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R.B.MADKHOLKAR MAHAVIDYALAYA, CHANDGAD
Department of Statistics/B.Sc.-I/ Expt. No.: 1
GRAPHICAL REPRESENTATION OF FREQUENCY DISTRIBUTION
Name of Student:
Date:
1. The marks in statistics of 100 students are given below.
Marks
:
0-10 10-20 20-30 30-40 40-50
50-80 60-70 70-80 80-90 90- 100
No. of students:
4
6
10
15
22
18
12
08
03
Draw the Histogram and find the value of mode,
ii) Draw frequency polygon and
frequency curve.
02
2. The weekly earning (Rs.) of 500 workers of a factory is as follows.
Weekly Earnings : 10-12 12-14 14-16 16-18 18-20 20-22 22-24 24-26 26-28 28-30 30-32 32-34 34-36 36-38 38-40
No. of Employees : 4
6
10
15
25
45
75
90
80
55
36
26
17
10
6
Draw i) Frequency polygon. ii) Frequency Curve. iii) Less than ogive curve and greater than ogive
curve, Hence show that they intersect at median.
3. Marks of 100 Students of a class are as follows:
Marks
: 10-19 20-29, 30-39, 40-49. 50-59, 60-69, 70-79. 80-89. 90-99
No. of Students:
1
2
4
7
15
30
22
12
7
Draw less than ogive curve and find,
i)
D3
ii]
D7
iii)
P85
iv)
Q1 and Q3
4. Following data gives the scores of two batches of students In an intelligence test:
Scores
: 0-10 10-20 20-30 30-40 40-50 50-60 60-70 70-80 80-90 90-100
No. of Students :
in Batch I
5
9
12
14
20
15
10
8
5
2
No. of Students :
In Batch II
7
8
15
19
20
14
10
04
2
1
Draw the less than ogive for both distributions and hence find:
i) Which batch is having higher value of median?
ii) Which batch is having smaller value of (Q3 – Q1) / (Q3 + Q1)?
5. Draw less than ogive curve from the following data.
Wt. in Lbs.
: 100-110 110-120 120-130 130-140 140-150
No. of Athletes :
8
11
12
17
19
150-160 160-170 170-180 180-190 190-200
10
8
7
5
3
Find i) The limits for middle 50 % Athletes. ii) Upper limit of the weight of first 35 %
Athletes. iii) Lower limit of the weight of last 30 % Athletes. iv) Number of Athletes whose weight
is less than 155 Lb.
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R.B.MADKHOLKAR MAHAVIDYALAYA, CHANDGAD
Department of Statistics/B.Sc.-I/ Expt. No.: 2
MEASURES OF CENTRAL TENDANCY - I
Name of Student:
Date:
1. Following figures are the heights in cms of 10 children chosen at random:
64 59,
67 69
65
70,
68, 58 66, 61
Calculate: i) Mean
ii) Median III) First & Third Quartiles
2. Calculate the Median, Quartiles, 5th Decile. 35th Percentile of the following series of marks
obtained by 15 candidates in an examination.
22, 26 , 14, 30, 18, 11, 35, 41, 49, 12, 32, 44, 28, 31, 20
3. Following data gives the number of children for 20 families selected at random.
3, 5, 1, 2, 3, 5, 5, 4, 4, 2, 3, 4, 1, 2, 3, 4, 5, 3, 3.
Find the Mean, Median and Mode of Number of Children,
4. Calculate the A. M. (A), G, M. (G. and H. M, (H) of the following observations:
32, 35, 36, 37, 39, 41, 43, 47, 48, 38
Hence show that A > = G > = H
5. A shoe owner sold the shoes of following number in a week.
Shoe Size
4
5
6
7
8
9
10
11
12
No. of Shoes
10
13
18
21
28
24
16
6
2
Find 1) Mean Value 2) Median Value 3) Geometric Mean 4) Harmonic Mean 5) Modal value
6. Following are the marks obtained by 12 students in two papers of Statistics.
Statistics I : 36, 56, 41,
46,
54,
59,
55,
51,
52,
Statistics II : 58,
54, 21,
51,
59,
46,
65,
31,
68,
Find – 1) The A.M. of the marks of each paper.
2) The combined mean of the marks of both papers.
