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Summary Statistics
1)!
MIS84-A6i
A factory which manufactures shoes sells direct to the public and records the sales of
different sizes over a period of one week. The results of this survey are shown in the table
below.
Shoe Size
Number Sold
a.
b.
c
d.
6
8
6·5
8
7
8
7·5
11
8
14
8·5
7
9
6
9·5
5
10
2
From this data, calculate the mean show size and the modal shoe size.
On the graph paper provided, draw a cumulative frequency polygon to represent the
data. Label your axes clearly.
Using the graph from part (b), or otherwise, find the median shoe size.
The manager of the factory wishes to know which size is sold most often. Which
measure - the mean, the median or the mode - will be of most value to him? Give a
reason for your answer.¤
70
60
50
CUMULATIVE
40
FREQUENCY
30
20
10
5 6 7 8 9 10
SHOE SIZE
« a) 7·70 (to 2 d.p.) b)
2)!
c) 7·5 d) The mode. This gives the
shoe size sold most often. »
MIS85-A6
The tables below show the scores compiled by three teams in ten innings in a series of oneday limited-over cricket matches.
Team A
206
212
287
209
205
257
205
261
277
290
i.
ii.
Team B
212
273
263
280
257
222
234
233
208
227
Team C
163
191
201
204
197
210
360
201
195
189
Calculate the range, mean and standard deviation for each set of team scores.
Which one of the three teams had the most consistent scores? Give a reason for
your answer.
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iii.
Which one of the above thirty scores was least representative of a team's actual
performance? Give a reason for your answer.
iv.
For team C, calculate the median score.
v.
For team C, which measure - the mean or the median - best describes the team's
performance? Give a reason for your answer.¤
« i)
TEAM A TEAM B TEAM C
RANGE
85
72
197
MEAN
240·9
240·9
211·1
STANDARD
DEVIATION
34·9
24·3
51·1
ii) Team B. Its range and standard deviation are smaller iii) Team C’s score of 360. It is almost
3 standard deviations above the mean. iv) 199 v) The median. The mean is affected by the
score of 360 »
3)!
MIS86-B6ii
A light bulb manufacturer tests light bulbs by determining the number of hours for which
bulbs will burn continuously. The results of testing 30 bulbs is presented in the histogram
below.
10
9
8
7
Number of 6
bulbs 5
4
3
2
1
a.
8 9 10 11 12 13 14 15
Life of bulb in hundreds of hours
Draw up a frequency table to represent this information using the headings shown
in the diagram below.
Life of bulbs
b.
Number of bulbs
For this distribution calculate
.
The mean life of a bulb.
.
The mode.
.
The range.
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.
.
The median.
The standard deviation.¤
« a)
Life of Bulbs Number of Bulbs
800
1
900
2
1000
4
1100
9
1200
6
1300
5
1400
2
1500
1
b) ) 1150 hours ) 1100 hours ) 700 hours ) 1100 hours ) 156·5 hours »
4)!
MIS89-A6b
The heights in centimetres of a sample of 25 girls in New South Wales are recorded in 7
classes in the table below.
HEIGHT
RANGE (cm)
140 - 149
150 - 159
160 - 169
170 - 179
180 - 189
190 - 199
200 - 209
MID POINT
(cm)
144·5
154·5
164·5
174·5
184·5
194·5
204·5
FREQUENCY
1
11
9
...........
1
0
1
i.
The frequency for the 170 - 179 cm class is missing. How do we know it must
be 2?
ii.
What is the modal class?
iii.
Calculate the mean.
iv.
Calculate the standard deviation.¤
« i) We are told there are 25 girls in the sample ie. the total frequency must be 25 ii) 150 - 159 cm
iii) 162·5 cm iv) 12 cm »
5)!
MIS90-A6c
Motorists ring a Road Service Patrol for assistance when their car breaks down. The number
of breakdown calls received by the Road Service Patrol was recorded over a period of
40 days. The number of breakdown calls is shown below:
70
54
65
48
i.
