[5 marks sum]

Question 15 Marks

40 students enter for a game of shot-put competition. The distance thrown (in metres) is recorded below:

Distance (in m)	12 - 13	13 - 14	14 - 15	15 - 16	16 - 17	17 - 18	18 - 19
Number of Students	3	9	12	9	4	2	1

Use a graph paper to draw an ogive for the above distribution.
Use a scale of 2cm = 1 m on one axis and 2cm = 5 students on the other axis.
Hence using your graph find:
(a) the median
(b) upper quartile
(c) the number of students who cover a distance which is above $16 \frac{1}{2} \mathrm{~m}$.

Answer

We first prepare a table as below:

Distance (in m)	No. of students	Cumulative Frequency
12 – 13	3	3
13 – 14	9	12
14 – 15	12	24
15 – 16	9	33
16 – 17	4	37
17 – 18	2	39
18 - 19	1	40

We plot the points $(12,0),(13,3),(14,12),(15,24),(16,33),(17,37),(18,39)$ and $(19,40)$.
Joint these points by a free hand curve to get the ogive as shown.

(a) Here, $n=40$
$\Rightarrow \frac{n}{2}=20$
From the graph, median $=14.6 m$
(b) $\frac{3 n}{4}=\frac{3 \times 40}{4}=30$
From the graph $Q _3=15.6 m$
(c) Number of students who cover a distance which is above $16 \frac{1}{2} m=40-35=5$.

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Question 25 Marks

Use Graph paper for this question.
A survey regarding height (in cm) of 60 boys belonging to Class 10 of a school was conducted. The following data was recorded:

Height (in cm)	135 - 140	140 - 145	145 - 150	150 - 155	155 - 160	160 - 165	165 - 170
No. of boys	4	8	20	14	7	6	1

Taking 2cm = height of 10 cm along one axis and 2cm = 10 boys along the other axis, draw an ogive for the above distribution. Use the graph to estimate the following:
(a) the median
(b) lower Quartile
(c) if above 158 cm is considered as the tall boys of the class, find the number of boys in the class who are tall.

Answer

We prepare a table as below :

Height (in cm)	No. of boys	Cumulative Frequency
135 – 140	4	4
140 – 145	8	12
145 – 150	20	32
150 – 155	14	46
155 – 160	7	53
160 – 165	6	59
165 - 170	1	60

Plot the points, $(135,0),(140,4),(145,12),(150,32),(155,46),(160,53),(165,59),(170,60)$.
Join these points by a free hand curve to get the ogive as shown.

Here, $n=60$
So, $\frac{n}{2}=30$
From the graph median $=149 cm$
(b) $\frac{n}{4}=\frac{60}{4}=15$
From the graph, $Q _1=146 cm$
(c) Number of boys in the class who are tall $=60-51=9 cm$.

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Question 35 Marks

The daily wages of 80 workers in a project are given below:

Wages in (₹)	400 - 450	450 - 500	500 - 550	550 - 600	600 - 650	650 - 700	700 - 750
No. of workers	2	6	12	18	24	13	5

Use a graph paper to draw an ogive for the above distribution. (Use a scale of 2cm = ₹50 on x-axis and 2cm = 10 workers on y-axis). Use your ogive to estimate:
(a) the median wage of the workers.
(b) the lower quartile wage of workers.
(c) the number of workers who earn more than ₹625 daily.

Answer

We prepare a table as below :

Wages (in ₹)	Upper Limit	No. of workers	Cumulative Frequency
400 – 450	450	2	2
450 – 500	500	6	8
500 – 550	550	12	20
550 – 600	600	18	38
600 – 650	650	24	62
650 – 700	700	13	75
700 – 750	750	5	80

Plot the points $(400,0),(450,2),(500,8),(550,20),(600,38),(650,62),(700,75),(750,80)$.
Joint these points by a free hand curve to get the ogive as shown.

(a) $n=80$
$\Rightarrow \frac{n}{2}=40$
From the graph, median $=600$
(b) Lower quartite, $Q _1=\frac{80}{4}=20$ th observation $=550$
(c) Number of workers who earn more than $₹ 625=80-51=29$.

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Question 45 Marks

The table shows the distribution of the scores obtained by 160 shooters in a shooting competition. Use a graph sheet and draw an ogive for the distribution. (Take 2 cm 10 scores on the x-axis and 2 cm 20 shooters on the y-axis).

Scores	0 - 10	10 - 20	20 - 30	30 - 40	40 - 50	50 - 60	60 - 70	70 - 80	80 - 90	90 - 100
No. of shooters	9	13	20	26	30	22	15	10	8	7

Use your graph to estimate the following:
(a) The median
(b) The interquartile range
(c) The number of shooters who obtained a score of more than 85%.

Answer

We prepare a table as below:

Scores	No. of Shooters	Cumulative Frequency (c.f.)
0 – 10	9	9
10 – 20	13	22
20 – 30	20	42
30 – 40	26	68
40 – 50	30	98
50 – 60	22	120
60 – 70	15	135
70 – 80	10	145
80 – 90	8	153
90 – 100	7	160

Now, we plot the points $(0,0),(10,9),(20,22),(30,42),(40,68),(50,98),(60,120),(70,135),(80,145),(90,153),(100, 160).$
Join these points by a free hand curve to get an ogive, as shown.

$\begin{array}{ll}\text {(a) } n=160 \\ \Rightarrow \frac{n}{2}=80\end{array}$
From the graph, Median $=44$
(b) Lower quartile $Q _1=\frac{n}{4}$ th observation
$=\frac{160}{4}=40 \text { th observation } \Rightarrow Q_1=30$
Upper quartile $Q_3=\left(\frac{3 n}{4}\right)$ th observation $=\frac{3 \times 160}{4}=120$ th observation $\Rightarrow Q_3=60$
Inter-quartile range $= Q _3- Q _1=60-30=30$
(c) Number of shooters getting more than $85 \%$ score
$=160-149=11$.

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Mathematics [5 marks sum]

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