44,
41,
37,
70,
7. The average weight of 15 persons is 53.4 Kg. and Median weight is 55 kg. At the time of
analysis two items are wrongly taken as 45 kg. and 56 kg. instead of 54 kg. and 65 kg.
Find the Corrected Value of Mean and Median Weight.
8. The weight of 14 new-born babies is as follows:
Weight in Lb (Xi): 8.5, 9.0, 7.8, 7.1, 8.0, 8.6, 9.3, 6.9, 7.7, 7.5, 7.7, 7.2, 8.5, 8.2
I) Show that
=0
ii) If Ui = Xi - 8 Find the mean value of Ui's. Hence, Verify that X = U + 8
9. Number of workers and mean wage in rupees per month in three sections of a factor are as
follows:
Section
No. of Workers
Mean Wage
A
50
113
B
60
120
C
90
115
Find the Mean wage of the workers in the factory.
59
36
R.B.MADKHOLKAR MAHAVIDYALAYA, CHANDGAD
Department of Statistics/B.Sc.-I/ Expt. No.: 3
MEASURES OF CENTRAL TENDANCY - II
Name of Student:
Date:
1. Following table gives the weights of 32 persons in a sample survey;
Weight in Lb : 130 135 140 145 146 147 148 149 150
No. of persons: 3
4
6
7
5
3
2
1
1
Find i) Mean Weigh! ii) Median Weight
iii) Mode Weight
2. A survey report on the effectiveness of the sleeping drug gives the following data for additional
sleep,:
Additional Sleep (hr.) :
1
2 3
4
5
6
7
8
No. of persons
: 10
13 18 23
29
24
16
8
Find
i) Mode ii) Median iii) A. M. iv) G. M,
v) H.M.
Hence show that H. M. < = G. M. < = A. M.
3. Data given below shows the waiting time of the customers for getting the services in an office.
Waiting Time (min.):
0
1
2
3
4
5
6
No. of Customers:
6
24
35
60
40
25
15
Find i) Median
ii) Quartiles
iii) D7 iv) P15
4. From the data given below find Mean, Mode and Median.
Age(yrs)
: 20-25, 25-30 30-35 35-40
40-45
No. of Persons : 50
69
81
179
151
45-50
119
50-55 55-60
71
40
5. Monthly Income of 160 families is given below.
Income (Rs)
: 400-799 800-1199 1200-1599 1600-1999 2000-2399 2400-2799 2800-3199
No. of Families :
04
12
41
51
27
16
09
Find i) mean ii) median iii) modal value of income.
6. The weekly wage of 800 workers of an industry is as follows.
Wage(Rs):
100-149 150-199 200-249 250-299 300-349 350-399 400-449 450-499
No. of Workers:
20
67
150
195
151
119
87
11
Find – i) Quartiles ii) Wage limit for middle 60% workers. iii) D5
iv)P75
7. Daily expenditure of 100 students is given below. The median is known to be Rs 50, find the
missing values
Expenditure (Rs.) :
0-20
20-40
40-60
60-80 80-100
No. of Students
:
14
--27
--15
8. Marks in Mathematics of a group of 125 students are as follows.
Marks : 0-10 10-20 20-30 30-40 40-50 50-60 60-70 70-80 80-90 90-100
Students: 1
9
15
19
26
23
17
10
3
2
Find A. M. and G. M. and hence show that AM > GM > HM.
9. A trains runs 25 miles at a speed of 30 m. p. h another 50 miles at a speed of 40 m.p.h. then due to
repair of track it travels for 6 minutes at a speed of 10 m.p.h. and finally it runs the remaining
distance of 24 miles at a speed of 24 m.p.h. What Is the average speed of the train per hour?
R.B.MADKHOLKAR MAHAVIDYALAYA, CHANDGAD
Department of Statistics/B.Sc.-I/ Expt. No.: 4
MEASURES OF DISPERSION – I
Name of Student:
Date:
1. Compute Quartile deviation (Q. D.) Inter Quartile Range (I. O. R.), Coefficient of
Q. D. from the following data.
a) 30,
14, 10,
17,
23,
20, 19,
26, 21, 18
21
b) 9.4
9.1 9.7
8.8
10.6 10 10.9 12.3 11.2 8.6 15.5
2. From the following data Verify that Mean Deviation about Median (M.D. about Median)
is less than M.D. about Mean.
