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MATHEMATICS [5 marks sum]

Statistics · ICSE STD 9 (ICSE - English Medium). 15 questions.

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[5 marks sum]

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15 questions · timed · auto-graded

Question 15 Marks
Draw frequency polygons for each of the following frequency distribution:$(a)$ using histogram$(b)$ without using histogram
$C.I$ $5 -15$ $15 -25$ $25 -35$ $35 - 45$ $45-55$ $55-65$
$ƒ$ $8$ $16$ $18$ $14$ $8$ $2$
Answer
Using Histogram:
$C.I.$ $f$
$5 - 15$ $8$
$15 - 25$ $16$
$25 - 35$ $18$
$35 - 45$ $14$
$45 - 55$ $8$
$55 - 65$ $2$
Steps:
  1. Draw a histogram for the given data.
  2. Mark the mid$-$point at the top of each rectangle of the histogram drawn.
  3. Also, mark the mid$-$point of the immediately lower class$-$interval and mid$-$point of the immediately higher class$-$interval.
  4. Join the consecutive mid$-$points marked by straight lines to obtain the required frequency polygon.

Image
Without using Histogram:
$1$. Find the class$-$mark $($mid$-$value$)$ of each given class$-$interval.
$2.$. On a graph paper, mark class$-$marks along $X$-axis and frequencies along $Y-$axis.
$3.$ On this graph paper, mark points taking values of class$-$marks along the $X-$axis an values of their corresponding frequencies along the $Y-$axis.
$4.$ Draw line segments joining the consecutive points marked in step $(3)$ above.
$C.I.$ Class$-$mark $f$
$-5 - 5$ $0$ $0$
$5 - 15$ $10$ $8$
$15 - 25$ $20$ $16$
$25 - 35$ $30$ $18$
$35 - 45$ $40$ $14$
$45 - 55$ $50$ $8$
$55 - 65$ $60$ $2$
$65 - 75$ $70$ $0$

