Name______Key 2______________
DSC 210 Exam 1—Fall 2014
Dr. Wells
(1) The data below shows the number of minutes it takes for the Dayton Fire Department to respond to a
call from UD’s campus for a sample of 5 calls.
Call
1
2
3
4
5
Time to Respond (minutes)
20
16
24
26
24
(a) Compute the average number of minutes to respond. Use our notation to label the average. (5 pts)
𝑥̅ =
20 + 16 + 24 + 26 + 24
= 22
5
(b) Compute the variance and standard deviation of the number of minutes to respond. Use our notation
to label the variance and the standard deviation. (10 pts)
𝑠2 =
(20 − 22)2 + ⋯ + (24 − 22)2
= 16
4
𝑠 = √𝑠 2 = √16 = 4
(c) In a separate study consisting of a sample of 100 calls, a researcher found that the average time to
respond was 14 minutes with a standard deviation of 5. Construct the interval that is 2.4 standard
deviations about the mean. What can we say about the number of sample values that fall in this interval?
(5 pts)
𝑥̅ ± 2.4(𝑠)
= 14 ± 2.4(5)
= 14 ± 12
= (2, 26)
1
At least 1 − 2.42 = 82.6% of the sample will fall in this interval. That means 83 or more of the 100 calls
will fall there.
IMPORTANT: YOU MUST SHOW YOUR WORK FOR CREDIT.
2
(2) A type of battery is sold in packages of five. Assume that the package in question contains two
defective batteries. We plan on taking a random sample of two batteries out of the package.
(a) Write out the outcome space. Carefully define what your symbols or terms mean. (5 pts)
Good batteries: G1, G2, G3
(G1, G2)
(G2, G3)
(G1, G3)
(G1, B1)
(G2, B1)
(G3, B1)
Bad Batteries: B1, B2
(G1, B2)
(G2, B2)
(G3, B2)
(B1, B2)
(b) Using the outcome space above, what is the probability of getting one defective battery in your
sample? Show how you got your result. (5 pts)
(G1, B1)
(G2, B1)
(G3, B1)
(G1, B2)
(G2, B2)
(G3, B2)
Assuming that the outcomes are equally likely the probability is 6/10 = 0.6.
(c) Using our counting rules, what is the probability of getting one defective battery in your sample? (5
pts)
3 2
( )( ) 3 ∙ 2
6
𝑃{1 𝑏𝑎𝑑 𝑏𝑎𝑡𝑡𝑒𝑟𝑦} = 1 1 =
=
5
10
10
( )
2
(d) Using our counting rules, what is the probability of getting two defective batteries in your sample? (5
pts)
3 2
( )( ) 1 ∙ 1
1
𝑃{2 𝑏𝑎𝑑 𝑏𝑎𝑡𝑡𝑒𝑟𝑖𝑒𝑠} = 0 2 =
=
5
10
10
( )
2
IMPORTANT: YOU MUST SHOW YOUR WORK FOR CREDIT.
(3) An insurance company collected data for 200 hospital patients who suffered a severe heart attack.
Information about the age at the time of the heart attack and the individual’s gender is shown below.
Age
Gender
Less than 30 (A)
30-60 (B)
More than 60 (C)
Total
Male (M)
15
90
35
140
Female (F)
5
40
15
60
Total
20
130
50
200
Assume that we select a patient at random. Answer the following in decimal or fractional form.
(a) What is the probability that the patient is more than 60 years old? (4 pts)
50
𝑃{𝐶} =
= .25
200
(b) What is the probability that the patient is a female? (4 pts)
60
𝑃{𝐹} =
= 0.3
200
(c) What is the probability that the patient is less than 30 and male? (4 pts)
60
𝑃{𝐹} =
= 0.3
200
(d) What is the probability that the patient is female or is more than 60 years old? (4 pts)
60 + 35 50 + 60 − 15
𝑃{𝐶 ∪ 𝐹} =
=
= .075
200
200
(e) If the patient is 30-60 years old, what is the probability that the patient is male? (4 pts)
90
𝑃{𝑀|𝐵} =
= .692
130
(f) If the patient is female, what is the probability that the patient is 30-60 years old? (4 pts)
40
𝑃{𝐵|𝐹} =
= .667
60
(g) Is the event C independent of the event F? Explain. (4 pts)
15
𝑃{𝐶|𝐹} =
= .25 = 𝑃{𝐶} by part (a) → 𝐶 𝑎𝑛𝑑 𝐹 𝑎𝑟𝑒 𝑖𝑛𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡
60
(h) Is the event C mutually exclusive of the event F? Explain. (4 pts)
15
𝑃{𝐶 ∩ 𝐹} =
≠ 0 → C and F are not mutually exclusive
200
3
IMPORTANT: YOU MUST SHOW YOUR WORK FOR CREDIT.
4
(4) A student who takes a DSC210 exam either crams (C) for the test or studies (S) throughout the term
and stays up with the homework. This results in the student receiving a good grade (G) or a bad grade
(B). If a student studies throughout the term and does the homework, the chance that she will receive a
good grade is .90. On the other hand, if a student crams for the test, the chance of receiving a good grade
is .20. In a given class it is known that 40% of the students crammed for the test.
(a) If a student receives a good grade on the test, what is the probability that he crammed? Be sure to
show all of your work. (10 pts)
𝑃{𝐺|𝑆} = 0.9
𝑃{𝐺|𝐶} = 0.2
𝑃{𝐶} = 0.4 →
𝑃{𝑆} = 0.6
𝑃{𝐺 ∩ 𝐶} = 𝑃{𝐺|𝐶}𝑃{𝐶} = (0.2)(0.4) = 0.08
𝑃{𝐺 ∩ 𝑆} = 𝑃{𝐺|𝑆}𝑃{𝑆} = (0.9)(0.6) = 0.54
C
S
Total
G
0.08
0.54
0.62
B
0.32
0.06
0.38
𝑃{𝐶|𝐺} =
. 08
= 0.129
. 62
Total
0.4
0.6
1.0
(b) If a student receives a bad grade on the test, what is the probability that he crammed? (5 pts)
𝑃{𝐶|𝐵} =
. 32
= 0.842
. 38
(c) What is the probability that a student will receive a good grade (5 pts)?
𝑃{𝐺} = 0.62
IMPORTANT: YOU MUST SHOW YOUR WORK FOR CREDIT.
(5) The data below represents the scores on a statistics test given to a class of 40 students.
Score
Frequencies
Cumulative
Frequencies
Below 60
2
2
60-69
6
8
70-79
16
24
80-89
9
33
90-100
7
40
(a) (5 pts) Use the frequencies to construct the cumulative frequencies in the cells provided.
(b) (3 pts) Describe exactly the meaning of this number.
There were 24 students who scored 79 or less.
5