Chapter 6 II Stastitic III ENHANCE (4)

7/31/2019 Chapter 6 II Stastitic III ENHANCE (4)

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CHAPTER 6: STATISTICS III

6.1 Understand the concept of class interval

The important concepts:

1. Data obtained from the measurement of certain quantities can be grouped and arrangedinto several classes. The range of each class is called the class interval.

Complete the following class interval.

Example: Exercise 8.1.1: Complete the tables below.

(a)

(b)(c)

Example: Exercise 8.1.2:

(a)

(b) (c)

Statistics

Class interval

31 40Class interval

0 9

10 19

20 2930 39

Class interval

20 21

Class interval

0 4

Class interval

1.1 1.5

1.6 2.0

2.1 2.5

2.6 3.0

3.1 3.5

Class interval

0.4 0.8

Class interval

2.0 2.9Class interval0.60 0.65

1


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2. The lower limit is the lowest value, whereas the upper limit is the highest value of theclass interval.

Example: Class interval of 1 5

Lower limit = 1

Upper limit = 5

3. Lower boundary of a class interval

=2

1 (lower limit of the class interval + upper limit of the class interval before it)


Lower boundary =1 0

0.52

+=

4. Upper boundary of a class interval

=2

1 (upper limit of the class interval + lower limit of the next class interval)


Upper boundary = 5.52

65=

+

5. Size of class interval = (Upper boundary Lower boundary )Example: Size of class interval of 1 5

= 5.5 0.5

= 5

Example: Complete the following tables.

Class interval Lower limit Upper limit Lowerboundary

Upperboundary

Size of classinterval

41 60 41 60 40.5 60.5 20

61 80 61 80 60.5 80.5 20

81 100 81 100 80.5 100.5 20

101 120 101 120 100.5 120.5 20

Class interval Lower limit Upper limit Lower

boundary

Upper

boundary

Size of class

interval

0.4 0.8 0.4 0.8 0.35 0.85 0.5

0.9 1.3 0.9 1.3 0.85 1.35 0.5

1.4 1.8 1.4 1.8 1.35 1.85 0.5

1.9 2.3 1.9 2.3 1.85 2.35 0.5

Statistics 2


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Exercise 6.1.3: Complete the following tables.

a)

b)

c)


boundary

Upper

boundary

Size of class

interval

10 16

17 23

24 30

31 37

d)


boundary

Upper

boundary

Size of class

interval

1.0 4.9

Statistics


boundary

Upper

boundary

Size of class

interval

3 9

10 16

17 23

24 30


boundary

Upper

boundary

Size of class

interval

4 8

9 13

14 18

19 23

3


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5.0 8.9

9.0 12.9

13.0 16.9

e)


boundary

Upper

boundary

Size of class

interval

2.0 2.9

3.0 3.9

4.0 4.9

5.0 5.9

Example:

In a Mathematics test, the marks scored by 40 students from Form 4 Beta are as follows:

78 98 62 54 73 68 82 89 49 80

75 57 87 45 97 78 67 63 56 78

86 89 95 90 76 67 55 45 61 58

85 80 94 93 91 73 75 83 67 40

Given the number of classes required is 6, determine the class interval.

Solution:

The highest value = 98

The lowest value = 40

The range = 98 40

= 58

The size of class interval = 106

58

The suggested class interval table is shown below:

Mark scored 40 - 49 50 - 59 60 - 69 70 -79 80 - 89 90 - 99

Exercise 6.1.4:

a) The collections of donations for the Tsunami victims from 30 classes in SMK Sri Ahmad

are as follows.

56 78 9

0

55 67 89 62 78 74 53 84 58 91 86 73

92 93 8

5

83 93 82 94 93 54 81 92 63 45 78 67

(i) Given the number of classes required is 5, determine the class interval.

(ii) Given the number of classes required is 8, determine the class interval.

Statistics 4

Range = the highest value

of the data the lowest

value of the data


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b) The following are the number of coconuts collected by a farmer in June.

47 45 63 60 61 51 56 54 57 62 58 50 40 44 56

68 53 58 43 65 51 47 50 66 65 54 52 60 53 42

Determine the class interval, given the number of classes required is 6.

Example:

Construct a frequency table for the following set of data.

Solution :

Class interval Tally Marks Frequency

27 32 5

33 38 7

39 44 4

45 50 5

51 56 3

Total 24

Exercise 6.1.5: Construct frequency tables for the following sets of data.

(a)

Statistics

56 44 33 29 36 47

28 37 32 50 40 27

43 38 31 48 44 42

35 51 46 34 52 36

100 129 130 101 105 111 115 108 125 129

104 115 126 117 103 107 129 125 104 128

130 116 119 126 118 125 105 109 114 116

5

Do not obtain the

frequency by counting the

number of data in the

given table. Use the tallymarks.


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(b)

(c)

Solution:

(a)

(b)


300 309

Statistics

315 345 350 307 308 315 340 330 327 318

349 336 333 326 319 342 322 328 317 341

309 333 316 348 303 316 341 327 338 304


100 105

Total 30

1.03 2.04 1.98 1.67 1.84

2.03 1.47 1.56 1.80 1.73

1.21 1.31 1.62 1.75 1.86

1.99 2.02 2.01 1.34 1.92

6


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Total

(c)


1.00 1.14

Total

6.2 Understand and use the concept of mode and mean of grouped data

1. Modal class is the class with the highest frequency.

2. Midpoint of class =2

1 ( Lower class limit + Upper class limit )

3. Mean =

Sum of the values of (midpoint frequency) of all the classes

Sum of frequencies of all the classes

or

fx

x f=

Example:

Find the modal class and the midpoint in the following tables.

Class interval Frequency Midpoint

20 23 2 21.5

24 27 5 25.5

28 31 7 29.5

32 35 9 33.5

36 39 8 37.5

40 43 6 41.544 47 3 45.5

Modal class is 32 35

Exercise 6.2.1: Find the modal class and the midpoints in the following tables.

(a)

Statistics 7

Note that midpoints

are also known as

class marks.


