Homework 5 Solutions-2
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School
University of California, Santa Barbara *
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Course
5A
Subject
Industrial Engineering
Date
Dec 6, 2023
Type
Pages
3
Uploaded by LieutenantGazellePerson897
HOMEWORK
5
PSTAT 5A – Fall 2023
1. Unit 5 Self-Assessment Quiz (Optional)
Solution:
Look for solutions in the textbook.
2. The length of a surfer’s ride (measured in seconds (s)) at Campus Point is normally distributed with
a mean time of 58.4 s and a variance of 158.76 s
2
. An oceanographer selects a surfer catching a wave
at random, and is interested in the length of the ride.
(a) Define the random variable of interest.
Solution:
X
= the length of a surfer’s ride.
(b) State the distribution of the random variable of interest.
Solution:
X
∼
(58
.
4
,
12
.
6)
(c) What is the probability the ride time is longer than 60.038 seconds?
Solution:
P
(
X >
60
.
038) =
P
(
Z >
0
.
13) = 1
-
0
.
5517 = 0
.
4483
(d) What is the probability that the ride time is between 50 seconds and 70 seconds?
Solution:
P
(50
≤
X
≤
70) =
P
(
Z
≤
0
.
92)
-
P
(
Z <
-
0
.
67) = 0
.
8212
-
0
.
2514 = 0
.
5698
(e) How long is a ride at the 67
th
percentile?
Solution:
P
(
Z
≤
0
.
44) = 0
.
6700
⇒
x
-
58
.
4
12
.
6
= 0
.
44
⇒
x
= 63
.
944
s
3. The length of a surfer’s ride (measured in seconds (s)) at Campus Point is normally distributed with a
mean time of 58.4 s and a variance of 158.76 s
2
. An oceanographer selects 16 surfers’ rides at random,
and is interested in the mean length of time the rides last.
(a) Define the random variable of interest.
Solution:
¯
X
= the sample mean duration of 16 rides.
(b) State the distribution of
¯
X
.
Solution:
¯
X
is
N
(58
.
4
,
3
.
15)
(c) What is the probability the mean ride time is longer than 60.038 seconds?
Solution:
P
(
¯
X >
60
.
038) =
P
(
Z >
0
.
52) = 1
-
0
.
6985 = 0
.
3015
(d) What is the probability that the mean ride time is between 50 seconds and 70 seconds?
Solution:
P
(50
≤
¯
X
≤
70) =
P
(
Z
≤
3
.
68)
-
P
(
Z <
-
2
.
67) = 1
-
0
.
0038 = 0
.
9962
(e) What is the probability the mean ride time is less than 54.4625 seconds?
Solution:
P
(
X <
54
.
4625) =
P
(
Z <
-
1
.
25) = 1
-
0
.
8944 = 0
.
1056
4. The NBA is interested in the number of fouls committed in a game (fouls by both teams are included
in the total). They select a sample of 9 games and record the number of fouls as follows:
31
,
34
,
26
,
51
,
37
,
43
,
34
,
42
,
48
(a) Calculate the sample mean.
Solution:
¯
x
=
31+34+26+51+37+43+34+42+48
9
= 38
.
4444
(b) Calculate the standard deviation.
Solution:
s
=
q
(31
-
38
.
4444)
2
+
···
+(48
-
38
.
4444)
2
9
-
1
=
q
534
.
2222
8
= 8
.
1718
(c) Find the mode(s)
Solution:
34
(d) Find the 5 number summary.
Solution:
26
,
31
,
34
,
34
,
37
,
42
,
43
,
48
,
51
Min = 26, Q1 = 32.5, Q2 = 37, Q3 = 45.5, Max = 51.
(e) Calculate the
IQR
.
Solution:
IQR = 45
.
5
-
32
.
5 = 13
5. According to data from Caj´
e Coffee, thirty-four percent of its customers order non-espresso based
drinks. The Caj´
e manager select 50 customers at random and finds that 21 of them order non-espresso
based drinks.
(a) Define the random variable of interest,
X
.
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