1.
One thousand candidates sit an examination. The distribution of marks is shown in the following grouped frequency table. Marks 1–10 Number of 15 candidates (a)
11–20
21–30
31–40
41–50
51–60
100
170
260
220
50
61–70
71–80
90
81–90
45
91–100
30
20
Copy and complete the following table, which presents the above data as a cumulative frequency distribution. (3)
Mark Number of candidates (b)
≤
10
15
≤
20
≤
30
≤
40
≤
50
≤
60
65
≤
70
≤
80
≤
90
≤
100
905
Draw a cumulati cumulative ve freque frequency ncy graph graph of the the distribu distribution, tion, using using a scale scale of 1 cm for for 100 candidates on the vertical axis and 1 cm for 10 marks on the horizontal axis. (5)
(c) (c)
Use Use your your gra graph ph to to answ answer er par parts ts (i) (i)–( –(ii iii) i) bel below ow,, (i) (i)
Find Find an esti estima mate te for for the the med median ian sco score re.. (2)
(ii)
Candidate Candidatess who who score scored d less less than than 35 were were required required to retake retake the examinatio examination. n. How many candidates had to retake? (3)
(iii)
The highest-s highest-scorin coring g 15% 15% of candidates candidates were awarded awarded a distinctio distinction. n. Find the mark above which a distinction was awarded. (3) (Total 16 marks)
2.
At a conference of 100 mathematicians there are 72 men and 28 women. The men have a mean height of 1.79 m and the women have a mean height of 1.62 m. Find the mean height of the 100 mathematicians. Working:
Answers:
....…………………………………….......... (Total 4 marks)
1
25
∑ x 3.
The mean of the population x1, x2, ........ , x25 is m. Given that
i =1
i
= 300 and
25
∑ ( x – m)
2
i
= 625, find
i =1
(a)
the value of m;
(b) (b)
the the stan standa dard rd dev devia iati tion on of of the the popu popula latio tion. n.
Working:
Answers:
(a) (a) ………… ……………… ………… ………… ………… ………… ………… …….. .. (b) .......... ............... .......... ........... ........... .......... .......... .......... .......... .......... .......... .......... ..... (Total 4 marks)
4.
A supermarket records the amount of money d spent by customers in their store during a busy period. The results are as follows:
(a)
Money in $ ( d )
0–20
20–40
40–60
60–80
80–100
100–120
120–140
Number of customers ( n)
24
16
22
40
18
10
4
Find an an estimate estimate for the the mean mean amount amount of money money spent spent by the custo customers, mers, giving giving your your answer to the nearest dollar ($). (2)
(b)
Copy and complete complete the the followin following g cumulat cumulative ive frequen frequency cy table table and use use it it to draw draw a cumulative frequency graph. Use a scale of 2 cm to represent $ 20 on the horizontal axis, and 2 cm to represent 20 customers on the vertical axis. (5)
Money in $ ( d )
<20
<40
Number of customers ( n)
24
40
< 60
<80
< 100
< 120
< 140
2
(c)
The time t (minutes), spent by customers in the store may be represented by the equation 2 3 t = 2d + 3.
(i)
Use this this equati equation on and and your your answe answerr to part part (a) (a) to esti estimat matee the mean mean time time in minu minutes tes spent by customers in the store. (3)
(ii)
Use the equat equation ion and and the the cumulativ cumulativee frequenc frequency y graph graph to to estimat estimatee the number number of customers who spent more than 37 minutes in the store. (5) (Total 15 marks)
5.
The table shows the scores of competitors in a competition. Score
10
20
30
40
50
Number of competitors with this score
1
2
5
k
3
The mean score is 34. Find the value of k. Working:
Answers:
....…………………………………….......... (Total 4 marks)
3
6.
A survey is carried out to find the waiting times for 100 customers at a supermarket.
(a)
waiting time (seconds)
number of customers
0–30
5
30– 60
15
60– 90
33
90 –120
21
120–150
11
150–180
7
180–210
5
210–240
3
Calculate Calculate an estima estimate te for for the the mean mean of the waiting waiting times, times, by using using an appropria appropriate te approximation to represent each interval. (2)
(b)
Constr Construct uct a cumul cumulati ative ve freq frequen uency cy tabl tablee for for these these data. data. (1)
(c)
Use the the cumulati cumulative ve frequ frequency ency table to draw, draw, on graph paper, paper, a cumulative cumulative frequency frequency graph, using a scale of 1 cm per 20 seconds waiting time for the horizontal axis and 1 cm per 10 customers for the vertical axis. (4)
(d)
Use the the cumulati cumulative ve frequen frequency cy graph graph to find find estimates estimates for the median median and the the lower lower and upper quartiles. (3) (Total 10 marks)
7.