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17 21 0

22 26 2

27 31 10

32 36 15

37 41 16

42 46 3

47 51 1

Modal class is _____________

(b)


10.5 10.9 12

11.0 11.4 15

11.5 11.9 17

12.0 12.4 20

12.5 12.9 13


(c )


Example:

Calculate the mean from the frequency table below.


0 9 5 4.5

10 19 7 14.5

20 29 9 24.5

30 39 10 34.5

Mean =(4.5 5) (14.5 7) (24.5 9) (34.5 10)

5 7 9 10

+ + +

+ + +

=31

5.689

= 22 .2

Statistics


121 124 20

125 128 35

129 132 33

133 136 28

137 140 18

8


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Exercise 6.2.2: Calculate the means from the frequency tables below.

a)

Class interval Frequency Midpoint12 15 4 13.5

16 19 6

20 23 7

24 27 3

Mean =

=

=

(b)


7.1 7.4 10 7.025

7.5 7.8 20

7.9 8.2 20

8. 3 8.6 5

Mean =

=

=

c)


45 49 5 47

50 54 8 52

55 59 9

60 - 64 10

65 - 69 4

Mean =

=

Statistics 9


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=

6.3 Represent and interpret data in histograms with class intervals of the same size to solveproblems

i. Draw a histogram based on the frequency table.

ii. Interpret information from the given histogram.

iii. Solve problems involving the histogram.

Examples:

1. Draw a histogram for each of the

following frequency tables.a )

b)

Length (cm) Frequency

20 29 5

30 39 15

40 49 20

50 59 25

60 69 30

70 79 10

2. The histogram shows the results of anobjective test in a certain examination.

Statistics


20 24 4

25 29 8

30 34 6

35 39 8

40 44 5

45 49 7

10

frequency

5.5 10.5 15.5 20.5 25.5 30.535.5 40.5 45.5

marks


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Answer

a)

b)

2. a) Modal class = 16 20 (with the

highest frequency.)

b) Total number of students = 15 + 30

+ 55 + 40 + 50 + 30 + 15 = 235.

c) Number of students who passed the

test = 40 + 50 + 30 + 15 = 135.

a) State the modal class.

b) How many students took this test?

c) If the passing mark is 20, how many

students passed the test?d) State the two class intervals that have

the same frequency.

d) The two class intervals are:

11 15 and 31 35,

or 6 10 and 36 40.

Statistics 11

Frequency

Length (cm)

19.5 29.5 39.5 49.5 59.5 69.5 79.5

Frequency

Length (cm)


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Exercise 6.3:

1. The histogram shows the number of books read by students in SMK Putra in a year.

a) State the modal class.b) Students who read more than 50 books received an award. Find the number of students

who received the award.

c) Find the mean number of books read by each student.

6.3 Represent and interpret data in frequency polygons to solve problems.

i. Draw a frequency polygon based on

a. a histogram.

b. a frequency table.ii Interpret information from a given frequency polygon.

iii Solving problems involving frequency polygons.

Examples

1. Draw a frequency polygon based on the given histogram.Length 10-14 15-19 20-24 25-29 30-34

Midpoint 12 17 22 27 32

Frequency 4 6 12 8 2

Statistics 12

Frequency

9.5 19.5 29.5 39.5 49.559.5 69.5

Number of books read

Frequency

9.5 14.5 19.524.5 29.5 34.5 Length

(cm)Histogram Frequency Polygon

7 12 17 22 2732 37

Frequency

MidpointLength(cm)


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2. Draw a frequency polygon for the given frequency table.

Length(cm) 8-10 11-13 14-16 17-19 20-22

Frequency 16 20 8 12 16

Answer:

Length (cm) 5-7 8-10 11-13 14-16 17-19 20-22 23-25

Frequency 0 16 20 8 12 16 0

Midpoint 6 9 12 15 18 21 24

Exercise 6.4:

1. The frequency polygon shows the ages

of the members of a club.

Find the

(a) total number of members in this

club.

(b) the modal class.

(c) the mean age.

(d) the percentage of the members who

are above 23 years old.

Statistics 13

Frequency

Length(cm)

Frequency

Age


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6.3 Understand the concept of cumulative frequency

The Cumulative Frequency is the sum of the frequencies of all the values of data or class intervals

before it.

Example:

Construct the cumulative frequency table for the following data:


2 6 6

7 11 10

12 16 15

17 21 4

Solution:

Length (cm) Frequency Cumulative frequency

2 6 6 6

7 11 10 6 + 10 = 16

12 16 15 6 + 10 + 15 = 21

17 21 4 6 + 10 + 15 + 4 = 25

Exercise 6.5:

1. Construct the cumulative frequency tables for the following:

a)

Time (min) Frequency8 11 2

11 13 4

14 16 8

17 19 10

20 22 12

b)

Weight (kg) Frequency

1 2 10

3 4 15

5 6 8

7 8 6

Statistics 14


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6.3 Understand and use the concept of measures of dispersion to solve problems.

The important concepts are :

a) Range(i) Range of the ungrouped data is the measure of dispersion which refers to the

difference between the highest value and the lowest value of the data.

(ii) Range for the grouped data is the measure of dispersion which refers to the

difference between the midpoint of the last class and the midpoint of the first

class.

b) Median

Median of a set of data is the value in the middle of the set after the data has been

arranged in numerical ascending/descending order.

c) First Quartile (Q1)First quartile is the value of data such that one-quarter of the set of data have values less

than or equal to it.

d) Third QuartileThird quartile is the value of data such that three-quarter of the set of data have values

less than or equal to it.

e) Inter-quartile RangeInter-quartile range is the difference between the third quartile and the first quartile.

Example:

1. Find the range of the following sets of data.

12, 14, 15, 13, 18, 17, 10

Solution:

Range = 18 - 10

= 8

2. Determine the range of the following sets of data.

Marks scored 40 - 49 50 -59 60 - 69 70 -79

Frequency 4 5 7 10

Solution:

Range = 74.5 - 44.5

= 30

Statistics 15


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3. A physical education teacher has recorded the time taken by 40 students for the 400 m

event during a sports practice.

