STA13 Winter 2023, Practice Midterm II Version A
Instructions:
(b) disjoint
• 50 minutes, closed book midterm
(c) dependent
• one letter-sized (handwritten, double-sided) “cheatsheet”, hand held calculator and scrap paper are allowed
(d) complementary
(e) independent
Answer: e
• 16 multiple choice problems, 1 points each
5. In a classes of 400 students, 220 are women, 100 are
earning an A, and 70 are women that are earning an
A. If a student is randomly selected from the class,
find the probability that the student is a women given
that the student is earning an A.
• put your name & ID on the scantron
• hand in the scantron at the end of the midterm
• grading is based on scantron which will not be returned to you
(a) 4/5
• this is version A, please mark the corresponding
circle on the scantran
1. An experiment consists of tossing 5 fair coins simultaneously. What is the probability of getting exactly
three heads?
(b)
5
16
(c)
1
2
(d)
2
5
(e)
(d) 5/11
(a) 0.728 (b) 0.714 (c) 0.465 (d) 0.275 (e) 0.135
Answer: a
11
16
7. Consider the given discrete probability distribution.
Find the probability that X exceeds 4.
x
2
4
7
9
p(x) 0.13 ? 0.02 0.09
5
Answer: b ( 16
)
2. A fair die is tossed twice and event A= the sum of
two tosses equals to 7, event B= the first toss is 3.
Which of the following is true?
(a) A and B are mutually exclusive
(a) 0.11
(b) 0.87
(c) 0.89
(e) Not enough information is given.
(b) A and Bc are independent
Answer: a
(c) A = Bc
(e) none of the above.
Answer: b (P(A) = 1/6, P(B) = 1/6, P(A ∩ B) =
1/36, thus A and B are independent. Therefore A
and Bc are independent.)
The mean µ of X is
(a) .75 (b) .9 (c) 1.9 (d) 3.1 (e) none of the above
Answer: c (µ = ∑ xp(x) = 1.9)
3. An urn contains 10 red balls and 10 yellow balls.
Two balls are randomly chosen without replacement,
what is the probability of getting two red balls?
9
38
(b)
1
4
(c)
1
10
(d)
(d) 0.76
8. A random variable X has the probability distribution
given below:
x
0
1
2
3
4
p(x) 0.1 0.3 0.3 0.2 0.1
(d) A is a subset of B
(a)
(e) 11/20
6. A teacher designs a test so a student who studies will
pass 84% of the time, but a student who does not
study will pass 9% of the time. A certain student
studies for 85% of the tests taken. On a give test,
what is the probability that a student passes?
Midterm begins here...
3
5
(c) 7/22
Answer: b
• good luck!
(a)
(b) 7/10
1
2
9. Refer to Problem 8. The variance σ2 of X is
(a) .75 (b) 1.29 (c) 1.9 (d) 1.5 (e) none of the above
Answer: b (Var(X) = ∑(x − µ)2 p(x) = 1.29)
(e) none of the above
Answer: a (A1 =first ball is red, A2 =second ball
9
is red, then P(A1 ) = 10
20 , P(A2 |A1 ) = 19 . Therefore
9
P(A1 A2 ) = P(A2 |A1 )P(A1 ) = 38
)
Answer: d 1/3
10. Let X denote a random draw from the following list
of 5 numbers: 0, 1, 1, 0, 1. Let Y be X − 1/5. What is
E(Y )?
4. P(A|B) = P(A) and P(B|A) = P(B), then events A
and B are said to be:
(a) 1
(a) mutually exclusive
(b) 3/5
Answer: c 2/5
1
(c) 2/5
(d) 1/2
(e) -1/5
11. Find E(X 2 /10) where X is a random variable with
PMF given by the following table.
x
−1
0
1
p(x) 0.2 0.1 0.7
(a) 0.9
(b) 0.09
(c) 0.5
(d) 0.05
(d) No, a binomial distribution requires only two
possible outcomes for each experimental unit
sampled.
(e) Not enough information is given.
(e) 1.58
Answer: d
Answer: b 0.09
12. Suppose X is a random variable with PMF
(
2( 13 )x if x = 1, 2, 3, ...
p(x) =
0
otherwise
Find P(X ≤ 2).
(a) 0.11
(b) 0.56
(c) 0.89
(d) 0.33
(e) 0.22
Answer: c 0.89
13. From past experience, it is known that 70% of oneyear-old children can distinguish their mother’s voice
from the voice of a similar sounding female. A random sample of 20 one-year-olds are given this voice
recognition test. What is the probability at least 15
children recognize their mother’s voice.
(a) 0.762 (b) 0.238 (c) 0.584 (d) 0.416 (e) none of
the above
Answer: d (n = 20, p = .7; P(X ≥ 15) = .416)
14. The standard deviation of a binomial distribution for
which n = 50 and p = 0.15 is (rounded to three digits):
(a) 50.15 (b) 7.082 (c) 6.375 (d) 2.525 (e) 12.5
p
Answer: d (σ = np(1 − p))
15. Suppose X ∼ Bino(2, 21 ). Find P(X = 0|X ≤ 1).
(a) 3/4
(b) 1/4
(c) 1/2
(d) 1/3
(e) 2/3
Answer: d
16. A recent article in the paper claims that business
ethics are at an all-time low. Reporting on a recent
sample, the paper claims that 35% of all employees
believe their company president possesses low ethical standards. Suppose 20 of a company’s employees
are randomly and independently sampled and asked
if they believe their company president has low ethical standards and their years of experience at the
company. Could the number of years of experience
be modeled by a binomial random variable? (Choose
the most appropriate one.)
(a) Yes, the sample is a random and independent
sample.
(b) Yes, the sample size is 20.
(c) No, the employees would not be considered independent in the present sample.
2