.      What is the probability that a household receives 10 pieces of mail on a given day?

d.     What is the probability that a household receives 5 pieces of mail on a given day?

e.     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.

a.     If a family traveled 600 miles, how many standard deviations are they away from the mean?

b.     Calculate the following probabilities:

                                       i.   

                                      ii.   

                                     iii.   

                                     iv.   

 

 

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.

a.    Calculate the mean and standard deviation for the random variable .

b.     Calculate the probability that the sample mean is between 26 and 33.

c.      Describe the central limit theorem in words.