Time (seconds) 50 - 64 65 - 79 80 - 94 95 - 109 110 - 124

No. of students 3 10 18 7 2

Construct a cumulative frequency table and draw an ogive.

Solution:

Time

(seconds)

Frequency Cumulative

frequency

Upper

Boundary

35 - 49 0 0 49.5

50 - 64 3 3 64.5

65 - 79 10 13 79.5

80 - 94 18 31 94.5

95 - 109 7 38 109.5

110 - 124 2 40 124.5

Solution:

a) The median occurs at ( of the total number of students) i.e. at ( x 40) studentsThe median occurs at the 20th student.

Median = 85.5 (read from the ogive)

b) The First Quartile occurs at ( of the total number of students) i.e. at ( x 40) studentsThe First Quartile occurs at the 10th student.

First Quartile = 76.5 (read from the graph)

c) The Third Quartile occurs at ( of the total number of students) i.e. at ( x 40) studentsThe Third Quartile occurs at the 30 th student.

Third quartile = 93 (read from the graph)

d) The interquartile range = third quartile first quartile= 93 - 76.5

= 16.5

Statistics 16

4. From the ogive, determine

a) the median,

b) the first quartile,

c) the third quartile,

d) the inter-quartile range.

An ogive is alsoknown as acumulative

frequency curve


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Exercise 6.6:

1. The data in the table shows the heights of a group of students in cm.

155 173 167 173 166 166 174 178176 165 153 169 162 160 151 163

160 176 167 175 174 156 172 164171 174 179 169 155 171 157 179

181 172 164 157 168 171 169 154

a) Construct a cumulative frequency table beginning with 145 149.

b) By using 2 cm to 5 cm on the x-axis and 2 cm to 5 students on the y- axis, plot an ogive.c) From the ogive , determine

i) the median

ii) the interquartile range

2. The data below shows the duration, in minutes, taken to solve mathematical problems bya group of 35 college students.

30 12 4 10 25 20 16

20 8 3 24 15 12 17

10 10 12 8 20 13 18

15 9 14 5 30 19 20

13 5 4 20 18 24 25

a) Construct a cumulative frequency table using 1 5, 6 10 and.

b) Plot an ogive using 2 cm to 5 minutes on the x-axis and 2 cm to 5 students on the

y-axis.

c) State the

(i) median,(ii) interquartile range.

Statistics 17


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6.7 Questions Based on the Examinations Format

STATISTICS Paper 2

1. The data above shows the pocket money, in RM, per week of a group of 40 students.

10 13 18 7 19 5 21 2118 16 18 10 15 19 20 8

4 14 12 6 9 20 20 15

8 10 16 11 13 17 9 1119 15 17 12 14 11 16 17

(a) Based on the data, complete the table below.

Pocket money (RM) Frequency Midpoint

1 3 0

4 6

(b) Based on the table,

(i) state the modal class,

(ii) calculate the mean,

of the data.

(c) For this part of the question, use the graph paper.

By using a scale of 2 cm to RM 3 on the x-axis and 2 cm to 1 student on the y-axis, draw a

histogram and a frequency polygon on the same graph paper.

2. (a)

Score 10 11 12 13 14 15

Frequency 4 7 3 x 5 8

The table above shows the scores of a group of players in the game.(i) Given the median is 12.5, find the value ofx.

(ii) Given the mode is 15, find the maximum value ofx.

(b)

49 45 47 66 66 62 68 43 46

53 50 61 65 67 63 68 65 44

48 63 62 45 61 67 56 43 44

57 64 60 52 58 46 59 41 43

64 65 47 46 52 50 54 42 60

The data shows the mass, in kg, of a group of students.

Statistics 18


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(i) Based on the data, complete the frequency table below.

Mass (kg) Frequency

41 44

65 68

(ii) For this part of question, use the graph paper.

(a) By using a scale of 2 cm to 4 kg on thex-axis and 2cm to 1 student on

they-axis, draw a histogram for the data.

(b) State one information obtained from the histogram.

4. The histogram shows the donations from a group of 40 students to a school fund.

(a) Based on the histogram, state the modal class of the data.

(b) Complete the frequency table below.

Donation (RM) Frequency Cumulative frequency Upper boundary

0 0

1 4 2

Statistics 19

0

2

4

6

8

10

12

14

0.5 4.5 8.5 12.5 16.5 20.5 24.5 28.5

Donation (RM)

Frequency

0.5 4.5 8.5 12.5 16.5 20.5 24.5 28.5


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(c) For this part of question, use a graph paper.

(i) By using a scale of 2 cm to RM4 on the x-axis and 2 cm to 5 students on the

y-axis, draw an ogive for the data.

(ii) From the ogive, find the inter quartile range.

5. The ogive shows the scores of a group of 48 students in a Mathematics test.

(a) Based on the ogive, calculate the percentage of students who scored more than 80.

(b) Complete the table below.

Score Cumulative frequency Frequency Midpoint

60 64 0

65 69

(c) Calculate the mean of the data.

(d) For this part of question, use the graph paper.

By using a scale of 2 cm to 5 scores on the x-axis and 2 cm to 2 students on the y-axis, draw a frequency polygon for the data.

Statistics 20

0

10

20

30

40

50

60

64.5 69.5 74.5 79.5 84.5 89.5 94.5 99.5

Score

Cumulative

frequency


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6. The frequency table above shows the masses, in kg, of watermelons in a lorry.

Mass (kg) Frequency

1.5 1.9 4

2.0 2.4 10

2.5 2.9 26

3.0 3.4 84

3.5 3.9 50

4.0 4.4 15

4.5 4.9 8

5.0 5.4 3

(a) From the table, state

(i) the size of the class interval,(ii) the midpoint of the modal class.

(b) (i) Based on the information, complete the table below.

Upper

boundary1.45 5.45

Cumulative

Frequency0 200

(ii) For this part of question, use a graph paper.

