Midterm 1
1. In a multiplechoice test, a student can answer any question correctly with probability
of 0.85, independently of all other questions. The test consists of 100 questions. Each correct
answer is worth 4 points, and for each incorrect answer 1 point is deducted. This student
always answers all the questions.
a) What is the student’s expected score (in points)?
b) What is the probability of the student’s score dropping below 275 points? Hint: use
normal approximation.
c) What is the probability of the student obtaining more than 350 points on the test?
Solution.
If we denote by
X
the number of questions that the students answers correctly,
and by
Q
his score in points, then, obviously,
X
and
Q
are related by
P
= 4
X

(100

X
) =
5
X

100.
a) Clearly,
X
∼
Bin
(100
,
0
.
85). Therefore,
E
(
Q
) = 5
E
(
X
)

100 = 5
·
100
·
0
.
85

100 =
325.
b) Pr(
Q <
275) = Pr(
X <
75) = Pr(
X
≤
74)
≈
Pr
‡
Z <
74
.
5

85
√
100
·
0
.
85
·
0
.
15
·
= Φ(

2
.
66) =
0
.
0016.
c) Pr(
Q <
350) = Pr(
X >
90) = Pr(
X
≥
91)
≈
Pr
‡
Z >
90
.
5

85
√
100
·
0
.
85
·
0
.
15
·
= 1

Φ(1
.
54) =
0
.
062.
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 Spring '08
 Perevalov
 Statistics, 130 hours, 18.2 meters, 18.6 meters

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