7,
4,
10,
9,
15,
12 , 7 ,
9,
7,
3. For the following data, show that variance is less than Mean Square Deviation about any point (a)
say a = 10
5, 6. 8, 9, 11, 10, 4,
11,
12, 14
4. Compute Standard Deviation and Mean Deviation about Mean from the following data.
57, 54, 39, 45, 47, 53, 55, 46,
66
If all the values in the data is increased by 8 then find the Standard Deviation and Mean deviation
about Mean and Coefficient of Mean Deviation.
5. The number of runs scored by batsman A and B during the test for each of 10 innings are given
below.
Batsman A:
5, 26, 97, 76, 112, 89, 6, 108, 24.
16
Batsman B: 51 47, 36 10. 58, 39, 44, 42. 71,
50
i) Who is more consistent?
ii) What is the C.V. of the combined set?
6. A crop cutting survey was carried in Solapur District with its two sub divisions on {Jowar} the
data obtained was given below:
Subdivision
Plot for selected
Yield of Jowar (in Quintals)
No. of Villages
Sum
Sum of squares
I
15
1034
75366
II
12
725
48331
Which subdivision In the district is homogenous with respect to yield of Jowar?
Compute the standard deviation of yield of Jowar for the district.
i)
ii)
7. The following table gives the relevant statistics for the data on fat contents In 3 brands of bacon:
Sample / Brand
Size
1
5
2
5
3
5
Mean
42.0
36.5
52.4
Variance
2.5
2.3
1.3
1) Compute the variance of the fat contents in 3 brands combined.
2) Which is more consistent in respect of fat content?
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R.B.MADKHOLKAR MAHAVIDYALAYA, CHANDGAD
Department of Statistics/B.Sc.-I/ Expt. No.: 5
MEASURES OF DISPERSION – II
Name of Student:
Date:
1. Following is the distribution of weight of newborn babies:
Weight (lb) : 6 7
8
9
10
11
12
No. of babies : 7 10
16
24
21
16
6
Compute 1) Mean Deviation about mode. 2) Q. D. 3) Coeff. of Q.D.
4) Coeff. of Range
2. The following table gives the cumulative frequency distribution of the daily wages {in Rs.) of
workers:
Wages above: 55
60
65
70
75
80
85
90
No. of workers: 164
153
141
122
99
67
40
31
Calculate 1) Q. D. and its coefficient. 2) Mean Deviation about Median and coefficient of Mean
Deviation.
3. Compare the effectiveness of two drugs which is given to two groups of patients and their
recovery period is as follows.
Length of illness (in days)
: 5 10
15
20 25
No. of patients given to drug A: 3 20
17
8 2
No. of patients given to drug B: 4 30
10
4 2
4. From the following frequency distribution find:
I) Quartile Deviation and its coefficient.
II) Mean Deviation about Mean and its Coefficient.
III) Coefficient of Variation (C.V.)
Classes
:
0-5
5-10 10-15
15-20
Frequencies:
13
15
17
13
20-25
8
5. Following is the distribution of gallons of Fizzy Cola Syrup sold by 50 employee’s B and D
Company in one month:
Midpoint of the gallons sold (m): 85
95 105
115
125 135
145
No. of employees (f)
: 2
6
10
14
9
7
2
Compute i) Coefficient of Q. D.
ii) If u = (m-115)/10. Find the variance of u.
6. The following is the distribution of resistance(in Ohm) of 500 units of certain electric products:
Resistance : 2.7-2.9 3.0-3.2 3.3-3.5 3.6-3.8 3.9-4-1 4.2-4.4 4.5-4.7 4.8-5.0 5.1-5.3
Frequency:
2
16
46
88
13
113
71
22
4
Compute i) Coefficient of Q.D.
ili) Coefficient of S. D.
7. The frequency distribution of marks obtained in Mathematics (M) and Starts (S) are as follows;
Whose marks are more variable?
Class of Marks
: 0-10
10-20 20-30
30-40
40-50
No. of Students (M) : 10
No of Students (S) : 5
16
9
20
15
14
11
10
10
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R.B.MADKHOLKAR MAHAVIDYALAYA, CHANDGAD
Department of Statistics/B.Sc.-I/ Expt. No.: 6
MOMENTS , STUDY SKEWNESS & KURTOSIS-I
Name of Student:
Date:
1. The weight [in Lbs] of randomly selected students were found to be,
109. 114, 120, 120, 117, 95, 96
Compute the Mean weight and first four moments about origin Also find, 3 (Mean-Median) / S.D.
