Exercise 31.2
Page no 31.36
Type 1
Question 1
Find the mean and variance of the following :
(i) $6,7,10,12,13,4,8,12$,
(ii) $65,58,68,44,48,45,60,62,60,50$
(iii) $8,9,11,13,14,15,21$
(iv) $11,14,15,17,18$
(v) $2,4,5,6,8,17$
Question 2
Find the mean, variance and standard deviation of the following marks scored by 10 students : $45,70,62,60,50,48,67,34,65,58$
Question 3
The mean and variance of 8 observations are 9 and $9.25$ respectively. If six of the observations are $6,7,10,12,12$ and 13 , find the remaining two observations.
Question 4
The mean and variance of 7 observations are 8 and 16 , respectively. If five of the observations are $2,4,10,12,14$, find the remaining two observations.
Question 5
Find the possible values of $x$ if standard deviation of the numbers $2.3 .2 \mathrm{x}$ and 11 is $3.5$.
Question 6
The variance of 20 observations is 5 . If each observation is multiplied by 2 , find the new variance of the resulting observations.
Question 7
The mean and standard deviation of 6 observations are 8 and 4 , respectively, If each observation is multiplied by 3 , find the new mean and new standard deviation of the resulting observations.
Question 8
Mean and standard deviation of 100 items are 50 and 4 respectively. Find the sum of all the items and also the sum of the squares of the items.
Question 9
While calculating the mean and variance of ten readings, a student wrongly used the figure 52 for the correct figure 25 . He obtained the mean and variance as 45 and 16 respectively. Find the correct mean and variance.
Question 10
The mean and standard deviation of 200 items are found to be 60 and 20 respectively. If at the time of calculation, two items were wrongly taken as 3 and 67 instead of 18 and 17, respectively, find the correct mean and the standard deviation.
Question 11
If for a distribution $\Sigma(x-5)=3, \Sigma(x-5)^{2}=43$ and total number of items $N=18$, find the mean and the standard deviation.
[HOTS]
Type 2
Question 12
$\begin{aligned}&\text { Find the mean and standard deviation of the following distribution : }\\&\begin{array}{|l|c|c|c|c|c|c|c|}\hline \text { Marks } & 10 & 20 & 30 & 40 & 50 & 60 & 70 \\\hline \text { No. of students } & 1 & 5 & 12 & 22 & 17 & 9 & 4 \\\hline\end{array}\end{aligned}$
Page no 31.37
Question 13
$\begin{aligned}&\text { Find the mean and standard deviation for the following data : }\\&\begin{array}{|c|c|c|c|c|c|c|c|}\hline x_{i} & 92 & 93 & 97 & 98 & 102 & 104 & 109 \\\hline f_{1} & 3 & 2 & 3 & 2 & 6 & 3 & 3 \\\hline\end{array}\end{aligned}$
Question 14
(i) Find the variance and standard deviation for the following data :
$\begin{aligned}&\begin{array}{|c|c|c|c|c|c|c|c|}\hline x_{i} & 4 & 8 & 11 & 17 & 20 & 24 & 32 \\\hline f_{i} & 3 & 5 & 9 & 5 & 4 & 3 & 1 \\\hline\end{array}\\&\text { (ii) }\\&\begin{array}{|c|c|c|c|c|c|}\hline x_{j} & 3 & 8 & 13 & 18 & 23 \\\hline f_{i} & 7 & 10 & 15 & 10 & 6 \\\hline\end{array}\end{aligned}$
Type 3
Question 15
The following frequency table given the ages of a group of 50 children invited to a birthday party. Find the mean and standard deviation of the distribution:
$\begin{array}{|l|c|c|c|c|c|}\hline \text { Age (in years) } & 5-7 & 7-9 & 9-11 & 11-13 & 13-15 \\\hline \text { Frequency } & 16 & 13 & 10 & 6 & 5 \\\hline\end{array}$
Question 16
$\begin{aligned}&\text { Find the S.D. of the following data : }\\&\begin{array}{|l|c|c|c|c|c|}\hline \text { Class interval } & 25-35 & 35-45 & 45-55 & 55-65 & 65-75 \\
\hline \text { Frequency } & 21 & 20 & 16 & 25 & 18 \\\hline\end{array}\end{aligned}$
Question 17
$\begin{aligned}&\text { (i) Find the mean, variance and S.D. using short-cut method : }\\&\begin{array}{|l|c|c|c|c|c|c|c|c|c|}\hline \begin{array}{l}\text { Class } \\\text { interval }\end{array} & 10-20 & 20-30 & 30-40 & 40-50 & 50-60 & 60-70 & 70-80 & 80-90 & 90-100 \\\hline \text { Frequency } & 3 & 4 & 7 & 7 & 15 & 9 & 6 & 6 & 3 \\\hline\end{array}\end{aligned}$
$\begin{aligned}&\text { (ii) Find the mean, variance and S.D. using short-cut method : }\\&\begin{array}{|l|c|c|c|c|c|c|c|c|c|}\hline \begin{array}{l}\text { Height } \\\text { in cm }\end{array} & 70-75 & 75-80 & 80-85 & 85-90 & 90-95 & 95-100 & 100-105 & 105-110 & 110-115 \\\hline \begin{array}{l}\text { No. of } \\\text { children }\end{array} & 3 & 4 & 7 & 7 & 15 & 9 & 6 & 6 & 3 \\\hline\end{array}\end{aligned}$
Question 18
$\begin{aligned}&\text { (i) Find the mean and variance of the following distribution : }\\&\begin{array}{|l|c|c|c|c|c|c|c|}\hline \text { Classes } & 0-30 & 30-60 & 60-90 & 90-120 & 120-150 & 150-180 & 180-210 \\\hline \text { Frequencies } & 2 & 3 & 5 & 10 & 3 & 5 & 2 \\\hline\end{array}\end{aligned}$
$\begin{aligned}&\text { (ii) Find the mean and variance of the following distribution : }\\&\begin{array}{|l|c|c|c|c|c|}\hline \text { Classes } & 0-10 & 10-20 & 20-30 & 30-40 & 40-50 \\\hline \text { Frequencies } & 5 & 8 & 15 & 16 & 6 \\\hline\end{array}\end{aligned}$
Question 19
9. Let $\bar{x}$ and $\bar{y}$ be the arithmetic means for the variables $x$ and $y$ respectively and $6 x, 6 y$ be their standard deviations.
If the variables $x$ and $y$ are related by $m y=n x+k$, where $m, n, k$ are constants show that
(i) $m \bar{y}=n \bar{x}+k$
(ii) $m^{2} \sigma_{y}^{2}=n^{2} \sigma_{z}^{2}$
Page no 31.38
Question 20
The marks obtained by 200 pupils of a class in their annual examination are given in the following table. Find the mean and standard deviation of these marks :
$\begin{array}{|l|c|c|c|c|c|c|c|c|c|c|}\hline \text { Marks } & 1-10 & 11-20 & 21-30 & 31-40 & 41-50 & 51-60 & 61-70 & 71-80 & 81-90 &91-100 \\\hline \begin{array}{l}\text { No. of } \\\text { students }\end{array} & 2 & 3 & 10 & 19 & 30 & 47 & 54 & 28 & 5 & 2 \\\hline\end{array}$
Question 21
$\begin{aligned}&\text { Find the S.D. of the following data : }\\&\begin{array}{|l|c|c|c|c|c|c|c|c|}\hline \text { Age (under) } & 10 & 20 & 30 & 40 & 50 & 60 & 70 & 80 \\\hline \text { No. of students } & 15 & 30 & 53 & 75 & 100 & 110 & 115 & 125 \\\hline\end{array}\end{aligned}$
Type 4
Question 22
If $N=10, \bar{x}=12, \Sigma x^{2}=1530$, find the coefficient of variation.
