MATH 251 Final

February 3, 2006

WARNİNG: Calculators, Z table and 2 pages of summaries are allowed to be used during the exam. Please type your answers between the gaps of two questions. If the gap is not enough for your answers you can use the back of the pages.

 

Questions

 

1.        The following data are for 15 observations on two qualitative variables, x and y. the categories for x are A, B, and C: the categories for y are 1 and 2.

               

Observation

X

Y

a)        Develop a Cross Tabulation for the data, with x in the rows and y in the columns.(9)

b)        Compute the row percentage.(8)

c)        What is the relationship, if any, between x and y. (8)

 

Solution: a)

x/y

1

2

Total

A

3

1

4

B

2

3

5

C

3

3

6

Total

8

7

15

 

b)

x/y

1

2

Total

A

%75

%25

%100

B

%40

%60

%100

C

%50

%50

%100

 

c) No Relation because one variable is quantitative the other one is qualitative since we do not have further information about the data we say no relation.

 

 

 

 

 

 

 

 

 

 

 

                

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

 

 

 

 

 

 

A

B

B

C

B

C

B

C

C

A

A

B

C

C

A

 

 

 

 

 

 

1

1

2

2

2

1

2

1

2

2

1

1

1

2

1

 

 

 

 

 

 

2.        Final examination scores for 25 statistics students follow.

 

121  35  98  107  32  90  73  155  18  36  154  29  25  121  98  15  10  47  24  40  49  130  68  91  14

 

a)        Provide a five-number summary (12p)

b)        Show a box-plot and detect the outliers if any (13p)

 

Solution:

a) 1. largest Value 155 2. Q1=29  Q2=49       Q3=107     Smallest Value=10

b) No outliers

 

3.        Assume that we have two events, A and B that are mutually exclusive. Assume further that we know P(A)=0,30 and P(B)=0,4

a)        What is P(AB)=?(7p)

b)        What is P(A|B)=?(8p)

c)        The Concept of mutually exclusive events and independence are really the same, and if events are mutually exclusive they must be independent. Do you agree with this statement? Use the probability information in this problem to justify your answer.(10p) 

 

Solution:

a)        P(AB)=0 if event A and event B is mutually exclusive then A and B has no common point. So it is zero

b)         

 

c)        The condition for the independence is P(A|B)=P(A). If A and B are mutually exclusive than the left hand side of this condition will be zero (explanation is in part b) and right hand side will be NOT Zero. So if set A and Set B are two non-empty and mutually exclusive than these two sets are DEPENDANT according to the definition of independence.

 

4.        During the period of time phone-in registrations are being taken at a local university, calls come in at the rate of one every 0,5 minute.

 

a)        What is the expected number of calls in half an hour?(7p)

b)        What is the probability of three calls in 5 minutes?(9p)

c)        What is the probability of no calls in 4 minutes?(9p)

 

Solution:

a) call/30min

b)  call/5min

c) call/4 min

 

5.        Most computer languages have a function that can be used to generate random numbers. In Excel, the RAND function can be used to generate random numbers between 0 and 1. If we let x denote a random number generated using RAND, then x is a continuous random variable with the following probability density function.

 

a)        Write and graph the density function (7p)

b)        What is the probability of generating a random number between 0,25 and 0,75?(6p)

c)        What is the probability of generating a random number with a value less than or equal to 0,30?(6p)

d)        What is the probability of generating a random number greater than 0,60?(6p)

 

Solution:

a)

 

b)      P(0,25<X<0,75)=(0,75-0,25)*1

         P(0,25<X<0,75)=0,5

d)        P(X<=0,30)=(0,3-0)*1

         P(X<=0,30)=0,3

e)        P(X>0,6)=(1-0,6)*1

         P(X>0,6)=0,4

 

6.        10% of students in a university got AA. A random sample of 3 students is selected. ‘X’ is the random variable to denote the number of students who got AA in this university.

 

a)        write down the probability distribution of x (13p)

b)        Calculate the mean and the variance of x (12p)

 

 

 

 

 

 

Solution:

a)

 

c) E(X)=3*0,1=0,03     V(X)=3*0,1*0,9=0,27

 

7.        The time needed to complete a final exam in Fatih University is normally distributed with a mean of 80 minutes and a standard deviation of 10 minutes. Answer the following questions.

a)        What is the probability of completing the exam in 1 hour or less?(8P)

b)        What is the probability that a student will complete the exam in more than 60 minutes but less than 75 minutes?(8p)

c)        Assume that class has 60 students and that the examination period is 90 minutes in length. How many students do you expect will be unable to complete the exam in the allotted time?(9p)

 

Solution:

a)

b)

c)

 

8.        A study of 31,000 hospital administrators in Istanbul found that 4% of admissions led to treatment-caused injuries. One-seventh of these treatment-caused injuries resulted in death, and one-fourth were caused by negligence. Malpractice claims were filed in one out of 7.5 causes involving negligence, and payments were made in one out of every two claims.

a)        What is the probability a person admitted to hospital will suffer a treatment-caused injury due to negligence?(8p)

b)         What is the probability a person admitted to hospital will die from a treatment-caused injury due to negligence?(8p)

c)        In the case of negligent treatment-caused injury, what is the probability a malpractice claim will be paid?(9)

 

Solution:

a)        0,04*0,25=0,01

b)        0,04*1/7*0,25=0,0014

c)        1/7,5*1/2=1/15