Statistics 230, Fall 2010

Midterm Test 1October 14, 2010Duration: 75 Minutes

Family Name:

Given Name:

ID #:

Signature:Section:

2 001 - LaCroix (12:30-1:20) 2 002 - Moshksar (12:30-1:20) 2 003 - Struthers (10:30-11:20)2 004 - LaCroix (1:30-2:20) 2 005 - Moshksar (1:30-2:20)

Instructions:

1. Please indicate your section above.

2. You may use a pink tie calculator for this exam, but a calculator is not required.

3. Show your work where possible. Part marks cannot be awarded for work that is not shown.

4. If you experience difficultly with one part of a question, but would like to use its answer ina subsequent part, use an unknown to represent the answer, and show how you would finishthe problem if you had this extra information.

5. You may tear off the attached piece of scrap paper. It should not be handed in.

1. [2 marks] Let A and B be two events with P(A) = 0.4 and P(B) = 0.3. Find P(A ∪ B) if

(a) A and B are mutually exclusive.

(b) A and B are independent.

2. [3 marks] A researcher has heard that 10% of students have cheated on an examination,and wishes to confirm this result with a confidential study. The researcher prepares a boxcontaining 100 cards, 75 of which contain Question A and 25 of which contain Question B.

Question A: “Have you ever cheated on an examination?”

Question B: “Do you think the Leafs will win the Stanley Cup this decade?”

Each student draws a card from the box, answers the question it contains and returns the cardback to the box without showing it to the researcher. Since only the student knows whichquestion is being answered, there is little incentive to lie.

A separate study determines that 20% of students think the Leafs will win this decade.

(a) What is the probability that a student answers ‘yes’?

(b) What is the probability that a student who answers ‘yes’ was responding to Question B?

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STAT 230 - Midterm 1 Name:

3. [5 marks] Andy, Beverly, and Chris go to a restaurant for dinner. Since the waiter forgotwho ordered which meal, their bills are given to them at random. Let C be the event thatexactly one person is given the correct bill.

(a) List the elements of a sample space, S, for this experiment such that all outcomes areequally likely. Explain your notation for full marks. (Your list should probably consistof 6 outcomes.)

(b) List the outcomes from S that are in C, and use your list to compute P(C).

(c) For each of the following parts, find an event with the listed properties, or prove that nosuch event exists.

i. an event E such that C and E are mutually exclusive and P(E) = 2

3,

ii. an event F such that C and F are independent and 1

2< P (F ) < 1,

iii. an event G such that C and G are independent and P(G) = 1

2,

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STAT 230 - Midterm 1 Name:

4. [4 marks] Suppose that you are studying a random variable X, and need to determineP (0 < X ≤ 1). A government employee provides you with graphs of three functions. Onefunction is the cumulative distribution function of the random variable you are interested in,but the other two were drawn by his three year old nephew.

For each of the following functions either:

• state a property of c.d.f.’s that is not satisfied by F (x), or

• compute P(0 < X ≤ 1), if F (x) is the c.d.f. of a random variable X.

(a)

y = 0

y = 1

1

F (x) =

{

0 if x < 0x2 if x ≥ 0

(b)

y = 0

y = 1

1

F (x) =arctan(x)

π+

1

2

(c)

y = 0

y = 1

21

F (x) =

0 if x < 0√x if 0 ≤ x < 1

1

2if 1 ≤ x < 2

1 if x ≥ 2

5. [3 marks] Let C and D be two events. Prove that C and D are independent events if andonly if C and D are independent events.

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STAT 230 - Midterm 1 Name:

6. [7 marks] Five tourists plan to attend Oktoberfest. Each attends a location selected atrandom from the choices of Alpine Club, Bingemans, Concordia Club, Kitchener MemorialAuditorium, Queensmount, Schwaben Club, Transylvania Club (7 locations in total).

(a) What is the probability they all attend different locations?

(b) What is the probability that they all attend the same location?

(c) What is the probability that 2 attend one location, and 3 attend a different location?

(d) What is the probability that at least one of the tourists attends Queensmount?

(e) What is the probability that both Queensmount and Concordia are unattended by thetourists?

(f) What is the probability that at least one of the tourists attends Queensmount and atleast one of the tourists attends Concordia Club?

(g) If at least one tourist attends Concordia Club, then what is the probability that at leastone tourist attends Queensmount?

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