The following diagram represents the lengths, in cm, of 80 plants grown in a laboratory.
f r e q
2
0
1
5
u 1
e n 0
c y
5 0
0
1
0
2
0
3
0
4
0 5 0 6 l e n g t h
0 7 0 ( c m
8 )
0
9
0 1
0
0
4
(a) (a)
How How man many y pla plant ntss hav havee len lengt gths hs in cm bet betwe ween en (i )
50 and 60?
(ii)
70 and 90? (2)
(b)
Calculate Calculate estima estimates tes for for the mean mean and and the standard standard deviation deviation of of the length lengthss of the the plants. plants. (4)
(c)
Explain Explain what what feature feature of the the diagram diagram suggests suggests that the median median is differ different ent from from the mean. mean. (1)
(d)
The foll followi owing ng is an extr extract act from from the the cumul cumulativ ativee freque frequency ncy table. table. length in cm
cumulative
less than
frequency
.
.
50
22
60
32
70
48
80
62
.
.
Use the information in the table to estimate the median. Give your answer to two significant figures. (3) (Total 10 marks)
5
8.
Given the following frequency distribution, find (a)
the median;
(b)
the mean. Number ( x)
1
2
3
4
5
6
Frequency ( f )
5
9
16
18
20
7
Working:
Answers:
(a) (a) ………… ……………… ………… ………… ………… ………… ………… …….. .. (b) .......... ............... .......... ........... ........... .......... .......... .......... .......... .......... .......... .......... ..... (Total 4 marks)
9.
The table below represents the weights, W , in grams, of 80 packets of roasted peanuts. Weight (W )
80 < W ≤ 85
85 < W ≤ 90
90 < W ≤ 95
95 < W ≤ 100
100 < W ≤ 105
105 < W ≤ 110
Number of packets
5
10
15
26
13
7
(a)
110 < W ≤ 115
4
Use the the midpoin midpointt of each interval interval to find an an estimat estimatee for the stand standard ard deviati deviation on of of the weights. (3)
(b)
Copy and complete complete the following following cumulative cumulative frequency frequency table for the above above data. data.
Weight (W )
W ≤ 85
W ≤ 90
Number of packets
5
15
W ≤ 95
W ≤ 100
W ≤ 105
W ≤ 110
W ≤ 115
80 (1)
6
(c)
N p
A cumulati cumulative ve frequen frequency cy graph graph of of the distributi distribution on is shown shown below, below, with with a scale 2 cm for 10 packets on the vertical axis and 2 cm for 5 grams on the horizontal axis.
8
0
7
0
6
0
5
0
u
m
o f a c k 4
0
3
0
2
0
1
0
8
b
e
e
t s
r
0
8
5
9
0
9
5 W
1 e i g
0 h
0 t
( g
1 0 5 r a m
1 1
0
1
1
5
s )
Use the graph to estimate (i )
the median;
(ii) (ii)
the upp upper er quar quartil tilee (tha (thatt is, is, the the third third quarti quartile) le)..
Give your answers to the nearest gram. (4)
(d)
Let W , W , ..., W be the individual weights of the packets, and let 1
2
80
W
be their mean.
What is the value of the sum (W 1 – W ) + (W 2 – W ) + (W 3 – W ) + ... + (W 79 – W ) + (W 80 – W ) ? (2)
7
(e)
One of of the 80 80 packet packetss is selec selected ted at at random random.. Given Given that that its weigh weightt satisfi satisfies es 85 < W ≤ 110 , find the probability that its weight is greater than 100 grams. (4) (Total 14 marks)
10.
–1
The speeds in km h
(a)
of cars passing a point on a highway are recorded in the following table. Speed v
Number of cars
v ≤ 60
0
60 < v ≤ 70
7
70 < v ≤ 80
25
80 < v ≤ 90
63
90 < v ≤ 100
70
100 < v ≤ 110
71
110 < v ≤ 120
39
120 < v ≤ 130
20
130 < v ≤ 140
5
v > 140
0
Calcul Calculate ate an esti estimat matee of of the the mean mean speed speed of the cars. cars. (2)
(b)
The follow following ing table table gives some of the the cumulat cumulative ive frequ frequencie enciess for for the informati information on above. above. Speed v
Cumulative frequency
v ≤ 60
0
v ≤ 70
7
v ≤ 80
32
v ≤ 90
95
v ≤ 100
a
v ≤ 110
236
v ≤ 120
b
v ≤ 130
295
v ≤ 140
300
(i )
Write do down the va value lues of a and b.
