Download Introduction to the Practice of Statistics

Introduction to the Practice of Statistics
Fifth Edition
Moore, McCabe
Section 4.5 Homework Answers to 86, 88, 89
4.86 Call a household prosperous if its income exceeds $100,000. Call the household educated if the
householder completed college. Select an American household at random, and let A be the event that the
selected household is prosperous and B the event that it is educated. According to the Current Population
Survey, P(A) = 0.138, P(B) = 0.261, and the probability that a household is both prosperous and
educated is P(A and B) = 0.082. What is the probability P(A or B) that the household selected is either
prosperous or educated?
P(A or B) = P(A) + P(B) – P(A and B)
= 0.138 + 0.261 – 0.082
= 0.317
4.88 Draw a Venn diagram that shows the relation between the events A and B in Exercise 4.86. Indicate
each of the following events on your diagram and use the information in Exercise 4.86 to calculate the
probability of each event. Finally describe in words what each event is.
a) {A and B} The event is a household is prosperous and that it is educated.
P(A and B) = 0.082
b) {A and Bc} The household is prosperous, but not educated.
P(A and Bc) = 0.138 – 0.082 = 0.056
c) {Ac and B} The household is not prosperous, but it is educated.
P(Ac and B) = 0.261 – 0.082 = 0.179
d) {Ac and Bc} The household is neither prosperous nor is it educated.
P (Ac and Bc) = 1 – 0.317
= 0.683
4.89 Motor vehicles sold to individuals are classified as either cars or light trucks (including SUVs) and
as either domestic or imported. In early 2004, 69% of vehicles sold were light trucks, 78% were
domestic, and 55% were domestic light trucks. Let A be the event that a vehicle is a light truck and B the
event that it is imported. Write each of the following events in set notation and give its probability.
I will do this problem twice. Once I will adhere to the definitions that were originally given. The second
below I will do my own way.
First let us write down the given probabilities using correct notation.
P(A) = 0.69
P(Bc) = 0.78
P(A and Bc) = 0.55 Ok that took me a few minutes to keep straight their own given definitions and
values.
(a) The vehicle is a car. P(Ac) = 1 – 0.69 = 0. 31
(b) The vehicle is an imported car. Here is what I want :P(Ac and B) .
I do not know if I have independence so the rule P(A and B) = P(A)P(B) can not be used.
Therefore lets create a Venn Diagram that depicts the area I am interested in.
Now that I have the picture I will go and find what
need to get the result.
I have P(B) = 1 – 0.78 = 0.22 using the above
information.
The probability I do have is given as
P(A and Bc) = 0.55
I can see that If I know P(B) and I know P(A and B) I can get
what I want, P(Ac and B).
P(A and B) = P(A) – P(A and Bc)
= 0.69 – 0.55
= 0.14
Thus, P(Ac and B) = P(B) – P(A and B)
= 0.22 – 0.14
= 0.08
I
4.89 Motor vehicles sold to individuals are classified as either cars or light trucks (including SUVs) and
as either domestic or imported. In early 2004, 69% of vehicles sold were light trucks, 78% were
domestic, and 55% were domestic light trucks. Let A be the event that a vehicle is a light truck and B the
event that it is imported. Write each of the following events in set notation and give its probability.
I will do this problem again, but this time I am going to ignore the definitions of A and B, and just write
the given probabilities using function notation using words or phrases.
P(light truck) = 0.69
P(domestic) = 0.78
P(domestic and light truck) = 0.55
(a) The vehicle is a car. First I rewrite the question using the given events I have already
written down.
P(not light truck) = 1 – P(light truck)
= 1 – 0.69
= 0.31
(b) The vehicle is an imported car. First I rewrite the question using the given events I have
already written down.
P(not domestic and not light truck) Next I create a Venn Diagram
to help me see what area we are considering; I shade in the corresponding area.
Now I can look at the information I have and determine if I can get the shaded area I have drawn.
P(not domestic and not light truck) = 1 – P(domestic OR light truck)
= 1 – [ P(domestic) + P(light truck) – P(domestic AND light truck)]
= 1 – [0.69 + 0.78 – 0.55]
= 0.08