Download NON-CALCULATOR REVIEW SET 23A

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DESCRIPTIVE STATISTICS (Chapter 23)
REVIEW SET 23A
NON-CALCULATOR
1 The data supplied below is the diameter (in cm) of a number of bacteria colonies as measured
by a microbiologist 12 hours after seeding.
0:4 2:1 3:4 3:9 4:7 3:7 0:8 3:6 4:1 4:9 2:5 3:1 1:5 2:6 4:0
1:3 3:5 0:9 1:5 4:2 3:5 2:1 3:0 1:7 3:6 2:8 3:7 2:8 3:2 3:3
a Find the
i median
ii range of the data.
b Group the data in 5 groups and display it using a frequency histogram.
c Comment on the skewness of the data.
2 The data set 4, 6, 9, a, 3, b has a mean and mode of 6. Find the values of a and b given
a > b.
Girls
frequency
6
5
4
3
2
1
0
32 33 34 35 36 37 38 39 40 41
time (s)
Boys
frequency
6
5
4
3
2
1
0
32 33 34 35 36 37 38 39 40 41
time (s)
3 The histograms alongside show the times for the
100 metre freestyle recorded by members of a
swimming squad.
a Copy and complete:
Distribution
Girls
Boys
shape
median
mean
modal class
b Discuss the distributions of times for the boys
and girls. What conclusion can you make?
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4 Find k given that 2, 5, k, k, 3, k, 7, and 4 have a mean of 6.
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DESCRIPTIVE STATISTICS (Chapter 23)
5 Consider the column graph alongside.
12
10
8
6
4
2
0
a Find the:
i mode
ii median
iii mean of the data.
b Describe the distribution of the data.
729
frequency
0
1
6 The table alongside shows the number of
appointments at a dentist surgery over 54 days.
2
3
4
5
6
7
Number of appointments
2
3
4
5
6
7
8
9
a Draw a column graph to display the data.
b Describe the distribution of the data.
c Find the:
i mode
ii mean of the data.
8
9
10 x
Frequency
1
2
5
7
11
15
9
4
7 Suppose a, b, c, d, and e have a mean of 8. Find the mean of 10 ¡ a, 10 ¡ b, 20 ¡ c,
20 ¡ d, and 50 ¡ e.
8 The data set f12, 13, 8, 10, 14, 7, a, bg has mean of 10, and a variance of 8:5 . Find a and b
given that a < b.
9 Consider the data set f8, 11, 12, 9, ag.
a Find the mean of the data set in terms of a.
b Given that the variance of the data set is 6, find the possible values of a.
10 Prove that the variance of any set of five consecutive integers is 2.
REVIEW SET 23B
CALCULATOR
1 The data below shows the
71:2 65:1
84:3 77:0
90:5 85:5
distance in metres that Thabiso threw a baseball:
68:0 71:1 74:6 68:8 83:2 85:0 74:5 87:4
82:8 84:4 80:6 75:9 89:7 83:2 97:5 82:9
90:7 92:9 95:6 85:5 64:6 73:9 80:0 86:5
a Determine the highest and lowest value for the data set.
b Choose between 6 and 12 groups into which all the data values can be placed.
c Prepare a frequency distribution table.
d Draw a frequency histogram for the data.
e Determine the:
i mean
ii median.
2 Consider the data set:
k ¡ 2, k, k + 3, k + 3.
a Show that the mean of the data set is equal to k + 1.
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b Suppose each number in the data set is increased by 2. Find the new mean of the data set
in terms of k.
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3 Consider the following distribution of continuous grouped data:
Scores
0 6 x < 10
10 6 x < 20
20 6 x < 30
30 6 x < 40
40 6 x < 50
Frequency
1
13
27
17
2
Estimate the mean and standard deviation of the data.
4 The number of litres of petrol purchased by a group
of motor vehicle drivers is shown alongside.
Estimate the mean and standard deviation of the
number of litres purchased.
5 The table alongside shows the number of
matches in a sample of boxes.
Number of vehicles
Litres
15 6 l < 20
5
20 6 l < 25
13
25 6 l < 30
17
30 6 l < 35
29
35 6 l < 40
27
40 6 l < 45
18
45 6 l < 50
7
Number
Frequency
47
21
48
29
49
35
50
42
51
18
a Find the mean and standard deviation for
this data.
b Does this result justify a claim that the average number of matches per box is 50?
