Mathematics - Statistics
Exam Duration: 45 Mins Total Questions : 30
The mean height of 25 male workers in a factory is 161cm and mean height of 35 female workers in the same factory is 158 cm. The combined mean height of 60 workers in the factory is
- (a)
159.25
- (b)
159.5
- (c)
159.75
- (d)
158.75
Combined mean height = \({25X161+35X158\over60}=159.25\)
In an english learning classes, there are 8 men, 7 women and 5 children whose mean ages separately are respectively 24,20 and 6 yr. The mean age of English learning classes candidate is
- (a)
18.0
- (b)
18.1
- (c)
18.2
- (d)
18.3
If g1 and g2 are the geometric means of two series of n1 and n2 items. Then, the GM of the series obtained on combining is
- (a)
\({ [(g_{ 1 })^{ n_{ 1 } }(g_{ 2 })^{ n_{ 2 } }] }^{ \frac { 1 }{ n_{ 1 }+n_{ 2 } } }\)
- (b)
\((g_1g_2)^{n_1\over{n_1+n_2}}\)
- (c)
\((g_1g_2)^{n_1\over{n_1+n_2}}\)
- (d)
\((g_1g_2)^{n_1n_2\over{n_1+n_2}}\)
Suppose g1=(X1X2....Xn1)1/n1 and g2=(y1y2.....yn2)1/n2
Let g is the GM of the combined series.
Now, \(g=([X_{ 1 }.X_{ 2 }.X_{ 3 }...X_{ { n }_{ 1 } }]\times [y_{ 1 }.y_{ 2 }.y_{ 3 }...y_{ n_{ 2 } }])^{ \frac { 1 }{ n_{ 1 }+n_{ 2 } } }\)
=\([(g_{ 1 })^{ n_{ 1 } }.(g_{ 2 })^{ n_{ 2 } }]^{ 1/(n_{ 1 }+n_{ 2 }) }\)
The mean deviation and coefficient of mean deviation about the median from the data of weight (in kg) 54,50,40,42,51,45,47,55,57 is
- (a)
4.5, 0.0900
- (b)
4.78, 0.0956
- (c)
3.8, 0.0056
- (d)
4.96, 0.0946
Increasing order of magnitude are 40,42,45,47,50,51,54,55,57
Number of terms, n=9
Median = \(({9+1\over2})\)th term =5th term =50kg [from given data]
Mean deviation from median = \(43\over9\)=4.78kg
Coefficient of mean deviation with the help of median = \({MD\over Median}={4.78\over50}={0.0956}\)
Find the mean deviation about the mean of the following data.
Size | 1 | 3 | 5 | 7 | 9 | 11 | 13 | 15 |
---|---|---|---|---|---|---|---|---|
Frequency | 3 | 3 | 4 | 14 | 7 | 4 | 3 | 4 |
- (a)
2.95
- (b)
3.24
- (c)
4
- (d)
2
The SD of a variate x is \(\sigma\). The SD of the variate \(ax+b\over c\), where a,b and c are constants is
- (a)
\(({a\over c})\sigma\)
- (b)
\(\begin{vmatrix} \frac { a }{ c } \end{vmatrix}\sigma \)
- (c)
\(({a^2\over C^2})\sigma\)
- (d)
None of these
If the standard deviation of y1,y2,...yn is 3.5, then the standard deviation of -2y1-3,-2y2-3,...,-2yn-3 is
- (a)
-7
- (b)
9
- (c)
7
- (d)
2.45
Let new observation be X1,X2,...,Xn i.e.,
Xi=-2yi-3
Yi=\(-{1\over2}x_i-{3\over2}\)
\(\sigma_y=\begin{vmatrix}{-1/2}\end{vmatrix}\sigma_x\)
\(\sigma_x=2\sigma_y\)=2X3.5=7
The means of two samples of size 200 and 300 were found to be 25 and 10, respectively. Their standard deviations were 3 and 4, respectively. The variance of the combined sample of size 500 is
- (a)
64
- (b)
65.2
- (c)
67.2
- (d)
642
The sum of squares of deviations taken from mean 50 is 250. The coeficient of variation is
- (a)
10%
- (b)
40%
- (c)
50%
- (d)
None of these
Which of the following statement regarding arithmetic mean is true?