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Question 25 Marks
Draw frequency polygons for each of the following frequency distribution: $(a)$ using histogram$(b)$ without using histogram
$C.I$ $10 - 30$ $30 - 50$ $50 - 70$ $70 - 90$ $90 - 110$ $110 - 130$ $130 - 150$
$ƒ$ $4$ $7$ $5$ $9$ $5$ $6$ $4$
Answer
using histogram
$C.I$ $ƒ$
$10 - 30$ $4$
$30 - 50$ $7$
$50 - 70$ $5$
$70 - 90$ $9$
$90 - 110$ $5$
$110 - 130$ $6$
$130 - 150$ $4$
Steps:
    1. Draw a histogram for the given data.
    2. Mark the mid$-$point at the top of each rectangle of the histogram drawn.
    3. Also, mark the mid$-$point of the immediately lower class$-$interval and mid$-$point of the immediately higher class$-$interval.
    4. Join the consecutive mid$-$points marked by straight lines to obtain the required frequency polygon.
Image
Without using Histogram:Steps:
  1. Find the class$-$mark $($mid$-$value$) $of each given class-interval.
    classmark $=$ mid$-$value $=\frac{\text { Upper limit }+ \text { Lower limit }}{2}$
  2. On a graph paper, mark class-marks along $X-$axis and frequencies along $Y-$axis.
  3. On this graph paper, mark points taking values of class-marks along the $X-$axis and the values of their corresponding frequencies along $Y-$axis.
  4. Draw line segments joining the consecutive points marked in step $(3)$ above.
    $C.I.$ Class$-$mark $f$
    $-10 - 10$ $0$ $0$
    $10 - 30$ $20$ $4$
    $30 - 50$ $40$ $7$
    $50 - 70$ $60$ $5$
    $70 - 90$ $80$ $9$
    $90 - 110$ $100$ $5$
    $110 - 130$ $120$ $6$
    $130 - 150$ $140$ $4$
    $150 - 170$ $160$ $0$
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Question 35 Marks
The daily wages in a factory are distributed as follows:
Daily wages $($in $Rs.)$ $125 - 175$ $175 - 225$ $225 - 275$ $275 - 325$ $325 - 375$
Number of workers $4$ $20$ $22$ $10$ $6$
Draw a frequency polygon for this distribution.
Answer
Steps:
  1. Draw a histogram for the given data.
  2. Mark the mid$-$point at the top of each rectangle of the histogram drawn.
  3. Also, mark the mid$-$point of the immediately lower class$-$interval and mid$-$point of the immediately higher class-interval.
  4. Join the consecutive mid$-$points marked by straight lines to obtain the required frequency polygon.
The required frequency polygon is as follows:
Image
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Question 45 Marks
Construct a frequency polygon for the following data:
Class$-$Intervals $10 - 14$ $15 - 19$ $20 - 24$ $25 - 29$ $30 - 34$
Frequency $5$ $8$ $12$ $9$ $4$
Answer
The class intervals are inclusive. We will first convert them into the exclusive form.
Class$-$Interval Frequency
$9.5 - 14.5$ $5$
$14.5 - 19.5$ $8$
$19.5 - 24.5$ $12$
$24.5 - 29.5$ $9$
$29.5 - 34.5$ $4$
Steps:
  1. Draw a histogram for the given data.
  2. Mark the mid$-$point at the top of each rectangle of the histogram drawn.
  3. Also, mark the mid$-$point of the immediately lower class$-$interval and mid$-$point of the immediately higher class$-$interval.
  4. Join the consecutive mid$-$points marked by straight lines to obtain the required frequency polygon.
The required frequency polygon is as follows:
Image
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Question 55 Marks
Construct a combined histogram and frequency polygon for the following frequency distribution:
Class$-$Intervals $10 - 20$ $20 - 30$ $30 - 40$ $40 - 50$ $50 - 60$
Frequency $3$ $5$ $6$ $4$ $2$
Answer
Steps:$1.$ Draw a histogram for the given data.
$2.$ Mark the mid$-$point at the top of each rectangle of the histogram drawn.
$3.$ Also, mark the mid$-$point of the immediately lower class$-$interval and mid$-$point of the immediately higher class$-$interval.
$4$. Join the consecutive mid$-$point marked by straight lines to obtain the required frequency polygon.
$5$. The require combined histogram and frequency polygon are shown in the following figure:
Image
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Question 65 Marks
Construct a frequency polygon for the following distribution:
Class$-$intervals $0-4$ $4 - 8$ $8 - 12$ $12 - 16$ $16 - 20$ $20 - 24$
Frequency $4$ $7$ $10$ $15$ $11$ $6$
Answer
The frequency polygon is shown in the following figure
Image
Steps:$(i)$ Drawing a histogram for the given data.
$(ii)$ Marking the mid$-$point at the top of each rectangle of the histogram drawn.
$(iii)$Also, marking mid$-$point of the immediately lower class$-$interval and mid$-$point of the immediately higher class$-$interval.
$(iv)$ Joining the consecutive mid$-$points marked by straight lines to obtain the required frequency polygon.
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Question 75 Marks
Construct a cumulative frequency distribution table from the frequency table given below:
Class Interval Frequency
$0 -8$ $9$
$8 - 16$ $13$
$16 - 24$ $12$
$24 - 32$ $7$
$32 - 40$ $15$
Answer
The cumulative frequency distribution table is
$C.I$ $c.f$
$0 -8$ $9$
$8 - 16$ $22$
$16 - 24$ $34$
$24 - 32$ $41$
$32 - 40$ $56$
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Question 85 Marks
Use the table given below to find:
$(a) $The actual class limits of the fourth class.$(b)$ The class boundaries of the sixth class.$(c)$ The class mark of the third class.$(d)$ The upper and lower limits of the fifth class.$(e)$ The size of the third class.