50
48
48
63
55
55
70
50
59
63
69
58
50
70
63
65
62
59
62
59
70
60
67
66
71
60
69
62
54
60
50
60
59
70
55
70
Copy the table below into your examination booklet and complete it.
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Class
Interval
47-51
52-56
57-61
62-66
67-71
ii.
iii.
Class
Centre
Tally
Write down the modal class.
Calculate the mean number of breakdown calls received each day.¤
Class
Interval
47-51
52-56
57-61
62-66
67-71
6)!
7)!
0
12
1
9
What was the average number of errors per page?
(A) 1·6
(B) 2
(C) 8
9)!
Class
Centre
49
54
59
64
69
Tally
Freq.
« i)
fx
7
343
5
270
9
531
9
576
10
690
TOTAL
40
2410
ii) 67 - 71 iii) 60·25 »
MIS93-A9
A spelling test was given to a group of Year Six students and to a group of Year Nine
students. The 30 Year Six students had an average score of 13, and the 60 Year Nine students
had an average score of 19. What was the average score of the combined groups?
(A) 16
(B) 17
(C) 32
(D) 45¤
« B »
MIS93-A20
The number of errors on each page of a document was recorded. The results are shown in
the table below.
Number of errors
Number of pages
8)!
Frequency
2
7
3
7
4
5
(D) 12·8¤
« A »
MIS95-A15
Three students have an average mass of 45 kg. A fourth student with a mass of 66 kg joins
the group. What is the average mass of the four students?
(A) 48 kg
(B) 51 kg
(C) 56 kg
(D) 62 kg¤
« B »
MIS96-A4
Which one of the following groups of scores has a mean of 60 and a median of 50?
(A) 10, 50, 60, 70, 80, 90
(B) 40, 40, 45, 55, 70, 90
(C) 40, 45, 45, 55, 85, 90
(D) 30, 40, 50, 50, 70, 80¤
« C »
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10)!
11)!
MIS96-A14
Three Mathematics classes did the same assessment task. The mean marks for the three
classes were 60, 75, and 76·5. The numbers of students in the three classes were 22, 18, and
20 respectively. What was the mean mark for all students on this assessment task?
(A) 45·25
(B) 70
(C) 70·5
(D) 75¤
« B »
MIS97-A8
Score
Frequency
21
2
22
4
23
6
24
1
25
1
A score of 25 is added to this sample. Which of these measures will change?
(A) Range
(B) Median
(C) Mode
(D) Mean¤
12)!
13)!
14)!
15)!
« D »
MIS97-A16
After five English tests, Sue’s mean mark was 65. In the next three English tests she scored
70, 75 and 80. Calculate Sue’s mean mark for all of these English tests.
(A) 68·75
(B) 70
(C) 72·5
(D) 75¤
« A »
MIS98-2
In which set of scores is the mode greater than the median?
(A) 3, 4, 4, 5, 5, 6, 6, 6, 7
(B) 3, 4, 4, 5, 5, 5, 6, 6, 7
(C) 3, 4, 4, 4, 5, 5, 6, 6, 7
(D) 2, 2, 2, 2, 3, 4, 4, 5, 6¤
« A »
MIS98-14
The mean height of five Sydney Flames basketball players at the start of a game is 1·88 m.
During the game, a player who is 1·74 m tall is injured and is replaced by a player who is
1·94 m tall. What is the mean height of the 5 players now?
(A) 1·89 m
(B) 1·91 m
(C) 1·92 m
(D) 2·08 m¤
« C »
MIS98-23b
The Cavetto Pasta Company employs seventy people. The annual income of the employees
is shown in the frequency distribution table below.
Annual income
($)
10 000 - 19 999
20 000 - 29 999
30 000 - 39 999
40 000 - 49 999
50 000 - 59 999
60 000 - 69 999
Class centre
x
15 000
25 000
35 000
45 000
55 000
65 000
Number of employees
F
16
24
11
9
7
3
f = 70
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fx
240 000
600 000
A
405 000
385 000
195 000
fx =
M_BANK\YR11-GEN\DATA04.HSC
i.
ii.
iii.
iv.
v.