By using a scale of 2 cm to 0.5 kg on the x-axis and 2cm to 20 watermelons

on the y-axis, draw an ogive for the data.

(iii) From the ogive,

a) state the median of the data,

b) find the number of watermelons with the weight less than 3.8 kg.

7. The cumulative frequency table shows the marks of a group of 40 students in a test.

Mark 6064 6569 7074 7579 8084 8589 9094Cumulative

frequency7 11 17 26 34 37 40

(a) Based on the information, complete the table in the answer corner.

(b) Find

(i) the size of the class interval,

(ii) the modal class of the data,

(iii) the midpoint of the modal class.

(c) Calculate the mean of the data.

(d) For this part of question, use the graph paper.By using a scale of 2 cm to 5 marks on the x-axis and 2 cm to 1 student on the

y-xis, draw a histogram for the data.

Statistics 21


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8. The data shows the marks of a group of 40 students in a Mathematics test.

71 20 44 48 47 37 70 56

33 61 52 21 36 31 55 56

53 66 72 38 45 49 50 50

83 29 55 46 59 63 60 48

42 69 43 44 58 78 40 38

(a) Using the data and a class interval of 10 marks, complete the table below.

Mark Frequency Cumulative frequency Upper boundary

1 10

81 - 90

(b) For this part of question, use a graph paper.

By using a scale of 2 cm to 10 marks on the x-axis and 2 cm to 5 students on the y-axis,

draw an ogive for the data.

(c) From your ogive,

(i) find the interquartile range,

(ii) find the median, hence, explain briefly the meaning of the median.

9. The data shows the ages of a group of 40 workers in a factory.

38 41 32 39 32 50 31 38

36 51 31 27 44 30 49 28

26 31 22 36 32 33 34 4331 27 46 41 35 35 34 31

40 35 25 27 50 37 33 45

(a) Find the range of the data.

(b) Based on the data and by using a class interval of 5 years, complete the table below.

Statistics 22


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Ages (years) Frequency Midpoint

21 25

26 30

(c) From the table,


(ii) calculate the mean,

of the data.

(d) For this part of question, use a graph paper.

By using a scale of 2 cm to 5 years on the x-axis and 2 cm to 2 workers on the y-axis,draw a frequency polygon for the data.

10. The frequency polygon shows the daily incomes, in RM, of 28 workers.

(a) Based on the frequency polygon, complete the table below.

Daily income

(RM) MidpointUpper

boundary FrequencyCumulative

frequency

16 20 18 20.5 0 0

Statistics 23

0

2

4

6

8

10

12

18 23 28 33 38 43 48

Daily income

F

requency


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(b) Based on the table,


(ii) calculate the mean, of the data.

(c) For this part of question, use a graph paper.

(i) By using a scale 2 cm to RM5 on the x-axis and 2 cm to 4 workers on the y-axis,

draw an ogive for the data.

(iii) From the ogive, find the number of workers whose daily income is more thanRM38.

6.8 PAST YEAR SPM QUESTIONS

1. SPM Nov 2003

The data below shows the donations, in RM, of 40 families to their childrens school

welfare fund.

40 24 17 30 22 26 35 19

23 28 33 33 39 34 39 28

27 35 45 21 38 22 27 35

30 34 31 37 40 32 14 28

20 32 29 26 32 22 38 44

(a) Using the data above and a class interval of RM5, complete the following table.

[4 marks]

Donation (RM) Frequency Cumulative Frequency

11 15

16 20

(b) For this part of question, use a graph paper.

By using a scale of 2 cm to RM5 on the x-axis and 2 cm to 5 families on the y-axis, draw

an ogive based on the data.

[6 marks]

(c) From your ogive in (b),

(i) find the third quartile,(ii) hence, explain briefly the meaning of the third quartile. [2 marks]

Statistics 24


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2. SPM July 2004

The data in Diagram 7 shows the number of durian trees planted by 44 farmers.

Diagram 7

(a) (i) Based on the data in Diagram 7 and by using a class interval of 10, complete Table 3

provided in the answer space.

(ii) Hence, state the modal class. [6marks]

Answer :

Class Interval Upper Boundary Frequency Cumulative Frequency

11 20 20.5 0 0

21 30

(b)For this part of the question, use the graph paper on page 41 .

By using a scale of 2 cm to 10 trees on the x-axis and 2 cm to 5 farmers on the y-axis,

draw an ogive for the data. [4marks]

(c) Based on the ogive in (b), Ahmad concludes that 50% of the farmers planted less than 52

durian trees.

Determine whether the conclusion is correct or not and give a reason.

[2marks]

3. SPM Nov 2004

The data below shows the masses, in kg, of suitcases for a group of tourists. Each tourist

has one suitcase.

27 10 22 28 21 14 29 25

29 18 22 13 20 21 24 27

27 25 16 19 16 24 26 27

29 19 33 25 23 24 26 31

Statistics

52 33 48 22 34 42 57 51 51 65 41

66 54 66 53 53 34 46 52 65 75 52

25 68 48 63 62 43 52 56 59 49 5843 58 36 72 68 54 62 40 73 38 63

25


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0

5

10

15

20

25

30

115 120 125 130 135 140 145 150

Height (in cm)

NumberofPupils

(a) Based on the data above and by using a class interval of 3, complete the table below.[4 marks]

Class Interval Frequency Midpoint

10 12

13 15

(b) Based on the table in (a), calculate the estimated mean mass of the suitcases.

[3 marks]

(c) For this part of the question, use a graph paper.

By using a scale of 2 cm to 3 kg on the x-axis and 2 cm to 1 suitcase on the y-axis, draw

the histogram for the data. [3 marks]

(d) State one information obtained based on the histogram in (c).

[2 marks]

4. SPM July 2005

Diagram 9 is a frequency polygon which represents the heights, in cm, for a group of 80 pupils.

(a) (i) Based on the information from the frequency polygon, complete Table 2 in theanswer space.