Class Interval Frequency
$30 - 34$ $7$
$35 - 39$ $10$
$40 - 44$ $12$
$45 - 49$ $13$
$50 - 54$ $8$
$55 - 59$ $4$
Answer
$(a)$ The actual class limit of the fourth class will be $44.5 - 49.5.$
$(b)$ The class boundaries of the sixth class will be $54.5 - 59.5$
$(c)$ The class mark of the third class will be the average of the lower bound and the upper bound of the interval. Therefore the class mark will be: $\frac{40+44}{2}=42$
$(d)$ The upper and lower limit of the fifth class is $54$ and $50$ respectively.
$(e)$ The size of the third class will be $44 - 40 + 1 = 5.$
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Question 95 Marks
Find the actual lower and upper-class limits and also the class marks of the classes:$1.1 - 2.0, 2.1 -3.0$ and $3.1 - 4.0.$
Answer
In the case of frequency $1.1 - 2.0$ the lower-class limit is $1.05,$ upper$-$class limit is $2.05$ and classmark.
is $1.05 - 2.05$
$\frac{1.05+2.05}{2}$
$=\frac{3.10}{2} $
$ =1.55$
In the case of frequency $2.1 - 3.0$ the lower$-$class limit is $2.05,$ upper$-$class limit is $3.05$ and classmark.
is $2.05 - 3.05$
$\frac{2.1+3.0}{2} $
$ =\frac{5.10}{2} $
$=2.55$
In the case of frequency $3.1 - 4.0$ the lower class limit is $3.05,$ the upper class limit is $4.05$ and classmark.
is $3.05 - 4.05$
$\frac{3.1+4.0}{2} $
$=\frac{7.10}{2}$
$=3.55$
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Question 105 Marks
Find the actual lower class limits, upper$-$class limits and the mid$-$values of the classes:$10 - 19, 20 - 29, 30 - 39$ and $40 - 49.$
Answer
$C. I.$ Exclusive $C. I.$
$10 - 19$ $9.5 - 19.5$
$20 - 39$ $19.5 - 29.5$
$30 - 39$ $29.5 - 39.5$
$40 - 49$ $39.5 - 49.5$
In case of frequency $9.5 - 19.5$ the lower class limit is $9.5$, the upper class limit is $19.5$ and mid$-$value is
$\frac{9.5+19.5}{2}=14.5$
In case of frequency $19.5 - 29.5$ the lower class limit is $19.5$, the upper class limit is $29.5$ and mid$-$value is
$\frac{19.5+29.5}{2}=24.5$
In case of frequency $29.5 - 39.5$ the lower class limit is $29.5$, the upper class limit is $39.5$ and mid$-$value is
$\frac{29.5+39.5}{2}=34.5$
In case of frequency $39.5 - 49.5$ the lower class limit is $39.5$, the upper class limit is $49.5$ and mid$-$value is
$\frac{39.5+49.5}{2}=44.5$
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Question 115 Marks
The marks of $24$ candidates in the subject mathematics are given below:
$45$ $48$ $15$ $23$ $30$ $35$ $40$ $11$
$29$ $0$ $3$ $12$ $48$ $50$ $18$ $30$
$15$ $30$ $11$ $42$ $23$ $2$ $3$ $44$
The maximum marks are $50.$ Make a frequency distribution taking class intervals $0 - 10, 10-20,\dots .......$
Answer
The frequency table for the given distribution is
Marks Tally Marks Frequency
$0 - 10$ $||||$ $4$
$10 - 20$ $6$
$20 - 30$ $|||$ $3$
$30 - 40$ $||||$ $4$
$40 - 50$ $7$
In this frequency distribution, the marks $30$ are in the class of interval $30 - 40$ and not in $20 - 30.$
Similarly, marks $40$ are in the class of interval $40 - 50$ and not in $30 - 40.$
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Question 125 Marks
Given below are the marks obtained by $30$ students in an examination:
$08$ $17$ $33$ $41$ $47$ $23$ $20$ $34$
$09$ $18$ $42$ $14$ $30$ $19$ $29$ $11$
$36$ $48$ $40$ $24$ $22$ $02$ $16$ $21$
$15$ $32$ $47$ $44$ $33$ $01$    
Taking class intervals $1 - 10, 11 - 20,,\dots ....., 41 - 50;$make a frequency table for the above distribution.
Answer
The frequency table for the given distribution is
Marks Tally Marks Frequency
$1 - 10$ $4$
$11 - 20$ $8$
$21 - 30$ $6$
$31 - 40$ $6$
$41 - 50$ $6$
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Question 135 Marks
Fill in the blank in the following table:
Class interval Frequency Cumulative Frequency
$25 - 34$ $......$ $15$
$35 - 44$ $......$ $28$
$45 - 54$ $21$ $......$
$55 - 64$ $16$ $......$
$65 - 74$ $......$ $73$
$75 - 84$ $12$ $......$
Answer
Class interval Frequency Cumulative Frequency
$25 - 34$ $15$ $15$
$35 - 44$ $13$ $28$
$45 - 54$ $21$ $49$
$55 - 64$ $16$ $65$
$65 - 74$ $8$ $73$
$75 - 84$ $12$ $85$
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Question 145 Marks
Construct the frequency distribution table from the following cumulative frequency table:
Ages No. of students
Below $4$ $0$
Below $7$ $85$
Below $10$ $140$
Below $13$ $243$
Below $16$ $300$
$(i) $State the number of students in the age group $10 - 13.(ii)$ State the age$-$group which has the least number of students.
Answer
The frequency distribution table is
$C. I$ $c.f$
$4 - 7$ $85$
$7 - 10$ $55$
$10 - 13$ $103$
$13 - 16$ $57$
$(i)$The number of students in the age group is $10 -13$ is $103.$
$(ii)$The age group which has the least number of students is $7 - 10$
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Question 155 Marks
Construct a frequency table from the following data:
Marks No. of students
less than $10$ $6$
less than $20$ $15$
less than $30$ $30$
less than $40$ $39$
less than $50$ $53$
less than $60$ $70$
Answer
The frequency table is
$C. I$ $c.f$
$0 - 10$ $6$
$10 - 20$ $9$
$20 - 30$ $15$
$30 - 40$ $9$
$40 - 50$ $14$
$50 - 60$ $17$
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[5 marks sum] - MATHEMATICS STD 9 Questions - Vidyadip