Calculate the value of A in the fx column.
Calculate fx.
Determine the mean annual income.
In which annual income class does the median of this distribution lie?
Using the information in the table and your calculator, find the standard deviation of
this distribution.
vi.
Cavetto Pasta wishes to employ another eight people. Four of these new employees
will earn $15 000 p.a. each, and the other four will earn $65 000 p.a. each. What
will be the effect of these new employees on the standard deviation of income
distribution at Cavetto Pasta? Give a brief reason for your answer.¤
« i) 385 000 ii) 2 210 000 iii) $31 571 iv) 20 000 - 29 999 v) 14 331 (to nearest whole number)
vi) The standard deviation will increase »
16)! MIS99-12
After six Mathematics tests, Nadine’s mean score was 71. She hopes to raise her mean score
to 75 after the next test. What mark will Nadine need to achieve on her seventh test if her
mean mark is to be 75?
(A) 75
(B) 79
(C) 95
(D) 99¤
« D »
17)! MIS99-16
50
No. of 40
students 30
20
10
18)!
0 1 2 3 4 5
Quiz mark
The graph shows the results of 150 students on a Maths quiz. Which statement is correct?
(A) The median is 3 and the mode is 2
(B) The median is 2 and the mode is 2
(C) The median is 2 and the mode is 3
(D) The median is 3 and the mode is 3¤
« B »
MIS99-23a
The maximum temperature for each day of September was recorded. The information is
listed in the table below.
Temperature (C)
x
19
20
21
22
23
24
i.
Frequency
f
4
6
9
0
7
4
fx
76
120
A
0
161
96
What is the mode of this distribution?
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Cumulative
Frequency
4
10
19
19
B
30
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ii.
iii.
iv.
v.
vi.
Find the value of A in the fx column.
Calculate the mean maximum temperature for the month of September.
Find the standard deviation of this distribution.
Find the value of B in the Cumulative Frequency column.
Complete the cumulative frequency histogram below.
35
30
25
20
Cumulative
frequency
15
10
5
0
vii.
viii.
19
20
21
22
23
Temperature (C)
Draw a cumulative frequency polygon on the same graph.
Use the graph to find the interquartile range of this distribution.¤
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24
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« i) 21 ii) 189 iii) 21·4C iv) 1·65 v) 26 vi) vii)
35
30
25
Cumulative frequency
20
15
10
5
0
19)!
20)!
19
23
22
21
Temperature (C)
20
24
viii) 2·9 »
MIS00-15
In order to pass his university course, Dimitri must average at least 50% over 5 assessment
tasks. After the first 4 assessment tasks, Dimitri has a mean mark of 45%. All tasks have
equal weight. What is the minimum mark, out of 100, that Dimitri must score in the fifth
assessment task to pass the course?
(A) 50
(B) 55
(C) 65
(D) 70¤
« D »
MIS00-23c
The following table contains data about the height of 40 year 12 students.
Height range
(cm)
155-159
Class centre
(cm)
157
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Frequency
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160-164
165-169
170-174
175-179
180-184
162
167
172
177
182
185-189
190-194
195-199
187
192
197
7
6
6
9
A
3
2
1
The frequency for the 180-184 height range is written as A in the table. What is
the value of A ?
ii.
What is the modal class?
iii.
What is the median class?
iv.
Use the class centres to calculate the approximate mean height. ¤
« i) 3 ii) 175 - 179 iii) 170 - 174 iv) 173 cm »
GEN01-8
The following frequency table shows Ravdeep’s scores on a number of quizzes.
i.
21)!
Score
1
2
3
4
5
Frequency
2
3
5
2
1
Which expression gives Ravdeep’s mean score?
2  6  15  8  5
2  6  15  8  5
(A)
(B)
13
5
1 2  3 4  5
1 2  3 4  5
(C)
(D)
¤
13
5
22)!
23)!
«A »
GEN02-26a
After three small quizzes, Vicki has an average mark of 5. She wants to increase her average
to 6. What mark must she score in the next quiz for her average mark to be exactly 6? ¤
« 9 »
GEN02-26b
Roxy selected 30 students at random from Year 12 at her high school, and asked each of
them how many text messages they had sent from a mobile phone within the last day. The
results are summarised in the following table.