(iv) Hence, calculate the mean height, in cm, of the pupils. Give your answer correct to 2

decimal places.Statistics 26

Diagram 9


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[6 marks]

Answer:

Height

(in cm)

Upper

Boundary

Midpoint Frequency Cumulative

Frequency118 122 122.5 120 2 2

123 127

128 132

133 137

138 142

143 147

(b) For this part of the question, use the graph paper on page 45. You may use a flexiblecurve rule.

By using a scale of 2 cm to a height of 5 cm on the horizontal axis and 2 cm to 10 pupils

on the vertical axis, draw an ogive for the data.

[5 marks]

(c) Give one information that can be obtained from the ogive in (b).[1 mark]

5. SPM Nov 2005

The data in Diagram shows the marks for an English Language monthly test for 42 pupils.

(a) Using data in

diagram and a class

interval of 5 marks,

complete Table in the

answer space.

[4marks]

Marks Midpoint Frequency

20 24 22

25 29

Statistics

51 20 45 31 26 40 30

25 32 37 41 21 36 38

46 38 28 37 39 23 39

33 35 42 29 38 31 23

42 34 26 35 43 28 22

25 47 31 48 44 34 54

27


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(b) Based on your table in (a),


(ii) calculate the mean mark for the English Language monthly test and give your answercorrect to decimal places.

[4marks]

(c)For this part of the question, use the graph paper provided .

By using a scale of 2cm to 5 marks on the horizontal axis and 2 cm to 1 pupil on the

vertical axis, draw a histogram for the data. [4marks]

6. SPM July 2006

Table 3 shows the distribution of age for a group of 40 tourists who visited a museum in a

particular day.

Age(Year) 20 24 25 29 30 34 35 39 40 44 45 49 50 54

Frequency 2 6 9 10 7 4 2

Table 3

(a) Based on the distribution of age in Table 3, complete Table 4 in the answer space.

[2marks]

Answer:

Age (Year) Midpoint Upper Boundary Cumulative Frequency

20 24 2225 29 27

30 34 32

35 39 37

40 44 42

45 49 47

50 - 54 52

Table 4

(b) Calculate the estimated mean age of the tourists. [3marks]

(c)For this part of the question, use the graph paper .

By using the scale of 2 cm to 5 years on the horizontal axis and 2 cm to 5 tourists on the

vertical axis, draw an ogive for the data in table 3. [5marks]

(d) From your ogive constructed in (c),

(i) find the first quartile,

(ii) state one piece of information about the first quartile [2marks]

Statistics 28


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7. SPM Nov 2006

The data in Diagram 7 show the donations, in RM, collected by 40 pupils.

49 26 38 39 41 45 45 43

22 30 33 39 45 43 39 31

27 24 32 40 43 40 38 35

34 34 25 34 46 23 35 37

40 37 48 25 47 30 29 28

Diagram 7

(b) Based on the data in Diagram 7 and by using a class interval of

5, complete Table 2 in the answer space.(3 marks)

Answer:

Class Interval Midpoint Frequency

21 25

26 30

Table 2

(c) Based on Table 2 in (a), calculate the estimated mean of thedonation collected by a pupil

(3 marks)

(d) For this part of the question, use the graph paper provided.

By using a scale of 2 cm to RM5 on the horizontal axis and 2 cm to 1 pupil on the vertical

axis, draw a frequency polygon for the data. (5 marks)

(d) Based on the frequency polygon in (c), state one piece of information about the

donations. (1 mark)

Statistics 29


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8. SPM Jun 2007, No 15

Diagram 8 shows the mass, in kg, of newspapers collected over the period of 40 days.

79 70 75 79 83 74 66 58 65 8584 76 56 80 64 57 78 67 52 59

72 58 57 75 69 76 55 75 60 7383 63 74 65 78 58 73 64 75 59

Diagram 8

Based on the data in Diagram 8, complete Table 2 in the answer space.

[4 marks]

(b) Hence, calculate the estimated mean of the masses of the newspapers collected.

[3 marks]

(c) For this part of the question, use the graph paper provided.

By using the scale of 2 cm to 5 kg on the horizontal axis and 2 cm to 1 day on the vertical

axis, draw a histogram for the data.

[4 marks]

(d) Based on your histogram in 15(c), give one information about the modal class of the data.

[1 mark]

Answer:

Mass (kg) Frequency Midpoint

52 56

57 61

9. SPM Nov 2007, No 16

Table 2 shows the frequency distribution of the mass, in kg, of a group of 80 students.

Mass (kg) Frequency

30 34 5

35 39 8

40 44 11

45 49 21

50 54 22

55 59 10

60 64 3

Table 2

(a) (i) State the modal class

(ii) Calculate the estimated mean of the mass of the group of students.[4 marks]

Statistics 30


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(b) Based on Table 2, complete Table 3 in the answer space to show the cumulative frequency

distribution of the masses.

[3 marks]

(c)For this part of the question, use the graph paper provided.By using the scale of 2 cm to 5 kg on the horizontal axis and 2 cm to 10 students on the

vertical axis, draw an ogive for the data.[4 marks]

(d) 25% of all the students in the group have a mass of less thanp kg. These students will be

supplied with nutritional food.

Using the ogive you had drawn in 16(c), find the value ofp.

[1 mark]

Answer:

(a) (i)

(ii)

(b)

Upper Boundary (kg) Cumulative frequency

29.5 0

34.5

Table 3

10. SPM Jun 2008, No 14

Data in Diagram 14 shows the number of books read by 40 students in a reading programme in a

particular class.

18 14 11 16 13 17 12 23

12 19 17 15 11 21 7 14

15 11 5 18 9 22 24 19

17 9 14 12 6 15 10 1319 15 20 10 14 8 12 16

Diagram 14

(a) Based on the data in Diagram 14 and by using the class interval of 3, complete Table 14

in the answer space.

[3 marks]

(b) Based on Table 14 in 14(a), calculate the estimated mean of the books read by a student.Statistics 31


32/55

[3 marks]

(c) By using the scale of 2 cm to 3 books on the horizontal axis and 2 cm to one student on

the vertical axis, draw the frequency polygon for this data.

