Homework
Wirtschaftsstatistik 1
1. (a) Let x1 , x2 , . . . , xn be a given data set with mean X̄. Now let yi = xi + c, for
i = 1, 2, . . . , n be a new data set with mean Ȳ , where c is a constant. What will be
the value of Ȳ compared to X̄?
(b) Let x1 , x2 , . . . , xn be a given data set with mean X̄. Now let yi = cxi , for i =
1, 2, . . . , n be a new data set with mean Ȳ , where c is a constant. What will be the
value of Ȳ compared to X̄?
(c) Calculate the mean of 2 4 10 5 1.
(a) compare it with the mean of 12 14 20 15 11
(b) compare it with the mean of 20 40 100 50 10.
Solution: 4.4, 14.4, 44.4
2. Sets A and B are given as follows: A = {1, 5, 7, 8, 10, 13} and B = {1, 2, 4, 7, 10, 15, 16}.
Construct the sets A [ B, A \ B, A \ B, B \ A and A 4 B, where 4 is called a symmetric
di↵erence and defined as A 4 B = (A \ B) [ (B \ A).
Solution: A[B = {1, 2, 4, 5, 7, 8, 10, 13, 15, 16} , A\B = {1, 7, 10} , A\B = {5, 8, 13} ,
B \ A = {2, 4, 15, 16} and A 4 B = {2, 4, 5, 8, 13, 15, 16} .
3. An experiment consists of rolling two dice. First define its sample space ⌦ and then
write the sets which describe the following events:
(a) A: The sum of numbers we get is 7.
(b) B: The sum of numbers we get is smaller then 4.
(c) C: The sum of numbers we get is larger then 12.
(d) D: The number we get for the first die is larger then the number we get for the
second die.
(e) E: The product of numbers we get is odd.
Calculate the probabilities of the events A, B, C, D, E, A [ B, E \ D and B \ E.
Solution:
1 1
, , 0, 15
, 1, 1, 1, 1 .
6 12
36 4 4 6 36
4. Three coins are tossed. First define its sample space ⌦ and then calculate the probability
that both heads and tails appear at least once.
Solution:
3
.
4
5. A prime number is a number that is divisible only by 1 and itself. The prime numbers
less than 100 are listed below.
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Homework
Wirtschaftsstatistik 1
2
31
73
3
37
79
5
41
83
7
43
89
11
47
97
13
53
17
59
19
61
23
67
29
71
Choose one of these numbers at random. Find the probability that
(a) The number is even.
(b) The sum of the number’s digits is even.
(c) The number is greater than 50.
Solution:
1 13 2
, , .
25 25 5
6. In a company the percent of employees who have the VISA credit card is 22%, the
percent of employees who have the MASTER credit card is 58%. The percentage of
employees who have both cards is 14%. Assume that no other credit cards are possible.
What is the percentage of people in this company who don’t have any credit card?
Solution: 0.34.
7. A sports club has 120 members, of whom 44 play tennis, 30 play squash, and 18 play
both tennis and squash. If a member is chosen at random, find the probability that this
person
(a) does not play tennis?
(b) does not play squash?
(c) does not play tennis nor squash?
Solution: 0.63, 0.75, 0.53.
8. A student has to sell 2 books from his collection of 6 mathematics books, 7 literature
books and 4 economics books. In how many ways can he do that if
(a) books must be on the same subject?
(b) books must be from di↵erent subjects?
Solution:
0.69.
6
2
+
7
2
+
4
2
= 42, 6 · 7 + 7 · 4 + 6 · 4 = 94. and probabilities are 0.31 and
9. A person has 8 college friends but only 5 of them can be invited to the Black & White
Party.
(a) In how many ways can this be done if we know that Kate and Serena are not on
speaking terms so both of them cannot be invited?
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Homework
Wirtschaftsstatistik 1
(b) In how many ways can this be done if we know that Kate and Neal are a couple
and will only attend a party if they are both invited?
Solution:
6
5
+
2
1
6
4
= 36,
6
5
+
6
3
= 26 and probabilities are 0.31 and 0.69.
A code consists of a digit chosen from 0 to 9 followed by a letter of the alphabet (A-Z).
What is the probability the code is 9Z?
Solution:
1
.
10·26
10. We randomly choose a number from the set {1, 2, . . . , 22}. What is the probability that
the number is even if we already know that
(a) the number is divisible by 3?
