Math 201: Statistics
December 4, 2007
Fall 2007 Midterm #2
Closed book & notes; only an A4-size formula sheet and a calculator allowed;
120 minutes. No questions accepted!
Instructions: There are nine pages (one cover and eight pages with questions) in this exam.
Please inspect the exam and make sure you have all 9 pages. You may only use your calculator
and your formula sheet. Do all your work on these pages. If you use the back of a page, make
sure to indicate that. You may not exchange any kind of material with another student.
The formula sheet should be handwritten; photocopies from others will not be accepted.
Remember: You must show all your work to get proper credit. Round your final answer to
two decimal places.
Academic Honesty Code: Koç University Academic Honesty Code stipulates that “copying
from others or providing answers or information, written or oral, to others is cheating.” By
taking this exam, you are assuming full responsibility for observing the Academic Honesty
Code.
NAME and SURNAME:_________KEY___________ SIGNATURE:______________________
INSTRUCTOR : ____________________
LECTURE TIME : ___________________
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Total:
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(1) Suppose there are two number wheels. Number wheel 1 consists of the four numbers
{1,2,3,4} and number wheel 2 consists of the three numbers {1,2,3}. Let X be the random
variable representing the outcome of spinning (spin=çevirmek, döndürmek) number wheel 1,
and Y be the random variable representing the outcome of spinning number wheel 2. The probability distribution functions (pdfs) for X and Y are given below:
x
p(x)
1
0.15
y
p(y)
2
0.2
1
0.3
3
0.35
2
0.4
4
0.3
3
0.3
(a) Based on the above pdfs, find E(X). (3 points)
(b) Based on the above pdfs, find Var(Y). (3 points)
(c) Spin the number wheels 1 and 2 separately (that is, spin one of the wheels and then spin the
other wheel), add the two numbers obtained, and subtract two from the sum. Let the random
variable W represent the outcome of this experiment. Find Var(W). (4 points)
If both wheels are connected mechanically together, they can spin at the same time. But in this
case, the (marginal) probability distributions for X and Y change. The joint probability distribution function of X and Y in this case is given below:
Y
1
2
3
1
0.03
0.12
0.07
X
0.02
0.13
0.11
2
0.01
0.13
0.14
3
0.01
0.09
0.14
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(d) Based on the above joint pdf, find the covariance between X and Y, i.e., Cov(X,Y). (4
points)
(e) Based on the above joint pdf, find the marginal distribution of X if both wheels are spun
(çevrilmek) at the same time. (3 points)
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3
(2) Consider the pdf for the random variable X in question (1) above. That is,
x
p(x)
1
0.15
2
0.2
3
0.35
4
0.3
(a) Let W be a random variable representing the number of times the outcome “more than 2”
has occurred in 18 spins of number wheel 1. Write the expression for the exact value of
P(7 ≤ W ≤ 11) . Do NOT evaluate this expression. (5 points)
(b) If you spin number wheel 1 sixteen times, what is the probability that in 8, 9, or 10 of those
sixteen times the outcome is more than two? Note: You MUST use a table provided with the
exam and there is only one answer, not three! (5 points)
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(3) It is given that X is a random variable representing the weight of a person in a population.
Suppose X has a normal distribution with mean 60 kg and standard deviation 5 kg.
(a) What is the probability that a person’s weight is within 20 kgs of 55 kgs?, i.e. (yani), calculate P(|X-55| < 20). (4 points)
(b) What is the 95th percentile of the random variable X? (4 points)
(c) Suppose we define another random variable Y as Y = X + 10. What is the mean, standard
deviation and the 95th percentile of Y? (4 points)
(d) What is the probability that a person selected randomly weighs less than 56kg? (4 points)
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(4) Suppose the amount of time Aylin spends for Math 201 midterm is represented by the random variable X; and the amount of time Burçin spends for the same midterm is represented by
the random variable Y. Assume X has a normal distribution with mean 10 hours and standard
deviation 2 hours and Y has a normal distribution with mean 11 hours and standard deviation 3
hours. Assume also that X and Y are independent from each other.
(a) What is the probability that Aylin will spend at least 11 hours for the midterm? (4 points)
(b) What is the probability that Aylin will spend more time than Burçin for the midterm? (5
points)
(c) What is the probability that the total time spent by Aylin and Burçin will exceed 15 hours?
(6 points)
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(5) The mean and standard deviation of the lifetime of a type of battery used in electric
cars are, respectively, 225 and 40 minutes. Find the probability that average lifetime for
100 batteries will be
(a) larger than 220 minutes? (4 points)
(b) less than 235 minutes? (4 points)
(c) between 220 and 235 minutes? (4 points)
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(6) Suppose that 12 percent of the members of a population are left-handed. In a random sample of 100 individuals from this population,
(a) Find the mean and standard deviation of the number of left-handed people. (4 points)
(b) Find the probability that the number of left-handed people is between 10 and 14 inclusive.
(6 points)
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(7) An airline is interested in determining the average amount of money spent by its customers
who are flying for reasons of business in a specified month. Assume the standard deviation of
all business customer expenditures (harcama) is 250 YTL for this month.
(a) If the airline wants to be 90 percent certain that its estimate will be correct within 20 YTL
(i.e., the length of the corresponding confidence interval is less than or equal to 40 YTL), how
large a random sample should it select? (5 points)
(b) If there are 518 customers who fly for reasons of business in a random sample of 1000 customers, and the amount of money spent by the business fliers is 1000000 YTL, construct a 95
percent confidence interval estimate for the average expenditure of all business fliers in the
specified month. (5 points)
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(8) A random sample from a certain population was asked to keep a record of the amount of
time spent watching television in a specified week.
(a) If the sample mean was 24.4 hours and the population standard deviation was 7.4 hours,
give a 95 percent confidence interval estimate for the average time spent watching television by
all members of the population that week, if the random sample was of size 50. State all the
assumption(s) you are making or theorem(s) you are using. (6 points)
(b) If the sample mean was 24.4 hours and the sample standard deviation was 7.4 hours, give a
95 percent confidence interval estimate for the average time spent watching television by all
members of the population that week, if the random sample was of size 20. State all the assumption(s) you are making or theorem(s) you are using. (6 points)
(c) Which confidence interval (i.e., confidence interval in part (a) or part (b)) is wider (daha
geniş)? Explain why. (3 points)
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