- (a)
When observation is increased or decreased (by a constant), mean is also increased or decreased by a same constant
- (b)
When observation is increased (by a constant), mean will decreased by a same constant
- (c)
When observation is decreased (by a constant), mean will increased by a same constant
- (d)
None of the above
According to definition of the arithmetic mean, when observation is increased or decreased (by a constant), then mean is also increased or decreased by a same constant.
Which of the following data is correct about the standard deviation?
- (a)
standard deviation can be calculated as positive or negative square root of squared deviation
- (b)
Standard deviation is calculated as positive square root of arithmetic mean of squared deviation only
- (c)
standard deviation is independent of change of scales
- (d)
None o the above
Standard deviation is calculated as positive square root of arithmetic mean of squares of deviation. Also, it is depend on change of scales.
Which of the following is a correct statement?
- (a)
Standard deviation is depend on change of origin.
- (b)
Standard deviation is depend on change of scales
- (c)
Median is independent of change of origin
- (d)
Mean is independent of change of scales
Effect on average and dispersion on change of origin and scale.
Change of origin | Change of scales | |
---|---|---|
Mean |
Dependent Dependent Dependent Not dependent Not dependent |
Dependent |
A sample of 90 values has standard deviation 3 and their mean is 55. A second sample of 110 values has mean 60 and its standard deviation is 2. The combined variance is equal to
- (a)
12.44
- (b)
13.24
- (c)
16.42
- (d)
13.65
Consider any set of 201 observations x1,x2,...x200,x201. It is given that x1<x2<,,,,<x200<x201. Then, the mean deviation about a point k is minimum, when k is equal to
- (a)
(x1+x2+...+x200+x201)/201
- (b)
x1
- (c)
x101
- (d)
x201
Given that, x1<x2<x3<...<x201
Hence, median of the given observation =(\(201+1\over2\))th item=X101
Now, as we know deviation will be minimum, if it is taken from the median.
Hence, mean deviation will be minimum, if k=X101
If the mean and standard deviation of the marks of 200 candidates of IIT entrance test were found to be 40 and 15, respectively. Later, it was wrongly read as 50. Then, the correct mean and standard deviation are
- (a)
39.95,14.98
- (b)
40.2,14.29
- (c)
30.9,15.9
- (d)
29.32,13.29
The average income of male employees in a financial sector of company 'A' was Rs.520 and that of emales was Rs.420. The mean income of all the employees was Rs.500. The percentage of male employees is
- (a)
50%
- (b)
80%
- (c)
40%
- (d)
20%
Life of bulbs produced by two actories A and B are given below
Length of life (in hours) |
Fatory A (number of bulbs) |
Factory B (Number of bulbs) |
---|---|---|
550-650 650-750 750-850 850-950 950-1050 |
10 22 52 20 16 |
8 60 24 16 12 |
Total | 120 | 120 |
The bulbs of which factory are more consistent from the point of view of length of life?
- (a)
A=B
- (b)
A>B
- (c)
B>A
- (d)
\(B\ge A\)
The mean and standard deviation of 20 observations are found to be 10 and 2, respectively. On rechecking, it was incorrect.
Match the ollowing cases in column I with their corresponding values in Column II and choose the correct option from the codes given below.
Column I | Column II |
---|---|
A. Correct mean, if wrong is omitted | p. 10.2 |
B. Correct standard deviation, i wrong item is omitted | q. 10.10 |
C. Correct mean, if it is replaced by 12 | r. 1.98 |
D. Correct standard deviation, if it is replaced by 12 | s. 1.99 |
- (a)
A B C D s q p r - (b)
A B C D q s p r - (c)
A B C D r p q s - (d)
A B C D q p s r
Find the mean deviation about the median for the following data.