52
31
6 Katie loves cats. She visits every house in her street
to find out how many cats live there. The responses
are given below:
Number of cats
0
1
2
3
4
5
Frequency
36
9
11
5
1
1
a Draw a graph to display this data.
b Describe the distribution.
c Find the:
i mode
ii mean
iii median.
d Which of the measures of centre is most appropriate for this data? Explain your answer.
7 The table shows the length of time cars spent in a particular
parking lot in one day.
a How many cars parked in the parking lot?
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b Estimate the mean and standard deviation of the data.
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Time (t hours)
0<t61
1<t62
2<t63
3<t64
4<t65
5<t66
Frequency
32
85
123
97
62
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DESCRIPTIVE STATISTICS (Chapter 23)
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8 Consider the data set: 42 58 74 62 51 45 73 54 66 84.
Find the:
a mean
b median
c standard deviation
9 Friends Kevin and Felicity each completed a set of 20 crossword puzzles. The time taken, in
minutes, to complete each puzzle is shown below.
37
49
39
48
53
37
42
33
Kevin
47 33
48 32
34 29
56 39
39
36
52
41
Felicity
36 41 26
49 57 39
25 34 27
34 35 50
33
38
48
38
52
44
53
31
a Find the mean of each data set.
b Find the standard deviation of each data set.
c Who generally solved the puzzles faster?
d Who was more consistent?
20
P
10 A set of 20 data values fx1 , x2 , ...., x20 g has
variance of the data set.
xi2 = 2872 and ¹ = 11. Find the
i=1
REVIEW SET 23C
1 The winning margin in 100 basketball games was recorded.
The results are given alongside:
Margin (points)
Frequency
1 - 10
11 - 20
21 - 30
31 - 40
41 - 50
13
35
27
18
7
a Is the winning margin discrete or continuous?
b Draw an appropriate graph to represent this
information.
c Can you calculate the mean winning margin exactly?
Explain your answer.
2 The following distribution has a mean score of 5:7:
Score
a Find the value of x.
b Hence find the variance of the distribution.
Frequency
3 The table alongside shows the number of customers
visiting a supermarket on various days.
Estimate the mean number of customers per day.
2
3
5
2
Number of customers
250
300
350
400
450
500
550
-
299
349
399
449
499
549
599
x
4
x+6
1
Frequency
14
34
68
72
54
23
7
4 The data set 33, 18, 25, 40, 36, 41, m, n has a mode of 36 and a mean of 32. Find m and
n given that m < n.
5 Consider the data set fm, m + 4, m ¡ 2, m + 1, m + 6, m ¡ 3g.
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a Find, in terms of m, the median and mean of the data set.
b Find the variance of the data set.
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DESCRIPTIVE STATISTICS (Chapter 23)
6 The weekly supermarket bills for a number of
families was observed and recorded in the table
given.
Estimate the mean bill and the standard deviation of
the bills.
Bill (E)
70 - 79:99
80 - 89:99
90 - 99:99
100 - 109:99
110 - 119:99
120 - 129:99
130 - 139:99
140 - 149:99
Number of families
27
32
48
25
37
21
18
7
7 Pratik is a quality control officer for a biscuit company. He needs to check that 250 g of biscuits
go into each packet, but realises that the weight in each packet will vary slightly.
a Would you expect the standard deviation for the whole population to be the same for one
day as it is for one week? Explain your answer.
b If a sample of 100 packets is measured each day, what measure would be used to check:
i that an average of 250 g of biscuits goes into each packet
ii the variability of the mass going into each packet?
c Explain the significance of a low standard deviation in this case.
8 A group of students were asked how long they slept for last
night. The results are shown in the table.
Estimate the mean and standard deviation of the data.
Time (t hours)
Frequency
66t<7
76t<8
86t<9
9 6 t < 10
10 6 t < 11
5
19
38
22
6
9 Roger and Clinton play golf together every Saturday. Their scores for the past 30 weeks are
given below.
Roger
Clinton
78 74 82 85 79 73
77 79 82 84 75 75
92 79 88 77 85 87
73 78 71 83 85 72
82 96 90 80 82 88
77 76 78 75 74 81
74 89 93 91 85 78
80 77 74 71 72 75
75 94 79 94 90 85
79 81 72 78 76 73
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a Find the mean and standard deviation of each data set.
b Which player generally has the lower score?
c Which player has the greater variation in their scores?
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PROBABILITY (Chapter 24)
REVIEW SET 24A
NON-CALCULATOR
1 List the different orders in which 4 people A, B, C, and D could line up. If they line up at
random, determine the probability that:
a A is next to C
b there is exactly one person between A and C.