2. Eight electronic tubes were tested and the following life times in hour) were recorded:
49.6, 7.2, 41.1, 37.8, 21.4, 3.8, 67.2, 8.3
i) Compute first four raw moments about a point 20 and find mean & variance of life times.
ii) Find first four moments, β1,β2 and comment.
3. Compute first four central moments from the following data; also find β1, β2 and γ1 γ2,
and give your conclusion: 3, 4, 5, 10, 12, 15
4. The marks of randomly selected students from a class were found to be:
66. 74. 44. 28, 80, 42, 69. 43
Compute first four raw moments and hence compute first four central moments, γ 1 γ2, and
comment.
5. The weight (in lb) of eight athletes are as follows:
190, 194. 185, 167, 210. 196, 210
Compute 1) First four moments about 200. 2) β1,β2 and γ1 γ2 and comment.
6. The birth weights (In pound) of 10 infant at civil hospital were found to be:
6.0.
7.8.
5.6, S.3,
4,2
7.6 7,1 7.5 6.7 9.1
Compute 1 ) mean 2) variance 3) coefficient of skewness and 4) coefficient of kurtosis
7. The following data are recorded from the frequency distribution of consumption of
electricity (X) used by 492 residential :
∑u = 210, ∑u2 = 2444, ∑u3 = 2142, ∑u4 = 31316 where u = (x-A)/h, A=87.5 and class
width h = 15 units,
Compute the mean consumption and first four moments about mean of x and then compute
β1, β2 coefficients and give your conclusion about the distribution.
8. The first four moments of a distribution about 2 are 1, 2.5, 5.5 and 16
Compute i) First four moments about mean ii) First four moments about origin iii) A. M. and
Variance.
9. For a distribution mean is 10, variance is 16, γ1 is 1, β2 is 4. Find first four raw moments.
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R.B.MADKHOLKAR MAHAVIDYALAYA, CHANDGAD
Department of Statistics/B.Sc.-I/ Expt. No.: 7
MOMENTS , STUDY SKEWNESS & KURTOSIS-II
Name of Student:
Date:
1. Following is the data on number of particles radiated (f) from a plate during different time (x):
X:
0.5
1.5 2.5
3.5
4.5
5.5
6.5
7.5
f:
11
40
74
100
103
79
50
26
Calculate first four raw moments about point 6, and hence compute the first four central moments
of the distribution, γ1 & γ2 and comment.
2. From the following frequency distribution of number of accidents in different age is as follows:
Age :
5
10
15
20
25
30
35
No, of accidents: 1
3
10
15
12
12
7
Compute first four raw moments of the distribution also find 3(Mean - Mode) / SD = Sk
3. The following is the distribution of marks obtained by 140 candidates in university examination.
Marks less than : 20 40
60
80
100
No, of students : 11
29
76
116
140
Compute first four moments about mean and then compute β1, β2, γ1, γ2 coefficients, hence give
your conclusion about the distribution,
4. Following is the distribution of gallons of fizzy cola syrup by 50 employees of B and D
Company.
Gallons sold
: 80-69 90-99 100-109 110-119 120-129 130-139 140-149
No. of employees : 2
6
10
14
9
7
2
Compute first four central moments. Apply Sheppard's correction then find corrected
moments β1, β2, and γ1, γ2 coefficients, hence comment.
5. Following is the frequency distribution of number of observations generated from X2 statistic:
C.I
: 0-10
10-20 20-30
30-40 40-50
50-60
Frequency:
10
13
15
16
13
7
Compute the mean, variance and first four moments about the mean of the distribution.
Also compute γ1, and γ2 coefficients and give comment on distribution.
6. Following is the distribution of lots according to number of defectives:
No. Of defectives:
10
11
13
14
15
16
17
No, Of lots :
7
10
16
24
21
16
6
Compute β1, β2, and γ1, γ2 coefficients from the moments and comment about the distribution.
7. The failure times of 100 transistor batteries of a certain kind in hours have the following frequency
distribution.