Number of text
messages sent
0
1
Frequency
3
3
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2
3
4
5
4
4
9
7
i.
24)!
Calculate the mean number of text messages sent. (Give your answer correct to two
decimal places.)
ii.
Calculate the sample standard deviation. (Give your answer correct to two decimal
places.)
iii.
Determine the median number of text messages sent.
iv.
Describe the skewness of the data.
v.
There are 150 students in Year 12 at Roxy’s school. Use the sample data in the
table to estimate how many of these Year 12 students would have sent more than
three text messages within the last day. ¤
« i) 313 ii) 166 iii) 4 iv) The data are negatively skewed v) 80 »
GEN03-12
Joy asked the Students in her class how many brothers they had. The answers were recorded
in a frequency table as follows:
Number of brothers
0
1
2
3
4
Frequency
5
10
3
1
1
What is the mean number of brothers?
(A) 115
(B) 2
(C) 23
25)!
(D) 4¤
«A »
GEN03-25a
A census was conducted of the 33 171 households in Sunnytown. Each household was asked
to indicate the number of cars registered to that household. The results are summarised in
the following table.
Number of cars
0
1
2
3
4
Total
i.
ii.
Frequency
2735
12 305
13 918
3980
233
33 171
1. Determine the mode number of cars in a household.
2. Explain what is meant by “the mode number of cars in a household”.
Sunnytown Council issued a ‘free parking’ sticker for each car registered to
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iii.
iv.
a household in Sunnytown. How many parking stickers were issued?
The council represented the results of the census in a sector graph. What is the
angle in the sector representing the households with no cars? Give your answer to
the nearest degree.
Visitors to Sunnytown Airport have to pay for parking. The following step graph
shows the cost of parking for t hours.
30
25
Parking
Fee ($) 20
15
10
5
26)!
27)!
28)!
0 1 3
6
12
What is the cost for a car that is parked one evening from 6 pm to 8:30 pm? ¤
« i) 1) 2 2) The most common number of cars per household ii) 53 013 iii) 30 iv) $10 »
GEN04-7
What are the median and the mode of the set scores 1, 3, 3, 3, 4, 5, 7, 7, 12?
(A) Median 3, mode 5
(B) Median 3, mode 3
(C) Median 4, mode 5
(D) Median 4, mode 3¤
« D »
Gen05-1
What is the mean of the set of scores?
3, 4, 5, 6, 6, 8, 8, 8, 15
(A) 6
(B) 7
(C) 8
(D) 9¤
« B »
Gen05-27d
Nine students were selected at random from a school, and their ages were recorded.
12
Ages
11
16
14
16
15
14
15
14
i.
What is the sample standard deviation, correct to two decimal places?
ii.
Briefly explain what is meant by the term standard deviation.¤
« i) 1∙69 ii) Standard deviation measures the average deviation of the scores from the mean »
29)! Gen06-4
A set of scores is displayed in a stem-and-leaf plot.
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1
2
3
4
2
5
8
1
2
8
9
3
What is the median of this set of scores?
(A) 28
(B) 30
(C) 33
3
9
(D) 47¤
« C »
30)!
31)!
Gen06-12
The mean of a set of 5 scores is 62.
What is the new mean of the set of scores after a score of 14 is added?
(A) 38
(B) 54
(C) 62
(D) 76¤
« B »
Gen06-23c
Vicki wants to investigate the number of hours spent on homework by students at her high
school.
i.
Briefly describe a valid method of randomly selecting 200 students for a sample.
ii.
Vicki chooses her sample and asks each student how many hours (to the nearest
hour) they usually spend on homework during one week. The responses are shown
in the frequency table.
Number of hours spent
on homework in a week
0 to 4
5 to 9
10 to 14
15 to 19
Frequency
69
72
38
21
What is the mean amount of time spent on homework?¤
« i) Assign each student a umber and draw numbers from a hat ii) 7∙275 hours »
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