4 Mark Question

Question 14 Marks

The mean of marks scored by 100 students was found to be 40. Later on it was discovered that a score of 53 was misread as 83. Find the correct mean.

Answer

We have,
N = The number of observations = 100, Mean = 40
$\text{Mean}=\frac{\text{Sum of the observations}}{\text{Total number of observations}}$
$\Rightarrow40=\frac{\text{Sum of the observations}}{100}$
Sum of the observations = 40 x 100
Thus, the incorrect sum of the observations = 40 x 100 = 4000
Now,
The correct sum of the observations = Incorrect sum of the observations – Incorrect observation + Correct observation.
The correct sum of the observations = 4000 - 83 + 53
The correct sum of the observations = 4000 - 30 = 3970
$\therefore\text{Correct mean}=\frac{\text{Correct sum of the observations}}{\text{Number of observations}}=\frac{3970}{100}=39.7$

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Question 24 Marks

Heights of 25 children (in cm) in a school are as given below:
168, 165, 163, 160, 163, 161, 162, 164, 163, 162, 164, 163, 160, 163, 163, 164, 163, 160, 165, 163, 162
What is the mode of heights?
Also, find the mean and median.

Answer

Arranging the data in tabular form, we get:

Here, $n =25$
Median $=$ value of $\frac{ n +1}{2}$ th observation $=$ value of the $13^{\text {th }}$ observation $=163 cm$.
Here, clearly, 163 cm occurs the most number of times. Therefore, the mode of the given data is 163 cm .
$\text { Mode }=3 \text { Median }-2 \text { Mean }$
$\Rightarrow 163=3 \times 163-2 \text { Mean }$
$\Rightarrow 2 \text { Mean }=326$
$\Rightarrow \text { Mean }=163 cm .$

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Question 34 Marks

The mean of the following data is 20.6. Find the value of p.

x:	10	15	p	25	35
f:	3	10	25	7	5

Answer

Calculation of mean:

$x_i$	$f_i$	$x_i f_i$
10	3	30
15	10	150
p	25	25p
25	7	175
35	5	175
Total	$\sum\text{f}_\text{i}=50$	$\sum\text{f}_\text{i}\text{x}_\text{i}=530\ +\ 25\text{p}$

We have,
$\therefore\text{Mean}=\frac{\sum\text{f}_\text{i}\text{x}_\text{i}}{\sum\text{f}_\text{i}}$
$\Rightarrow20.6=\frac{530+25\text{p}}{50}$
$\Rightarrow530 + 25\text{p} = 20.6 × 50$
$\Rightarrow25\text{p} = 1030 - 530$
$\Rightarrow\text{p}=\frac{500}{25}$
$\Rightarrow\text{p} = 20$

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Question 44 Marks

If the mean of the following data is 15, find p.

x:	5	10	15	20	25
f:	6	p	6	10	5

Answer

Calculation of mean:

$x_i$	$f_i$	$x_if_i$
5	6	30
10	p	10p
15	6	90
20	10	200
25	5	125
Total	$\sum\text{f}_\text{i}=27+\text{p}$	$\sum\text{f}_\text{i}\text{x}_\text{i}=445+10\text{p}$

We have,
$\sum\text{f}_\text{i}=27+\text{p},\sum\text{f}_\text{i}\text{x}_\text{i}=445+10\text{p}$
$\therefore\text{Mean}=\frac{\sum\text{f}_\text{i}\text{x}_\text{i}}{\sum\text{f}_\text{i}}$
$\Rightarrow20.6=\frac{445+10\text{p}}{27+\text{p}}$
$\Rightarrow445 + 10\text{p} = 405 + 15\text{p}$
$\Rightarrow5\text{p} = 445 - 405$
$\Rightarrow\text{p}=\frac{40}{5}$
$\Rightarrow\text{p} = 8$

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Question 54 Marks

The mean of 200 items was 50. Later on, it was discovered that the two items were misread as 92 and 8 instead of 192 and 88. Find the correct mean.

Answer

N = Number of observations = 200
$\text{Mean}=\frac{\text{Sum of the observations}}{\text{Number of observations}}$
$\Rightarrow50=\frac{\text{Sum of the observations}}{200}$
⇒ Sum of the observations = 50 × 200 = 10,000.
Thus, the incorrect sum of the observations = 50 × 200
Now,
The correct sum of the observations = Incorrect sum of the observations - Incorrect observations + Correct observations.
⇒ Correct sum of the observations = 10,000 - (92 + 8) + (192 + 88)
⇒ Correct sum of the observations = 10,000 - 100 + 280
⇒ Correct sum of the observations = 9900 + 280
⇒ Correct sum of the observations = 10,180

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