Hour :
0-25 25-50 50-75 75-100 100-125 125-150 150-175
Frequency : 33
28
20
12
4
2
1
Apply Sheppard's correction then find β1, β2, γ1, γ2 coefficients. Hence comment about the
distribution.
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R.B.MADKHOLKAR MAHAVIDYALAYA, CHANDGAD
Department of Statistics/B.Sc.-I/ Expt. No.: 8
CORRELATION COEFFICIENTS-I
(Karl Pearson & Charl Spearman)
Name of Student:
Date:
1. a) The age in years of 15 young couples is given below ;
Age of husband: 21 25 26 24 22 30 19 24 28 32 31 29 21 18 23
Age of Wife : 19 20 24 21 21 24 18 22 19 30 27 26 19 18 24
Draw the scatter diagram and give your conclusion.
b) Compute Karl Pearson’s correlation coefficient.
2. The table below gives the total grain production and net production of cereals in lakh of tonnes.
Compute correlation coefficient between total production and Cereal production.
Total production: 400
440 550
620 650
560
740 760
Cereal production: 50
70
65
95 100
105
116
120
(apply change of origin and scale technique)
3. A sample of size 10 from a bivariate distribution having variables X and Y gives the following
values:
∑X = 800, ∑Y = 720, ∑X2 = 68400, ∑y2 = 55750,
∑xy = 60350
Find i) Correlation coefficient between X and-Y ii) Correlation coefficient between X and 72-Y
iii) Correlation coefficient between X and X + Y iv) Correlation coefficient between X and X - Y
4. From the following data, calculate coefficient of rank correlation between X and Y.
X: 36
56
20
65
42
33
44
50
15
60
Y: 50
35
70
25
58
75
60
45
80
38
5. Following are The marks in three subjects A, B and C.
Marks in A: 35 65 32
30
35
65
65
58
40
54
Marks in B: 34 39 40
40
59
54
58
54
32
51
Marks in C: 65 28 59
57
52
28
35
52
65
48
Find the rank correlation coefficient between i) A and B ii) B and C iii) C and A
6.Ten Competitors in a musical test were ranked by the three Judges X,Y and Z in the following.
Rank X: 1
6
10
3
2 4
9
7
8
Rank Y: 3
5
4
7
10 2
1
6
9
Rank Z : 6
4
8
1
2 3
10
5
7
Using rank correlation method, discuss which pair of judges has the nearest approach to common
liking in music.
7. The rank correlation coefficient between the score in Maths and Stats of a group of 16 students is
0.8. But at the time of computation two of di’ s are wrongly taken as 5 and -4 instead of 6 and -3. Find
the corrected value of the Rank correlation coefficient.
8. The coefficient of rank correlation between marks in Physics and Chemistry obtained by a certain
group of students is 0.8. If the sum of squares of differences in ranks is given lo be 33. Find the
number of students in the group.
R.B.MADKHOLKAR MAHAVIDYALAYA, CHANDGAD
Department of Statistics/B.Sc.-I/ Expt. No.: 9
CORRELATION COEFFICIENTS-II
(Karl Pearson & Charl Spearman)
Name of Student:
Date:
1. The following table gives the number of candidates obtaining different marks in A/C (X) &
statistics (Y).
Marks in Statistics
30-40
40-50
50-60
60-70
Marks in A/c
25-35
3
1
1
-35-45
2
6
1
2
45-55
1
2
2
1
55-65
-1
1
1
i) Obtain the product moment correlation coefficient between the marks in the two subjects and
comment on its value. ii) Find average marks in A/c of the student whose score in statistics are 50-60.
2. In a survey of factories, the number of days with double duty per month by workers of different
age groups was as below.
No. of days with double duty
Age group
1 -3
3-5
5-7
7-9
9-11
20-30
----1
2
9
30-40
—
2
2
7
5
40-50
—
2
4
3
—
50-60
2
2
1
—
—
Compute Karl Pearson coefficient of correlation.
3. Find I) Correlation coefficient between the marks in English and marks in Mathematics.
II) Average marks In English of the student whose score in Mathematics are 30 – 40.