(ii) (ii)
On graph graph paper, paper, constr construc uctt a cumula cumulativ tivee freq frequen uency cy curve to represent this –1
information. Use a scale of 1 cm for 10 km h 1 cm for 20 cars on the vertical axis.
on the horizontal axis and a scale of (5)
8
(c)
Use Use you yourr gra graph ph to deter eterm mine ine (i)
the percen percentag tagee of cars cars trav travell elling ing at a speed speed in in exces excesss of of 105 105 km km h –1;
(ii) (ii)
the speed speed which which is exceed exceeded ed by 15% of the cars. cars. (4) (Total 11 marks)
11.
From January to September, the mean number of car accidents per month was 630. From October to December, the mean was 810 accidents per month. What was the mean number of car accidents per month for the whole year? Working:
Answer :
………………………………………….. (Total 6 marks)
9
12.
A taxi company has 200 taxi cabs. The cumulative frequency curve below shows the fares in dollars ($) taken by the cabs on a particular morning. 2
0
0
1
8
0
1
6
0
1
4
0
1
2
0
1
0
0
s b a c f
o r e b
m u N
8
0
6
0
4
0
2
0
1
0
2
0
3
0
4
0 F
(a)
a
5 r e s
0
6 ( $
0
7
0
8
)
Use Use the the curv curvee to to est estim imat atee (i )
the median fare;
(ii) (ii)
the numbe numberr of cabs cabs in which which the the fare fare taken taken is is $35 $35 or less. less. (2)
10
The company charges 55 cents per kilometre for distance travelled. There are no other charges. Use the curve to answer the following. (b)
On that that mornin morning, g, 40% 40% of of the the cabs cabs travel travel less less than than a km. Find the value of a. (4)
(c)
What What percen percentag tagee of the the cabs cabs trave travell more more than than 90 km on on that that morni morning? ng? (4) (Total 10 marks)
13.
Three positive integers a, b, and c, where a < b < c, are such that their median is 11, their mean is 9 and their range is 10. Find the value of a. Working:
Answer :
………………………………………….. (Total 6 marks)
14.
In a suburb of a large city, 100 houses were sold in a three-month period. The following cumulative frequency table shows the distribution of selling prices (in thousands of dollars). Selling price P P ≤ 100 ($ 1000) Total number of houses (a)
12
P ≤ 200
P ≤ 300
P ≤ 400
P ≤ 500
58
87
94
100
Repres Represent ent this this info informa rmatio tion n on on a cumul cumulati ative ve frequ frequenc ency y curve, using a scale of 1 cm to represent $ 50000 on the horizontal axis and 1 cm to represent 5 houses on the vertical axis. (4)
(b)
Use you yourr curv curvee to to find find the interq interquar uartil tilee rang range. e. (3)
11
The information above is represented in the following frequency distribution. Selling price P 0 < P ≤ 100 100 < P ≤ 200 200 < P ≤ 300 300 < P ≤ 400 400 < P ≤ 500 ($ 1000) Number of houses (c)
12
46
29
a
b
Find the value of a and of b. (2)
(d)
Use mid-in mid-interv terval al values values to calcul calculate ate an an estimat estimatee for for the the mean mean selling selling price. price. (2)
(e)
Houses Houses which which sell sell for more more than than $ 350 000 are descri described bed as De Luxe. (i) (i)
Use Use you yourr gra graph ph to esti estima mate te the the num numbe berr of of De De Luxe houses sold. Give your answer to the nearest integer.
(ii)
Two De Luxe houses are selected at random. Find the probability that both have a selling price of more than $ 400 000. (4) (Total 15 marks)
12
15.
The number of hours of sleep of 21 students are shown in the frequency table below Hours of sl sleep
Number of st students
4
2
5
5
6
4
7
3
8
4
10
2
12
1
Find (a)
the median;
(b)
the the lo lower quartile tile;;
(c)
the the in interq terqua uart rtil ilee ran rang ge.
Working:
Answers:
(a) (a) ………… ……………… ………… ………… ………… ………… ………… …….. .. (b) (b) ………… ……………… ………… ………… ………… ………… ………… …….. .. (c) .......... ............... .......... ........... ........... .......... .......... .......... .......... .......... .......... .......... ..... (Total 6 marks)
13
16.
A student measured the diameters of 80 snail shells. His results are shown in the following cumulative frequency graph. The lower quartile (LQ) is 14 mm and is marked clearly on the graph.
y c n e u q e r f
9
0
8
0
7
0
6
0
5
0
4
0
3
0
2
0
1
0
e v i t a l u m u C
0
5
0
1
0 L
(a)
(b) (b)
1 Q
5
2 =
D
0 1
2 4
i a m
5 e t e r
3
0
3
( m
m
5
4
0
4
)
On the the graph graph,, mark mark clearl clearly y in the the same same way way and and write write down down the the value value of of (i )
the median;
(ii) (ii)
the uppe pper quart artile ile.
Write Write down down the the int inter erqu quar arti tile le rang range. e.
Working:
Answer :
(b) (b) ………… ……………… ………… ………… ………… ………… ………… …….. .. (Total 6 marks)
14
17.