2 Given P(A) = m is the probability of event A occurring in any given trial:
a Write P(A0 ) in terms of m.
b State the range of possible values of m.
c Suppose two trials are performed independently. Find, in terms of m, the probability of A
occurring:
i exactly once
ii at least once.
3 A coin is tossed and a square spinner labelled A, B, C, D, is twirled. Determine the probability
of obtaining:
a a head and consonant
b a tail and C
c a tail or a vowel or both.
4 The probability that a man will be alive in 25 years is 35 , and the probability that his wife will
be alive is 23 . Determine the probability that in 25 years:
a both will be alive
b at least one will be alive
c only the wife will be alive.
5 Given P(Y ) = 0:35 and P(X [ Y ) = 0:8, and that X and Y are mutually exclusive events,
find:
a P(X \ Y )
b P(X)
c the probability that X occurs or Y occurs, but not both X and Y .
6 What is meant by:
a independent events
b mutually exclusive events?
7 Graph the sample space of all possible outcomes when a pair of dice is rolled. Hence determine
the probability of getting:
a a sum of 7 or 11
b a sum of at least 8.
8 In a group of 40 students, 22 study Economics, 25 study Law, and 3 study neither of these
subjects. Determine the probability that a randomly chosen student studies:
a both Economics and Law
b at least one of these subjects
c Economics given that he or she studies Law.
9 The probability that a particular salesman will leave his sunglasses behind in any store is 15 .
Suppose the salesman visits two stores in succession and leaves his sunglasses behind in one of
them. What is the probability that the salesman left his sunglasses in the first store?
10 Each time Mae and Ravi play chess, Mae has probability
determine the probability that:
a Mae wins 3 of the games
4
5
of winning. If they play 5 games,
b Mae wins either 4 or 5 of the games.
11 Suppose P(X 0 j Y ) = 23 , P(Y ) = 56 , and X 0 \ Y 0 = ?. Find P(X).
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12 The diagram alongside shows an electrical circuit with
switches. The probability that any switch is open is 13 .
Determine the probability that the current flows from
A to B.
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PROBABILITY (Chapter 24)
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13 One letter is randomly selected from each of the names JONES, PETERS, and EVANS.
Find the probability that:
a the three letters are the same
b exactly two of the letters are the same.
REVIEW SET 24B
CALCULATOR
1 Niklas and Rolf play tennis with the winner being the first to win two sets. Niklas has a 40%
chance of beating Rolf in any set. Draw a tree diagram showing the possible outcomes and
hence determine the probability that Niklas will win the match.
2 If I buy 4 tickets in a 500 ticket lottery, and the prizes are drawn without replacement, determine
the probability that I will win:
a the first 3 prizes
b at least one of the first 3 prizes.
3 The students in a school are all vaccinated against
measles. 48% of the students are males, of whom
16% have an allergic reaction to the vaccine. 35% of
the girls also have an allergic reaction. A student is
randomly chosen from the school. Find the probability
that the student:
a has an allergic reaction
b is female given that a reaction occurs.
4 On any one day it could rain with 25% chance and be windy with 36% chance.
a Draw a tree diagram showing the possibilities with regard to wind and rain on a particular
day.
b Hence determine the probability that on a particular day there will be:
i rain and wind
ii rain or wind or both.
c What assumption have you made in your answers?
5 A, B, and C have 10%, 20%, and 30% chance of independently solving a certain maths problem.
If they all try independently of one another, what is the probability that at least one of them will
solve the problem?
6 Two events are defined such that P(A) = 0:11 and P(B) = 0:7 . n(B) = 14.
i P(A0 )
a Calculate:
ii n(U )
i P(A \ B)
b If A and B are independent events, find:
ii P(A j B)
c If instead, A and B are mutually exclusive events, find P(A [ B).
7 Let C be the event that “a person has a cat” and D be the event that “a person has a dog”.
P(C) = 37 , P(D j C 0 ) = 25 , and P(D0 j C) = 34 .
a Copy and complete the tree diagram by marking a
probability on each branch.
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b If a person is chosen at random, find the probability
that the person has:
i a cat and a dog
ii at least one pet (cat or dog).
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PROBABILITY (Chapter 24)
8 A survey of 200 people included 90 females. It found that 60 people smoked, 40 of whom were
male.
a Use the given information to complete the
Female Male Total
table:
Smoker
b A person is selected at random. Find the
Non-smoker
probability that this person is:
Total
i a female non-smoker
ii a male given the person was a non-smoker.
c If two people from the survey are selected at random, calculate the probability that:
i both of them are non-smoking females
ii one is a smoker and the other is a non-smoker.