Marks in English
Marks in Maths
0-10
10-20
20-30
30-40
40-50
0-10
--
-
4
30
7
10-20
--
5
17
35
6
20-30
-
22
45
19
4
30-40
5
11
15
12
—
40-50
1
4
8
-
--
4. For a sample of married women, the level of education and marriage adjustment score is a follows:
Marriage Adjustment(y)
1
2
3
4
5
Education level (X)
1
12
50
31
20
22
2
11
24
15
16
15
3
10
14
6
12
8
4
6
5
5
6
3
Find the correlation coefficient between education level and marriage adjustment level and interpret
your findings.
R.B.MADKHOLKAR MAHAVIDYALAYA, CHANDGAD
Department of Statistics/B.Sc.-I/ Expt. No.: 10
REGRESSION -I
Name of Student:
Date:
1. Following data given price (X in Rs.) and supply (Y in ton) of a commodity.
X
Y:
47
64
51
62
52
63
51
61
53
59
50
58
46
66
Find: i) Equations of the line of regression.
ii) Acute angle between two regression lines.
iii)
Estimate the value of Y when X = 48
2. From a bivariate distribution a sample of size 40 gives the following values:
∑x = 1183, ∑Y = 3748, ∑X2 = 53948, ∑y2 = 30812, ∑xy = 34969
Find i) Equations of the line of regression ii) The correlation coefficient between X and Y
iv) Acute angle between two regression lines. iv) Estimate the Value of Y when X = 30
3. If the equations of the lines of regression are 2X - 3Y + 4 = 0 and 4Y- 5X + 10= 0
Find i) mean values of X and Y ii) Correlation coefficient between X and Y iii) variance of X
when variance of Y = 25 iv) Acute angle between the two regression lines.
4. The equations of the lines of regression based on two variables U and V are
0.5 V – 1.204 U - 5.2461 = 0 and 2.0 U - 0.05 V - 25.9778 = 0
if U = (X -10) / 2 and V = (Y-10)/0.5,
Find i) Equation of the regression of Y on X and hence estimate the value of Y when X = 140
ii) The correlation coefficient between X and Y iii) The mean values of X and Y
5. Find the correlation coefficient between price (X) and supply (Y in ton) of a commodity
using following data:
X: 71
66
66
67
70
72
69
73
70
65
Y: 59
64
65
63
62
61
60
58
57
65
Also Obtain regression equations of Y on X and X on Y.
9. A computer while calculating the correlation between 25 pairs of observations obtained the
following constants:
∑X=125, ∑Y = 100, ∑X2 = 750, ∑y2 = 560, ∑xy = 568,
It was later found that two pairs were wrongly copied as (6, 4) and (8, 4) instead of (5, 6) and (7, 5).
Obtain the corrected value of the correlation coefficient.
R.B.MADKHOLKAR MAHAVIDYALAYA, CHANDGAD
Department of Statistics/B.Sc.-I/ Expt. No.: 11
REGRESSION -II
Name of Student:
Date:
1. The following table gives a bivariate frequency distribution of 50 clerks according to age in year
and pay in Rupees.
Age (X)
Pay(Y)
250-300
300-350
350-400
400-450
20-30
8
3
—
—
30-40
2
5
2
2
40-50
-2
9
6
50-60
—
-5
6
Find 1) Equations of two regression lines, 2) Acute angle between two regression lines,
3) Estimate the pay when Age = 45 years
2) From the bivariate distribution having two variables U and V gives the following data:
∑uifi. = 73, ∑vjfj = 59, ∑ui2fi = 281, N = 100, ∑vj2fj = 249, ∑uivjfij= 195, If U = (X - 55) /10 and V
= (Y - 56) /5 find
i) Correlation coefficient between U and V ii) Regression equation of U on V and V on U,
iii) Regression equation of Yon X and X on Y iv) Find the angle between above pair of lines.
v) Variance of X + Y
3. A sample of 150 observations from a bivariate distribution having variables X and Y gives
the following values:
∑xifi= 3010,
∑yjfj = 4575,
∑xi2fi = 61415
2
∑yj fj = 145975,
∑ xiyjfij = 93805
lf U = X + Y and V--Y - X find
I) Equations of the lines of regression of U on V and V on U
II) Estimate the value of V when U = 50
4. Following data gives the age of wives and husbands of 100 couples selected at random from
certain area.