The cumulative frequency curve below shows the marks obtained in an examination by a group of 200 students.
N
u o
s t u
m f d
2
0
0
1
9
0
1
8
0
1
7
0
1
6
0
1
5
0
1
4
0
b
e r 1 3 0 e n t s 1
2
0
1
1
0
1
0
0
9
0
8
0
7
0
6
0
5
0
4
0
3
0
2
0
1
0 0
1
0
2
0
3
0
4 M
0
5
0 a r k
6
0 7 0 8 0 9 0 1 o b t a i n e d
0
15
(a)
Use the the cumul cumulati ative ve frequ frequenc ency y curve curve to comp complete lete the the frequ frequenc ency y table table below below.. Mark ( x x) Number of students
(b)
0 ≤ x < 20
20 ≤ x < 40
40 ≤ x < 60
60 ≤ x < 80
22
80 ≤ x < 100 20
Forty Forty perc percent ent of the the stud student entss fail. fail. Find Find the the pass pass mark mark..
Working:
Answer:
(b) (b) ………… ……………… ………… ………… ………… ………… ………… …….. .. (Total 6 marks)
18.
10
When the expression (2 + ax) value of a.
3
is expanded, the coefficient of the term in x is 414 720. Find the
Working:
Answer :
………………………………………….. (Total 6 marks)
16
19.
The cumulative frequency curve below shows the heights of 120 basketball players in centimetres.
N
u
m
b
e
1
2
0
1
1
0
1
0
0
9
0
8
0
7
0
6 r
0 o
5
0
4
0
3
0
2
0
1
0
f
p
l a y
e r s
0 1
6
01
6
51
7 H
01
7 e
51
8
i g
h
01 t
8 i n
51
9
01
c
e n
9
52
0
t i m
17
Use the curve to estimate (a)
the the median ian height;
(b)
the the inte interq rqua uart rtil ilee ran rang ge.
Working:
Answers:
(a) (a) ………… ……………… ………… ………… ………… ………… ………… …….. .. (b) (b) ………… ……………… ………… ………… ………… ………… ………… …….. .. (Total 6 marks)
20.
Let a, b, c and d be integers such that a < b, b < c and c = d . The mode of these four numbers is 11. The range of these four numbers is 8. The mean of these four numbers is 8. Calculate the value of each of the integers a, b, c, d . Working:
Answers:
............................., b = ............................. ............................. a = ............................., c = ............................., ............................., d = ............................. ............................. (Total 6 marks)
18
21.
A test marked out of 100 is written by 800 students. The cumulative frequency graph for for the marks is given below.
N u m o f c a n d
8
0
0
7
0
0
6
0
0
b
e r 5 0 0 i d a t e s 4
0
0
3
0
0
2
0
0
1
0
0
1
0
2
0
3
0
4
0
5 M
0
6
0
7
0
8
0
9
0 1
0
0
a r k
19
(a)
Write Write down down the the number number of of studen students ts who who scored scored 40 40 marks marks or or less less on the the test. test.
(b)
The middle middle 50 % of of test test result resultss lie lie betw between een marks marks a and b, where a < b. Find a and b. ................................................ ...................................................................... .............................................. .............................................. ....................................... ................. ................................................ ...................................................................... .............................................. .............................................. ....................................... ................. ................................................ ...................................................................... .............................................. .............................................. ....................................... ................. ................................................ ...................................................................... .............................................. .............................................. ....................................... ................. ................................................ ...................................................................... .............................................. .............................................. ....................................... ................. ................................................ ...................................................................... .............................................. .............................................. ....................................... ................. ................................................ ...................................................................... .............................................. .............................................. ....................................... ................. ................................................ ...................................................................... .............................................. .............................................. ....................................... ................. ................................................ ...................................................................... .............................................. .............................................. ....................................... ................. ................................................ ...................................................................... .............................................. .............................................. ....................................... ................. ................................................ ...................................................................... .............................................. .............................................. ....................................... ................. ................................................ ...................................................................... .............................................. .............................................. ....................................... ................. (Total 6 marks)
22.
The 45 students in a class each recorded the number of whole minutes, x, spent doing experiments on Monday. The results are (a)
x = 2230.
Find the mean number number of of minutes minutes the students students spent spent doing doing exper experiments iments on Monda Monday. y.
20
Two new students joined the class and reported that they spent 37 minutes and 30 minutes respectively. (b)
Calcul Calculate ate the new mean mean incl includ uding ing these these two studen students. ts.
Working:
Answers:
(a) (a) ………… ……………… ………… ………… ………… ………… ………… …….. .. (b) .......... ............... .......... ........... ........... .......... .......... .......... .......... .......... .......... .......... ..... (Total 6 marks)
21