Question 23
For a distribution, the coefficient of variation is $22.5 \%$ and the value of the arithmetic average is $7.5$. Find out the value of standard deviation.
Question 24
Coefficient of variation of two series are $75 \%$ and $90 \%$ and their standard deviations 15 and 18 respectively. Find their mean
Question 25
The following values are calculated in respect of heights and weights of the students of a section of class $\mathrm{XI}$.
$\begin{array}{|c|c|c|}\hline & \text { Height } & \text { Weight } \\\hline \text { Mean } & 162.6 \mathrm{~cm} & 52.36 \mathrm{~kg} \\\hline \text { Variance } & 127.69 \mathrm{~cm}^{2} & 23.136 \mathrm{~kg} \\\hline\end{array}$
Can we say that the weights show greater variation than the heights ?
Question 26
The mean and standard deviations of heights and weights of 50 students of a class are as follows :
$\begin{aligned}&\begin{array}{|l|c|c|}\hline & \text { Weights } & \text { Heights } \\\hline \text { Mean } & 63.2 \mathrm{~kg} & 63.2 \text { inch } \\\hline \text { Standard deviation } & 5.6 \mathrm{~kg} & 11.5 \text { inch } \\\hline\end{array}\\&\text { Which shows more variability, heights or weights? }\end{aligned}$
Question 27
An analysis of monthly wages paid to workers in two firms $A$ and $B$. belonging to the same industry, gives the following results :
$\begin{array}{|l|c|c|}\hline \text { No. of wage camers } & \text { Firm A } & \text { Firm B } \\\hline \text { Mean of monthly wages } & 586 & 648 \\\hline \text { Variance of the distribution of wages } & \text { Rs. } 5253 & \text { Rs. } 5253 \\\hline\end{array}$
(i) Which firm $A$ or $B$ pays out larger amount as monthly wages ?
(ii) Which firm $A$ or $B$, shows greater variability in individual wages?
Page no 31.39
Question 28
Following are the marks obtained out of 100 marks by two students in 10 tests. Find who is more intelligent and who is more consistent.
$\begin{array}{|l|l|l|l|l|l|l|l|l|l|l|}\hline A & 25 & 50 & 45 & 30 & 70 & 42 & 36 & 48 & 35 & 60 \\\hline B & 10 & 70 & 50 & 20 & 95 & 55 & 42 & 60 & 48 & 80 \\\hline\end{array}$
Question 29
Goals scored by teams $A$ and $B$ in a football season are as follows. Find which team is more consistent in its performance :
$\begin{array}{|c|c|c|c|c|c|c|}\hline \multicolumn{1}{|c|}{\begin{array}{c}\text { Number of goals scored in one } \\\text { match }\end{array}} & 0 & 1 & 2 & 3 & 4 \\\hline \multirow{2}{*}{\begin{array}{c}\text { No. of matches } \\\text { played }\end{array}} & \text { Team } A & 27 & 9 & 8 & 5 & 4 \\\cline { 2 - 6 } & \text { Team } B & 17 & 9 & 6 & 5 & 3 \\\hline\end{array}$
Question 30
From the prices of shares $X$ and $Y$ below, find out which is more stable in value :
$\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|}\hline X & 35 & 54 & 52 & 53 & 56 & 58 & 52 & 50 & 51 & 49 \\\hline Y & 108 & 107 & 105 & 105 & 106 & 107 & 104 & 103 & 104 & 101 \\\hline\end{array}$
Question 31
From the data given below state which group is more variable , A or B ?
$\begin{array}{|l|c|c|c|c|c|c|c|}\hline \text { Marks } & 10-20 & 20-30 & 30-40 & 40-50 & 50-60 & 60-70 & 70-80 \\\hline \text { Group } A & 9 & 17 & 32 & 33 & 40 & 10 & 9 \\\hline \text { Group } B & 10 & 20 & 30 & 25 & 43 & 15 & 7 \\\hline\end{array}$
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