9 In a certain class, 91% of the students passed Mathematics and 88% of the students passed
Chemistry. 85% of students passed both Mathematics and Chemistry.
a Show that the events of passing Mathematics and passing Chemistry are not independent.
b A randomly selected student passed Chemistry. Find the probability that this student did
not pass Mathematics.
10 A group of ten students includes three from Year 12 and four from Year 11. The principal calls
a meeting with five students randomly selected from the group. Calculate the probability that
exactly two Year 12 and two Year 11 students are called to the meeting.
11 A person with a university degree has a 0:33 chance of getting an executive position. A person
without a university degree has a 0:17 chance of getting an executive position. If 78% of
all applicants for an executive position have a university degree, find the probability that the
successful applicant does not have one.
12 A team of five is randomly chosen from six doctors and four dentists. Determine the probability
that it consists of:
a all doctors
b at least two doctors.
13 With each pregnancy, a particular woman will give birth to either a single baby or twins. There
is a 15% chance of having twins during each pregnancy. Suppose that after 2 pregnancies she
has given birth to 3 children. Find the probability that she had twins first.
14 Four different numbers are randomly chosen from the set S = f1, 2, 3, 4, 5, ...., 10g.
X is the second largest of the numbers selected.
Determine the probability that X is:
a 2
b 7
c 9.
REVIEW SET 24C
1 Systematically list the possible sexes of a 4-child family. Hence determine the probability that
a randomly selected 4-child family has two children of each sex.
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2 A bag contains 3 red, 4 yellow and 5 blue marbles. Two marbles are randomly selected from
the bag without replacement. What is the probability that:
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PROBABILITY (Chapter 24)
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3 A class contains 25 students. 13 play tennis, 14 play volleyball, and 1 plays neither of these
sports.
a A student is randomly selected from the class. Determine the probability that the student:
i plays both tennis and volleyball
ii plays at least one of these sports
iii plays volleyball given that he or she does not play tennis.
b Three students are randomly selected from the class. Determine the probability that:
i none of these students play tennis
ii at least one of these students plays tennis.
4 An urn contains three red balls and six blue balls.
a A ball is drawn at random and found to be blue. What is the probability that a second
draw with no replacement will also produce a blue ball?
b Two balls are drawn without replacement and the second is found to be red. What is the
probability that the first ball was also red?
c Based on the toss of a coin, either a red ball or a blue ball is added to the urn. A ball is
then drawn at random and found to be blue. Find the probability the added ball was red.
5 A and B are independent events where P(A) = 0:8 and P(B) = 0:65 .
d P(B j A).
Determine:
a P(A [ B)
b P(A j B)
c P(A0 j B 0 )
6 A school photocopier has a 95% chance of working on any particular day. Find the probability
that it will be working on at least one of the next two days.
7 Jon goes cycling on three random mornings of each week. When he goes cycling he has eggs
for breakfast 70% of the time. When he does not go cycling he has eggs for breakfast 25% of
the time. Determine the probability that Jon:
a has eggs for breakfast
b goes cycling given that he has eggs for breakfast.
8 A survey of 50 men and 50 women was conducted to see how many people prefer coffee or
tea. It was found that 15 men and 24 women prefer tea.
a Display this information in a two-way table.
C
b Let C represent the people who prefer coffee and
M represent the men. Hence complete the Venn
diagram.
i P(C 0 )
c Calculate:
ii P(M j C)
M
U
9 T and M are events such that n(U ) = 30, n(T ) = 10, n(M ) = 17, and n((T [ M )0 ) = 5.
a Draw a Venn diagram to display this information.
i P(T \ M )
b Hence find:
ii P((T \ M ) j M )
10 Answer the questions in the Opening Problem on page 734.
11 The independent probabilities that 3 components of a TV set will need replacing within one
1
1
1
, 50
, and 100
respectively. Calculate the probability that there will need to be a
year are 20
replacement of:
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PROBABILITY (Chapter 24)
12 When Peter plays John at tennis, the probability that Peter wins his service game is p and the
probability that John wins his service game is q, where p > q and p + q > 1. Which is more
likely:
A Peter will win at least two consecutive games out of 3 when he serves first
B Peter will win at least two consecutive games out of 3 when John serves first?
13 Two different numbers were chosen at random from the digits 1 to 9 inclusive. It was observed
that their sum was even. Determine the probability that both numbers were odd.
14 Using a 52 card pack, a ‘royal flush’ consists of the 10, J, Q, K, A of one suit. Find the
probability of dealing:
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