Xi | 3 | 6 | 9 | 12 | 13 | 15 | 21 | 22 |
fi | 3 | 4 | 5 | 2 | 4 | 5 | 4 | 3 |
- (a)
5.97
- (b)
4.97
- (c)
6.27
- (d)
5.21
Find the mean deviation about the mean for the following data
Marks obtained | Number of students |
---|---|
10-20 20-30 30-40 40-50 50-60 60-70 70-80 |
2 |
- (a)
20
- (b)
25
- (c)
21
- (d)
10
The variance of first 50 even natural number is
- (a)
\(833\over4\)
- (b)
833
- (c)
437
- (d)
\(437\over4\)
All the students of a class performed poorly in Mathematics. The teacher decided to give grace marks of 10 to eah of the students. Which of the following statistical measures will not change even after the grace marks were given?
- (a)
Mean
- (b)
Median
- (c)
Mode
- (d)
Variance
Let x1,x2,...,xn be n observations and if \(\bar{x}\) be their arithmetic mean and \(\sigma^2\) be the variance.
Statement I: Variance of 2x1,2x2,....,2xn is \(4\sigma^2\)
Statement II: Arithmetic mean of 2x1,2x2,....,2xn is \(4\bar{x}\)
- (a)
Statement I is true, statement II is true; Statement II is the correct explanation for statement I.
- (b)
Statement I is true, statement II is true; Statement II is not the correct explanation for statement I
- (c)
Statement I is true, statement II is false
- (d)
Statement I is false, statement II is true
If the mean deviations about the median of the numbers a,2a,....,50a is 50, then \(\begin{vmatrix}a\end{vmatrix}\) equals to
- (a)
3
- (b)
4
- (c)
5
- (d)
2
A scientist is weighting each of 30 fishes. Their mean weight worked out is 30g and a standard deviation of 2g. Later, it was found that the measuring scale was misaligned and always under reported every fish weight by 2g. The correct mean and standard deviation (in gram) of fishes are respectively
- (a)
28,4
- (b)
32,2
- (c)
32,4
- (d)
28,2
Correct mean=Old mean+2=30+2=32
As, standard deviation is independent of change o origin.
So it remains same.
Standard deviation=2
For two data sets, each of size 5, the variances are given to be 4 and 5 and the corresponding means are given to be 2 and 4, respectively. The variance of the combined data set is
- (a)
5/2
- (b)
11/2
- (c)
6
- (d)
13/2
Statement I: The variance of first n even natural numbers is \(n^2-1\over4\)
Statement II: The sum of first n natural numbers is \(n(n+1)\over2\) and the sum of squares of first n natural numbers is \(n(n+1)(2n+1)\over6\)
- (a)
Statement I is true, statement II is true; Statement II is the correct explanation for statement I.
- (b)
Statement I is true, Statement II is true; Statement II is not a correct explanation for statement I
- (c)
Statement I is true, Statement II is false
- (d)
Statement I is false, statement II is true
The mena of the numbers a,b,8,5,10 is 6 and the variance is 6.80. Then, which one of the following gives possible values of a and b?
- (a)
a=3, b=4
- (b)
a=0, b=7
- (c)
a=5,b=2
- (d)
a=1,b=6
Suppose population A has 100 observations 101,102,...,200 and another population B has 100 observations 151,152,....,250. If VA and VB represent the variances of the two populations respectively, then \(V_A\over V_B\) is
- (a)
9/4
- (b)
4/9
- (c)
2/3
- (d)
1
Since variance is independent of change of origin. Therefore, variance of observations, 101,102,...200 is same as variance of 151,152,...,250
VA=VB
\(V_A\over V_B\)=1
In a frequently distribution, the mean and median are 21 and 22 respectively, then its mode is approximately
- (a)
24.0
- (b)
25.5
- (c)
20.5
- (d)
22.0
Given that, mean=21 and median=22
Using the relation, mose=3 median-2 mean
Mode =3(22)-2(21)=66-42=24