[5 marks]

(d) Based on the frequency polygon in 14(c), give one information about this readingprogramme.

[1 mark]

Answer:

(a)


5 7 6

8 10

Table 14

(b)

(c)

(d)

11. SPM Nov 2008, No 14.

The data below shows the payment, in RM, by 40 drivers at a toll booth.

38 34 41 25 19 32 28 42

32 25 32 27 42 23 18 46

21 42 36 30 33 37 43 25

24 18 26 35 47 22 38 33

30 23 37 48 39 34 41 27

(a) Based on the data, complete Table 14 in the answer space.

[3 marks]

(b) Based on Table 14 in 14(a), calculate the estimated mean of the toll paid by a driver.

[3 marks]

(c) For this part of the question, use the graph paper provided.By using the scale of 2 cm to RM5 on the horizontal axis and 2 cm to 1 driver on the

vertical axis, draw a frequency polygon for the data.

[5 marks]

(d) Based on the frequency polygon in 14(c), state the number of drivers who paid more than

RM34 for the toll.

[1 mark]Statistics 32


33/55

Answer:

a)


15 19 17

b)

d)

Statistics 33


34/55

ANSWERS

Chapter 6 : Statistics III

Exercise 8.1.1:

a)class interval b)class interval c)class interval

31-40 0-4 20-21

41-50 5-9 22-23

51-60 10-14 24-25

61-70 15-19 26-27

71-80 28-29

Exercise 8.1.2

(a) (b) (c)

Exercise 8.1.3 :

(a)

Class

interval

Lower limit Upper limit Lower

boundary

Upper

boundary

Size of the

class interval

3 9 3 9 2.5 9.5 7

10 16 10 16 9.5 16.5 717 23 17 23 16.5 23.5 7

24 30 24 30 23.5 30.5 7

(b)


boundary

Upper

boundary

Size of the

class interval

4 8 4 8 3.5 8.5 5

9 13 9 13 8.5 13.5 5

14 18 14 18 13.5 18.5 5

19 23 19 23 18.5 23.5 5

(c)Class interval Lower limit Upper limit Lower

boundary

Upper

boundary

Size of the

class interval

10 16 10 16 9.5 16.5 7

17 23 17 23 16.5 23.5 7

24 30 24 30 23.5 30.5 7

31 37 31 37 30.5 37.5 7

Statistics 34

Class interval

2.0 2.9

3.0 3.94.0 4.9

5.0 5.9

6.0 6.9

Class interval

0.60 0.65

0.66 0.71

0.72 0.77

0.78 0.83

0.84 0.89

Class interval

0.4 0.8

0.9 1.3

1.4 1.8

1.9 2.3

2.4 2.8


35/55

(d)


boundary

Upper

boundary

Size of the

class interval

1.0 4.9 1.0 4.9 0.95 4.95 4

5.0 8.9 5.0 8.9 4.95 8.95 4

9.0 12.9 9.0 12.9 8.95 12.95 4

13.0 16.9 13.0 16.9 12.95 16.95 4

(e)


boundary

Upper

boundary

Size of the

class interval

2.0 2.9 2.0 2.9 1.95 2.95 1

3.0 3.9 3.0 3.9 2.95 3.95 1

4.0 4.9 4.0 4.9 3.95 4.95 1

5.0 5.9 5.0 5.9 4.95 5.95 1

Exercise 8.1.4 :

(a) The highest value is 94The lowest value is 45

The range is 94 45 = 49(i)

Donations collected 45 - 54 55 - 64 65 - 74 75 - 84 85 - 94

(ii)Donatio

ns

collected

44 -50 51 - 57 58 - 64 65 - 71 72 - 78 79 - 85 86 - 92 93 - 99

(b)

Number of coconut 40 -

44

45 - 49 50 - 54 55 - 59 60 - 64 65 - 69

Exercise 8.1.5 :

(a)

Class interval Frequency

100 -105 7

106 - 111 4

112 - 117 6

118 - 123 2

124 - 129 9

130 - 135 2

Total 30

(b)


300 309 5

310 319 7

320 329 5

330 339 5

340 349 7

350 359 1

Total 30

Statistics 35


36/55

(c)


1.00 1.14 1

1.15 1.29 1

1.30 1.44 2

1.45 1.59 2

1.60 1.74 3

1.75 1.89 4

1.90 2.04 7

Total 20

Exercise 8.2.1 :

(a) Modal class : 37 41

(b) Modal class : 12.0 12.4

(c) Modal class : 125 128

Exercise 8.2.2 :

a)

Mean =3764

)35.25()75.21()65.17()45.13(

+++

+++

=20

386

= 19.3

b)

Mean =5202010

)545.8()2005.8()2065.7()1025.7(

+++

+++

=55

75.428

= 7.795

c)

Statistics 36

Class interval Frequency Midpoint12 15 4 13.5

16 19 6 17.5

20 23 7 21.5

24 27 3 25.5


7.1 7.4 10 7.25

7.5 7.8 20 7.65

7.9 8.2 20 8.05

8.3 8.6 5 8.45


37/55

Mean =410985

)467()1062()957()852()547(

++++

++++

=36

2052

= 57.0

Exercise 8.3:

1a) Modal class = 20 29

b) Student who received the award = 30 +20

= 50c)

Class interval Frequency Midpoints

10 19 60 14.5

20 29 120 24.5

30 39 80 34.5

40 49 100 44.5

50 59 30 54.5

60 - 69 20 64.5

Mean =20301008012060

)5.6420()5.5430()5.24120()5.1460(

+

=410

13945

= 34.01

= 34

Exercise 8.4:

1.a) Total number of members = 4 + 16 + 36 + 24 = 80

b) Upper boundary =2

2520 += 22.5

Lower boundary =2

2515 += 17.5

Upper limit = 22Lower limit = 18

Modal class = 18 22

c) mean age =80

)2524()2036()1516()104(

= 20

d) percentage of members who are above 23 years old = 10080

24

Statistics 37


45 - 49 5 47

50 54 8 52

55 - 59 9 57

60 - 64 10 62

65 - 69 4 67


38/55

= 30%

Exercise 8.5:

1 a) 2, 6, 14, 24, 36, 40b) 10, 25, 33, 39

Exercise 8.6:

1 a)

Height (cm) frequency Upper boundary

Cumulativefrequency

145 - 149 0 149.5 0

150 - 154 3 154.5 3

155 - 159 5 159.5 8

160 - 164 6 164.5 14

165 - 169 9 169.5 23

170 - 174 10 174.5 33

175 - 179 6 179.5 39

180 - 184 1 184.5 40

c i ) median = 168

ii) interquartile range = 173 161.5 = 11.5

2 a)Duration (mm) frequency Upper

boundary

Cumulative

frequency

1 - 5 5 5.5 5

6 - 10 6 10.5 11

11 - 15 8 15.5 19

16 - 20 10 20.5 29

21 - 25 4 25.5 33

26 - 30 2 30.5 35

Statistics 38


39/55

c i) median = 15

ii} interquartile range = 19.25 8.5 = 10.75

8.7 Examination Format Questions

2. (a)

Pocket money(RM) Frequency Midpoint

1 3 0 2

4 6 3 5

7 9 5 8

10 12 8 11

13 15 7 14

16 18 9 17

19 21 8 20

(b) (i) Modal class = 16 18

(ii) Mean =8978530

)820()917()714()811()58()35(

+

=40

554

= RM 13.85

(iii)

Statistics 39


40/55

0

1

2

3

4

5

6

7

8

9

10

3.5 6.5 9.5 12.5 15.5 18.5 21.5

Pocket m oney

Frequency

3.5 6.5 9.5 12.5 15.5

3. (a) (i) x = ( 4+ 7 +3 ) ( 5 + 8 )= 14 13

= 1

(ii) Maximum value of x = 7

(b) (i)

Mass (kg) Frequency

41 44 7

45 48 8

49 52 5

53 56 3

57 60 561 64 8

65 68 9

(ii)

(iii) The modal class of the data is 65 68 kg.

Statistics 40

01

2

3

4

5

6

7

8

9

10

44.5

52.5

60.5

68.5

Mass(kg)

Frequency

40.5 44.5 48.8 52.5 56.5 60.5


41/55

4. (a) Modal class = 9 12

(b)

Donation (RM) Frequency Cumulative

frequency

Upper boundary

0 0 0 0.5

1 4 2 2 4.55 8 5 7 8.5

9 12 12 19 12.5

13 16 8 27 16.5

17 20 5 32 20.5

21 24 3 35 24.5

25 28 1 36 28.5

(c) (i)

0

5

10

15

20

25

30

35

40

0.5 4.5 8.5 12.5 16.5 20.5 24.5 28.5

Donation (RM)

Cumulativefrequency Third quartile

First

9. 16.5

(ii) Interquartile range = 16.5 9.3

= RM 7.20

5. (a) The percentage of students who scored more than 80

= 48

1948 x 100%

=48

29x 100%

= 60.42 %

Statistics 41


42/55

(b)

Score Cumulative frequency Frequency Midpoint

60 64 0 0 62

65 69 2 2 67

70 74 7 5 72

75 79 17 10 77

80 84 34 17 82

85 89 43 9 87

90 94 47 4 92

95 99 48 1 97

(c) Mean =1491710520

)197()492()987()1782()1077()572()267(

+

=48

3906

= 81.375

(d)

0

2

4

6

8

10

12

14

16

18

62 67 72 77 82 87 92 97 102

Score

Frequency

6. (a) (i) Size of class interval = 1.95 1.45

= 0.5 kg

(ii) Modal class = 3.0 3.4

Midpoint of the modal class =2

4.30.3 +

= 3.2

Statistics 42


43/55

(b) (i)

Upper boundary Cumulative frequency

1.45 0

1.95 4

2.45 14

2.95 40

3.45 124

3.95 174

4.45 189

4.95 197

5.45 200

(ii)

0

20

40

60

80

100

120

140

160

180

200

220

1.45 1.95 2.45 2.95 3.45 3.95 4.45 4.95 5.45

Mass (kg)

Cummulativefrequency

Median

164

3.4

0

(iii) (a) Median = 3.40 kg

(b) The number of watermelons with the weight less than 3.8 kg = 200 164= 36.

Statistics 43


44/55

7. (a)

Mark Cumulative

frequency

Frequency Midpoint

55 59 0 0 57

60 64 7 7 62

65 69 11 4 67

70 74 17 6 72

75 79 26 9 77

80 84 34 8 82

85 89 37 3 87

90 - 94 40 3 92

(b) (i) Size of class interval = 59.5 54.5

= 5 marks

(ii) Modal class = 75 79

(iii) Midpoint of the modal class =2

7975+

= 77

(c) Mean =33896470

)392()387()882()977()672()467()762(+

=40

3020

= 75.5 marks

(d)

0

1

2

3

4

5

6

7

8

9

10

54.5 59.5 64.5 69.5 74.5 79.5 84.5 89.5 94.5

Mark

Frequency

54.5 59.5 64.5 69.5 74.5 79.5 84.5 89.5 94.5

Statistics 44


45/55

8. (a)

Mark Frequency Cumulative

frequency

Upper boundary

1 10 0 0 10.5

11 20 1 1 20.5

21 30 2 3 30.5

31 40 7 10 40.5

41 50 12 22 50.5

51 60 9 31 60.5

61 70 5 36 70.5

71 80 3 39 80.5

81 90 1 40 90.5

(b)

0

5

10

15

20

25

30

35

40

45

10.5 20.5 30.5 40.5 50.5 60.5 70.5 80.5 90.5

Mark

Cumulativefrequency

First quartile

Median

Third quartile

4

9.5

59.5

(c) (i) Interquartile range = 59.5 40.5

= 19.0 marks

(ii) Median = 49.5 marks

The median means 20 of the students scored less than 49.5 marks.