(b) the number is divisible by 4?
(c) the number is divisible by 5?
Solution:
3
, 1, 12 .
7
11. A bag contains 3 red and 4 blue marbles. Two marbles are drawn at random without
replacement. If the first marble drawn is blue, what is the probability the second marble
is also blue?
Solution:
1
.
2
12. The coin is tossed, then a die is rolled and then the coin is tossed again. What is the
probability that you get the same side of the coin both times and as well that the number
you get on the die is divisible by 3?
Solution:
1
.
6
13. An unfair coin (0.4 probability for heads, 0.6 probability for tails) is tossed 5 times.
(a) What is the probability that you get exactly 3 tails?
(b) What is the probability that you get at least 3 tails?
Solution:
5
3
(0.6)3 (0.4)2 ,
5
3
(0.6)3 (0.4)2 +
5
4
(0.6)4 (0.4) + (0.6)5 .
14. In a class of 24 girls, 7 have blonde hair, 5 have red hair and the rest have black hair.
I select a committee of 5 from the class completely at random. Given that exactly two
are red heads, find the probability that my committee contains exactly one blonde.
3
Homework
Solution:
Wirtschaftsstatistik 1
154
.
323
15. Two fair dice are rolled.
(a) What is the conditional probability that at least one lands on 6 given that the dice
land on di↵erent numbers?
(b) What is the conditional probability that the first one lands on 6 given that the sum
of the dice numbers is 7?
(c) What is the conditional probability that the first one lands on 6 given that the sum
of the dice numbers is 10?
(d) What is the conditional probability that the first one lands on 6 given that the sum
of the dice numbers is 12?
Solution:
1 1 1
, , , 1.
3 6 3
16. Let X be a random variable with the following distribution.
✓
◆
2
4
6
X:
.
1/3 1/3 1/3
Calculate EX, Var(X) and SD(X).
Solution: 4, 2.67, 1.63.
17. Let X be a random variable with the following distribution.
✓
◆
2 1
4
X:
,
a 3/8 1/8
where a 2 R. Find a and calculate EX.
Solution: a = 1/2, EX =
1
.
8
18. A box contains 5 red and 5 blue marbles. Two marbles are withdrawn randomly at
the same time. If they are the same colour, then you win 1.1e. If they are of di↵erent
colours, then you win -1e. (I.e, you lose 1e). Calculate the expected value of the
amount you win.
Solution:
1
e.
15
4
Homework
Wirtschaftsstatistik 1
19. It is known that DVDs produced by a certain company will be defective with probability
0.01, independently of each other. The company sells the DVDs in packages of 10 and
o↵ers a money-back guarantee when at least 2 DVDs of the 10 DVDs are defective.
What is the probability that company must replace a sold package?
Solution: 1
(0.99)10
10 · 0.01(0.99)9 = 0.004.
20. Let f be defined as follows:
f (x) =
⇢
x3 ) : if 0 < x <
0 : otherwise
c(2x
5
2
where c 2 R. Can f be a density function?
Solution: No, c would have to be 64/225, but then the function f (x) is no longer
non-negative everywhere. For example, look at f (1).
21. Let f be a density function of a random variable X defined as follows:
⇢
a + bx2 : if 0  x  1
f (x) =
,
0 : otherwise
where a, b 2 R. If EX = 35 , find a and b.
Solution: a = 3/5, b = 6/5.
22. Let f be a function such that
f (x) =
⇢
1
x2
:
0:
if x 1
otherwise.
(a) Show that f is a density function.
(b) If X is a random variable with density f , calculate EX.
Solution:
(a)
(b) EX = 1.
23. Let X be a durability of a light bulb (in hours). The density of X is given by
8
< 2x : if 0  x  12
3
: if 2  x  3 .
f (x) =
: 4
0 : otherwise
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Homework
Wirtschaftsstatistik 1
(a) Which percent of light bulbs lasts longer than 15 minutes?
(b) Calculate EX and Var X.
Solution:
(a) 93.75% =
(b) EX =
47
24
15
16
· 100%
hours, which is 117.5 minutes. Var X =
545
576
hours.
24. Let Z be a random variable such that Z ⇠ N (0, 1). Find the following probabilities:
(a) P(Z < 0)
(b) P(Z 2 > 4)
(c) P(Z <
1.75)
(d) P( 2 < Z < 2)
(e) P(|Z| > 2.8).
