2nd Nine Weeks Study Guide
Math Studies
Answer Key
1) The scores awarded by two judges at a diving competition are shown in the table.
Competitor
P
Q
R
S
T
U
V
W
X
Y
Judge A
5
6.5
8
9
4
2.5
7
5
6
3
Judge B
6
7
8.5
9
5
4
7.5
5
7
4.5
a) Describe the association (positive/negative; strong/moderate/weak)
Positive, strong
b) Find r.
0.981
c) Find the line of best fit.
y = .799x + 1.87
d) If I add competitor Z and Judge A scored 4 & Judge B scored 5, how does this change the association?
What is the difference in r?
0.982 – 0.981 = 0.001
e) Use the equation of least squares regression to predict Judge B if Judge A gives a score of 4.5.
y = .799(4.5) +1.87
y = 5.4655
5.5
2) For his Mathematical Studies project, Marty set out to discover if stress was related to the amount of time
that students spent travelling to and from school. The results of one of his surveys are shown in the table below.
Travel time (t mins)
t ≤ 15
15 < t ≤ 30
30 < t
Number of students
high stress
moderate stress
9
5
17
8
18
6
low stress
18
28
7
He used the χ2 test at the 5% level of significance to find out if there was any relationship between
student stress and travel time.
a) State the Null (H0) and Alternative (H1) hypothesis. Null – There is no association between stress and
travel time. They are independent. Alt – There is any association between stress and travel time. They
are dependent.
b) Find the degrees of freedom.
(r – 1)(c – 1) = (3 – 1)(3 – 1) = (2)(2) = 4
c) Find Chi-Squared (χ2).
9.83
d) Compare Chi-Squared (χ2) to the given critical value (9.488).
9.83 > 9.488
e) What is the p-value.
0.0434
f) What conclusion can Marty draw from this test?
Reject the null and accept the alternative.
Math Studies
2nd Nine Weeks Study Guide
Answer Key
3) The given table shows complaints received by the Telecommunications Ombudsman concerning internet
services over a four year period.
a) What is the probability that a complaint received in 2000/01 is about customer service?
1181
= 0.148 = 14.8%
7965
b) What is the probability that a complaint received at any time during the 4 year period related to
billing?
10642
= 0.430 = 43.0%
24759
c) What is the probability that a complaint received in 2001/02 did not relate to either billing or faults?
3531
= 0.372 = 37.2%
9497
4) A mother gave birth to quadruplets.
a) Show the sample space. BBBB, BBBG, BBGG, BBGB, BGBB, BGBG, BGGG, BGGB, GGGG,
GGGB, GGBB, GGBG, GBBG, GBGB, GBBB, GBGG
b)What is the probability of having all girls?
1
= 0.0625 = 6.25%
16
c)What is the probability of having at least one boy?
15
= .9375 = 93.75%
16
5) The probability that it will snow tomorrow is 0.3.
If it snows tomorrow the probability that Chuck will be late for school is 0.8.
If it does not snow tomorrow the probability that Chuck will be late for school is 0.1.
(a) Complete the tree diagram below.
0.2
0.1
0.7
0.9
2nd Nine Weeks Study Guide
Math Studies
Answer Key
(b) Find the probability that it does not snow tomorrow and Chuck is late for school.
0.7(0.1) = 0.07 or 7%
(c) Find the probability that Chuck is late for school.
0.3(0.8) + 0.7(0.1) = 0.31 or 31%
6) Amos travels to school either by car or by bicycle. The probability of being late for school is
1
if he travels by car
10
1
if he travels by bicycle. On any particular day he is equally likely to travel by car or by bicycle.
5
(a) Draw a probability tree diagram to illustrate this information.
and
1
10
1
2
9
10
1
5
1
2
4
5
(b)
(c)
Find the probability that
(i) Amos will travel by car and be late.
1 1
1
( )=
2 10
20
(ii) Amos will be late for school.
1 1
1 1
3
( ) + 2 (5) = 20
2 10
Given that Amos is late for school, find the probability that he travelled by bicycle.
1 1
1
3
20
2
( ) = 10 ÷ 20 = 30 = 3
2 5
7) In a class of 40 students, 19 play tennis, 20 play netball, and 8 play neither of these sports. A student is
randomly chosen from the class. Determine the probability that the student: (Hint: Venn diagram)
a) Plays tennis
19
40
= 0.475
b) Does not play netball
20
40
1
= = 0.5
2
c) Plays at least one of the sports
32
40
4
= = 0.8
5
d) Plays one and only one of the sports
25
40
5
= = 0.625
8
e) Plays netball, but not tennis
13
40
= 0.325
f) Plays tennis knowing he/she plays netball
7
20
= 0.35
2nd Nine Weeks Study Guide
Math Studies
Answer Key
8) In a certain town 3 newspapers are published. 20% of the population read A, 16% read B, 14% read C, 8%
read A and B, 5% read A and C, 4% read B and C and 2% read all 3 newspapers. A person is selected at
random. Determine the probability that the person reads: (Hint: Venn diagram)
a) None of the papers
65
100
=
13
20
= 65%
b) At least one of the papers
35
100
=
7
20
= 35%
c) Exactly one of the papers
22
100
=
11
50
= 22%
d) Either A or B
28
100
=
7
25
= 28%
e) A, given that the person read at least one paper
20
35
4
= = 57.1%
7
9) If P(A) = 0.4, P(B) = 0.3, and P(A U B) = 0.5, find:
a) P(A ∩ B)
P(AUB) = P(A)+P(B)-P(A∩B)
0.5 = 0.4 + 0.3 – x
0.5 = 0.7 – x
-0.2 = -x
0.2
b) P(A│B)
P(A∩B) = 0.2 = 0.667
P(B)
0.3
10) Construct a truth table for the following and state whether the statement is a tautology, a logical contradiction or
neither: (you must show your truth tables)
b)¬(p⋁q) ⋁ (p⋁q)
a) (p⋁q)⋀(¬p⋀¬q)
p⋁q
¬q
¬p⋀¬q
T
T
T
F
F
T
F
T
F
F
F
T
¬(p⋁q)
(p⋁q)⋀(
¬p⋀¬q)
F
F
F
F
 p  q  q
¬q
F
T
F
T
(p⋁¬q)
T
T
F
T
q
T
F
T
F
T
T
F
T
tautology
11) Construct truth tables for the following:
a)
p⋁¬q
¬(p⋁q)
⋁ (p⋁q)
T
T
T
T
F
F
F
T
logical contradiction
c)(p⋁¬q)⋀p
 p  q  q
T
F
T
F
neither
b) p  q
¬p
F
F
T
T
q
T
F
T
F
p  q
F
T
T
F
(p⋁¬q)
⋀p
T
T
F
F
Math Studies
2nd Nine Weeks Study Guide
Answer Key
12) Write the converse, inverse, and contrapositive for each proposition (make sure you label each one).
a) If I love swimming, then I live near the sea.
Converse – If I live near the sea, then I love swimming.
Inverse – If I do not love swimming, then I do not live near the sea.
Contrapositive – If I do not live near the sea, then I do not love swimming.
b) If I like food, then I eat a lot.
Converse – If I eat a lot, then I like food.
Inverse – If I do not like food, then I do not eat a lot.
Contrapositive – If I do not eat a lot, then I do not like food.