Statistics 45


46/55

9. (a) Range = 51 22

= 29 years.

(b)

Ages (years) Frequency Midpoint

21 25 2 23

26 30 6 2831 35 15 33

36 40 7 38

41 45 5 43

46 50 4 48

51 55 1 53

(c) (i) Modal class = 31 35

(ii) Mean =14571562

)153()448()543()738()1533()628()223(

+

=40

1435

= 35.875 years

(d)

0

2

4

6

8

10

12

14

16

18 23 28 33 38 43 48 53 58

Age (years)

Frequenc

y

Statistics 46


47/55

10. (a)

Daily

income(RM)

Midpoint Upper

boundary

Frequency Cumulative

frequency

16 20 18 20.5 0 0

21 25 23 25.5 4 4

26 30 28 30.5 10 14

31 35 33 35.5 8 22

36 40 38 40.5 4 26

41 45 43 45.5 2 28


(ii) Mean =2481040

)243()438()833()1028()423(

+

=28

874

= RM 31.21

(c) (i)

0

4

8

12

16

20

24

28

20.5 25.5 30.5 35.5 40.5 45.5

Daily income (RM)

Cumulativefrequency

(ii) The number of workers whose daily income is more than RM 38 = 28 24

= 4

8.8 SPM PAST YEAR QUESTIONS

Statistics 47


48/55

1. SPM 2003

(a)

Donation (RM) Frequency Cumulative frequency

11 15 1 1

16 20 3 4

21 25 6 10

26 30 10 20

31 35 11 31

36 40 7 38

41 45 2 40

(b)

0

5

10

15

20

25

30

35

40

45

10.5 15.5 20.5 25.5 30.5 35.5 40.5 45.5

Donation (RM)

Cumulativefreque

ncy

(c) (i) 35

(ii) 30 families donated RM 35 or less

3. (SPM 2004)

(a)


10 12 1 11

13 15 2 14

16 18 3 17

19 21 5 2022 24 6 23

25 27 9 26

28 30 4 29

31 - 33 2 32

(b) Mean =24965321

)232()429()926()623()520()317()214()111(

+

Statistics 48


49/55

=32

742

= 23.19

(c)

0

1

2

3

4

5

6

7

8

9

10

12.5 18.5 24.5 30.5

Mass (kg)

Frequency

9.5 12.5 15.5 18.5 21.5 24.5 27.5 30.5 33.5

(d) The modal class for the mass of suitcases is 25 27

6. (SPM 2006J)

(a)

Age ( Year) Midpoint Upper boundary Cumulative frequency

20 24 22 24.5 2

25 29 27 29.5 830 34 32 34.5 17

35 -39 37 39.5 27

40 - 44 42 44.5 34

45 49 47 49.5 38

50 - 54 52 54.5 40

(b) Mean =40

)522()474()427()3710()329()276()222(

=

40

10418829437028816244 +

=40

1450

= 36.25

5. SPM 2005N

(a)

Marks Midpoint Frequency

Statistics 49


50/55

20 24

25 29

30 34

35 39

40 44

45 49

50 54

22

27

32

37

42

47

52

5

7

8

10

6

4

2


(ii) Mean =24610875

)52()474()426()3710()328()277()225(

+

=42

1469

= 34.98 ( 2 d.p)

(c)

0

1

2

3

4

5

6

7

8

9

10

11

19.5 24.5 29.5 34.5 39.5 44.5 49.5 54.5

Mark

Frequency

19.5 24.5 29.5 34.5 39.5 44.5 49.5 54.5

Statistics 50


51/55

7. November 2006

a)


21 25 23 5

26 30 28 6

31 35 33 8

36 40 38 10

41 45 43 7

46 50 48 4

b) mean =23 5 28 6 33 8 38 10 43 7 48 4

40

+ + + + +

=1420

40= 35.5

c)

d) Any correct information that can be obtained from the frequency polygon, eg. 4 students

donated more than RM48.

Statistics 51


52/55

8. SPM June 2007

a)

Mass (kg) Frequency Midpoint Freq x midpoint

52 56 3 54.5 163.5

57 61 8 59.5 47662 66 6 64.5 387

67 71 3 69.5 208.5

72 76 11 74.5 819.5

77 81 5 79.5 397.5

36 2452

ii. Mean =sum of (frequency midpoint)

sum of data

2452

36

68.11

=

=c)

d) Between 72 76 kg of newspapers was collected over the most number of days (11 days)

Statistics 52


53/55

9. SPM Nov 2007

a) i) Modal class is 50 54 kgii)

Mass (kg) Midpoint Frequency Midpoint x

frequency

30 34 32 5 160

35 39 37 8 296

40 44 42 11 462

45 49 47 21 987

50 54 52 22 1144

55 59 57 10 570

60 64 62 3 186

80 3805

Mean =

3805

80

47.56

=

=b)

Boundaries Frequency Cumulative

frequency

29.5 0 0

34.5 5 5

39.5 8 13

44.5 11 24

49.5 21 45

54.5 22 67

59.5 10 77

64.5 3 80

c)

d) at 25%, 20 of the students have a mass of less than p = 43 kg

Statistics 53


54/55

10. SPM June 2008 Q14

a)

Class interval Frequency Midpoint Freq x midpoint

5 7 3 6 18

8 10 5 9 45

11 13 9 12 108

14 16 10 15 150

17 19 8 18 144

20 22 3 21 63

23 25 2 24 48

40 576

b)

576mean

40

14.4

=

=

c)

d) The modal class for the number of books read by these students is 14 16 books.

Statistics 54


55/55

11. SPM Nov 2008 Q14

a)

Class Interval Midpoint Frequency Midpoint x frequency

15 19 17 3 51

20 24 22 5 110

25 29 27 7 189

30 34 32 9 288

35 39 37 7 259

40 44 42 6 252

45 49 47 3 141

40 1290

b)

1290Mean =

40

= 32.25

c)

d) Th b f d i h id th RM34 f th t ll

Documents

Chapter 6 II Stastitic III ENHANCE (4)