Solution:
(a)
1
2
(b) 0.0455
(c) 0.0401
(d) 0.9545
(e) 0.0051
25. Let Z be a random variable such that Z ⇠ N (15, 4). Find the following probabilities:
(a) P(Z < 23)
(b) P(Z < 11)
(c) P(11 < Z < 19)
(d) P(Z > 27).
Solution:
(a) 1
(b) 0.0228
(c) 0.9545
(d) 0
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Homework
Wirtschaftsstatistik 1
26. Let Z be a random variable such that Z ⇠ N (0, 1). Find the value a, such that it holds:
(a) P(Z < a) = 0.05
(b) P(Z < a) = 0.5
(c) P(|Z| < a) = 0.99.
Solution:
(a)
1.64
(b) 0
(c) 2.57
27. The time required to complete a certain loan application form is a normal random
variable with mean 90 minutes and standard deviation 15 minutes. Find the probability
that an application form is filled out in
(a) Less than 75 minutes
(b) More than 100 minutes
(c) Between 90 and 120 minutes.
Solution:
(a) 0.1587
(b) 0.2525
(c) 0.4772
28. To qualify for a police academy, candidates must score in the top 10% on a general
abilities test. The test has a mean of 200 and a standard deviation of 20. Find the
lowest possible score to qualify. Assume the test scores are normally distributed.
Solution: 225.63
29. An IQ test produces scores that are normally distributed with mean value 100 and
standard deviation 14.2. The top 1 percent of all scores is in what range?
Solution: Larger and equal than 133.03.
30. A producer of cigarettes claims that the mean nicotine content in its cigarettes is 2.4
milligrams with a standard deviation of 0.2 milligrams. Assuming these figures are
correct, find the expected value and variance of the sample mean nicotine content of
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Homework
Wirtschaftsstatistik 1
(a) 36
(b) 100
(c) 900
randomly chosen cigarettes.
Solution:
2
0.22
36
= 0.001.
2
0.22
100
= 0.004.
2
0.22
900
= 0.0004.
(a) X̄ ⇠ N (2.4, 0.2
). EX̄ = 2.4, Var X̄ =
36
(b) X̄ ⇠ N (2.4, 0.2
). EX̄ = 2.4, Var X̄ =
100
(c) X̄ ⇠ N (2.4, 0.2
). EX̄ = 2.4, Var X̄ =
900
31. Consider a sample of size 16 from a population having mean 100 and standard deviation
. Approximate the probability that the sample mean lies between 96 and 104 when
(a)
=1
(b)
= 16.
Data is assumed to be normal.
Solution:
(a) Approximately 1.
(b) 0.68.
32. A magazine reported that 6% of their readers read their newspapers at work. If 300
readers are selected at random, find the probability that exactly 25 say they read the
newspaper at work.
Solution: 0.0227
33. Of the members of a tennis club, 10% are lawyers. If 200 tennis club members are
selected at random, find the probability that 10 or more will be lawyers.
Solution: 0.9934
34. Suppose that we conduct a survey of 19 millionaires to find out what percent of their
income the average millionaire donates to charity. We discover that the mean percent
is 15. Assume that the standard deviation is 5. Find a 95% confidence interval for
the mean percent. Assume that the distribution of all charity percent is approximately
normal.
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Homework
Wirtschaftsstatistik 1
Solution: [12.75, 17.25].
35. The following data represent the number of drinks sold from a vending machine based
on a sample of 20 days:
56
51
44
50
53
72
40
45
65
69
39
38
36
40
41
51
47
47
55
53
Data is assumed to be normal.
(a) Determine a 95 percent confidence interval estimate of the mean number of drinks
sold daily if the standard deviation is assumed to be 10.
(b) Repeat part (a) for the 90 percent confidence interval.
Solution:
(a) [45.22, 53.98]
(b) [45.92, 53.28]
36. A survey of 1721 people found that 15.9% of individuals purchase cooking books. Find
the 95% confidence interval of the true proportion (expressed in percentages) of people
who buy these books.
Solution: [14.2%, 17.6%].
37. Out of a random sample of 100 students at a university, 82 stated that they were nonsmokers. Based on this, construct a 99 percent confidence interval estimate of the
proportion of all the students at the university who are non-smokers.
Solution: [0.721, 0.919].
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