MATH
251 Final
February 20, 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.
A Survey of
commercial buildings served by the ISKI was concluded in 2002. One question
asked what main heating fuel was used and another asked the year the commercial
building was constructed. A partial cross-tabulation of the findings follows.
|
|
Fuel
Type |
||||
|
Year Constructed |
Electricity |
Natural Gas |
Oil |
Propane |
Other |
|
1993
or Less 1994-1996 1997-1999 2000-2002 |
40 24 37 48 |
183 26 38 70 |
12 2 1 2 |
5 2 0 0 |
7 0 6 1 |
a)
Complete the
cross-tabulation by showing the row totals and column totals.(8P)
b)
Show the frequency
distributions for year constructed and for fuel type. (8p)
c)
Prepare cross-tabulation
showing column percentages. (9p)
Solution: a)
|
|
Fuel
Type |
|
||||
|
Year Constructed |
Electricity |
Natural Gas |
Oil |
Propane |
Total |
|
|
1993
or Less 1994-1996 1997-1999 2000-2002 |
40 24 37 48 |
183 26 38 70 |
12 2 1 2 |
5 2 0 0 |
7 0 6 1 |
247 54 82 121 |
|
Total |
149 |
317 |
17 |
7 |
14 |
504 |
b)
|
Year Constructed |
FD |
FUEL TYPE |
FD |
|
1993
or Less 1994-1996 1997-1999 2000-2002 |
247/504 54/504 82/504 121/504 |
Electricity Natural Gas Oil Propane Other |
149/504 317/504 17/504 7/504 14/504 |
c)
|
|
Fuel
Type |
||||
|
Year Constructed |
Electricity |
Natural Gas |
Oil |
Propane |
Other |
|
1993
or Less 1994-1996 1997-1999 2000-2002 |
40/149*100 24/149*100 37/149*100 48/149*100 |
183/317*100 26/317*100 38/317*100 70/317*100 |
12/17*100 2/17*100 1/17*100 2/17*100 |
5/7*100 2/7*100 0 0 |
7/14*100 0 6/14*100 1/14*100 |
|
Total |
100 |
100 |
100 |
100 |
100 |
2.
|
15,8 17,3 12,8 5,0 |
19,6 52,7 31,1 12,2 30,3 |
22,9 17,3 9,6 14,5 14,7 |
a)
Provide a five-number summary (6p)
b)
Compute the lower and upper limits (6p)
c)
Do there appear to be outlier? (6p)
d)
Show a box-plot (7p)
Solution:
a) 5 9 9,6 12,2 12,8 14,5 14,7 15,8 17,3 17,3 19,6 22,9 30,3 31,1 52,7
We can order
them like above so:
Smallest
Number: 5 Q1:12,2 Q3:22,9 Q2(median)=15,8
b)
LL=Q1-1,5*IQR=12,2-1,5*10,7=12,2-16,05=-4,3 UL=Q3+1,5*10,7=22,9+1,5*9,7=21,9+14,55=38,95
c) yes There
is an outlier and it is 52,7 since it is bigger than the UL
3.
The manager of a furniture store sells from 0 to
4 armchairs each week. On the basis of past experience, the following probabilities
are assigned to sales of 0,1,2,3, or 4 armchairs: P(0)=0,08; P(1)=0,32;
P(3)=0,30; and P(4)=0,12
a)
Are these valid probability assignments? Why or
Why not? (9p)
b)
Let A be the event that 2 or fewer are sold in
one week, find P(A) (8p)
c)
Let B be the event that 4 or more are sold in
one week, find P(B) (8p)
Solution:
a)
if P(2)=0,18 then these are valid probability
assignments because the total of all probabilities must be equal 1 in order to
make a valid probability assignment.
b)
P(X<=2)=P(0)+P(1)+P(2)=0,08+0,32+0,18=0,58
c)
P(X>=4)=P(4)=0,12 since the probability is 0
when X>4
4.
A highway engineer who is studying the number of
accidents at a busy intersection has determined that accidents occur at the
rate of 2,5 per month
a)
Find the probability that none occur in 2 month.
(8p)
b)
Find the probability that more than one occurs
in a given month (8p)
c)
Find the mean, variation and standard deviation
(9p)
Solution:
a)
b)
c)
5.
Suppose we are interested in bidding on a piece
of land and we know there is one other bidder. The seller has announced that
the highest bid in excess of $10,000 will be accepted. Assume that the
competitor’s bid x is a random variable that is uniformly distributed between
$10,000 and $15,000. ( this problem belongs to Prof. Roger Myerson of
a)
Suppose you bid $12,000. What is the probability
that your bid will be accepted? (8p)
b)
Suppose you bid $14,000. What is the probability
that your bid will be accepted? (8p)
c)
What amount should you bid to maximize the
probability that you will get the property? (9p)
Solution:
a)
If the competitor’s bid X is a uniform
distribution and changes between $10,000 and $15,000 than the probability of
winning of the bid with $12,000 is equivalent to Competitors bids less then
$12,000 i.e. P(X<$12,000)=($12,000-$10,000)/($15,000-$10,000)=$2,000/$5,000=0,4
b)
Same approach as part a.
P(X<($14,000-$10,000)/$15,000-$10,000)=4/5=0,8
c)
$15,000 bids guarantees the property since
P(X<=15,000)=1
6.
10% of students in a university are married. A
random sample of 3 students is selected. ‘x’ is the random variable to denote
the number of married students in the sample
a)
Write down the probability distribution of x
(13p)
b)
Calculate the mean and the variance of x (12p)
Solution
a) 
b) E(X)=3.0,1=0,3 V(X)=3.0,1.0,9=0,21
7.
According to the DİE, the average weekly pay for
a Turkish production worker was 112 YTL in 1998. Assume that available data
indicate that wages are normally distributed with 21YTL
a)
What is the probability that the worker earn
between 100YTL and 140YTL (8p)
b)
How much does a production worker have to earn
to be in the top 20% of wage earners? (9p)
c)
For a randomly selected production worker, what
is the probability the worker earns less than 71YTL per week (8p)
Solution
a)
Let the average weekly payment be X then X
N(112,21)
b)
c)
8.
An auto insurance
company charges younger drivers a higher premium than it does older drivers
because younger drivers as a group tend to have more accidents. The company has 3 age groups: Group A
includes those under 25 years old, 22% of all its policyholders. Group B includes those 25-39 years old, 43%
of all its policyholders, Group C includes those 40 years old and older. Company records show that in any given
one-year period, 11% of its Group A policyholders have an accident. The percentages for groups B and C are 3% and
2%, respectively.
a)
What percent of the company’s policyholders are expected to have an accident during
the next 12 months? (12p)
b)
Suppose Mr. X has just had a car accident. If he is one of the company’s policyholders, what is the probability that he is under
25? (13p)
Solution
a)
b)