Age of Wives
Age of
Husbands
20-29
30-39
40-49
50-59
60-69
70-79
80-89
20-29
4
3
2
—
—
—
—
30-39
40-49
5
—
3
5
11
10
5
8
2
3
—
—
—
—
50-59
—
6
10
5
3
1
—
60-69
—
—
—
2
2
3
—
70-79
—
—
—
—
1
2
2
80 -S9
—
—
—
—
—
1
1
Find I) The correlation coefficient between the age of wives and age of husband.
II) Equations of two regression lines, III) Acute angle between two regression lines.
R.B.MADKHOLKAR MAHAVIDYALAYA, CHANDGAD
Department of Statistics/B.Sc.-I/ Expt. No.: 12
ATTRIBUTES
(Calculation of frequencies & Consistency of data)
Name of Student:
Date:
1. Given the following ultimate class freq. (AB) = 270, (Aβ) = 130, ( B) = 190, ( β) = 60. Find the
frequencies of the positive and negative classes and the total number of observations. Examine
consistency.
2. 100 children took three examinations, 40 passed the first, 39 passed the second and 48 passed the
third. 10 passed all three. 21 failed all three, 9 passed the first two and failed the third, and 19
failed the first two and passed the third. Find how many children passed the at least two
examinations.
3. Find the remaining class frequencies from the following:
a) N = 1800, (A)= 850, (B)=780, (C) = 326, (AB) =250,( AC) = 140, (BC) = 122, (ABC) =50
b) (ABC) =15, (AB ) = 25, (AβC) =22, ( BC) =31, (Aβ ) =8, ( B ) =12. ( βC) =15, ( β ) =9
4. In 1700 car accidents it was observed that 650 of the drivers were males, 480 were drunk and
570 accidents took place at night after dark. In 360 cases the drivers were male & drunk, In 240
the time was after dark and drivers were drunk, while in 255 accidents drunker males were driving
after dark. While in 155 accidents drunker male were drivers after dark. Show that this
information must be correct.
5. Prove that the following data are inconsistent N = 1000, (A) = 525, (B) =312, (C) = 470,
(Aβ) = 483, (A ) = 378, (B ) = 226, (ABC) = 25.
6. Among the producers of cosmetics, 38 advertise their product in newspaper, 12 advertise on TV
and radio but not in newspaper and 17 advertise in newspaper only. Find the number of producers
using at least two media for advertising their product.
7. Out of 100 students interviewed, 40 liked Indian music, 70 liked Western music and 20 liked both.
Find the number of students who did not like music.
8. In an examination 60% passed in Physics, 52% passed in Statistics, while 32% failed in both the
subjects. Using relations among class frequencies in attributes find the percentage of students
passed in the both the subjects.
9. In a morbidity survey of 100 families, it was found that 50 were infected with Cholera, 25 were
infected with Typhoid and 20 were affected by both. Find the number of families infected by
either of the disease.
10. In a group of 100 employees in a firm there were 80 male. The number of married employees was
60 among whom 30 were male. Examine whether the information as it stands is correct.
11. Examine whether the data are consistent. N= 100, (A)=88, (B)=109,(C)= 20, (AB)=34, (βC)=13,
(AC)=14 and (ABC)=6
12. A survey by newspaper agency reveald that out of 1000 people, 811 read Marathi newspaper, 752
read English and 418 read Hindi newspaper. People reading both Marathi and English were 570.
Further 356 read English and Hindi, 348 read Marathi and Hindi. Only 297 read all the three. Show
that the information is inconsistent.
R.B.MADKHOLKAR MAHAVIDYALAYA, CHANDGAD
Department of Statistics/B.Sc.-I/ Expt. No.: 13
ATTRIBUTES
(Independence and Association)
Name of Student:
Date:
1. Find association between intelligence of husband and intelligence of wile from the following
data. Intelligent husbands with intelligent wives 50, Dull husbands with intelligent wives 100,
intelligent husbands with dull wives 200, and dull husbands with dull wives 300.
2. Find if A and B are independent, positively associated or negatively associated in each of the
following cases.
I) N = 1000
(A) = 470
(B) = 620
(AB) = 320
II) (A) = 490
(AB) = 294
( ) = 570
( B) = 380
III) (AB) = 256
( B ) = 768
( A β ) = 48
( β ) = 144
3. Compute the coefficient of association (Q) and coefficient of colligation (Y) and comment on the
association between A and B for each of the following.
I) N = 100 (A)=55
(B)=40
(AB)=45
II) N = 1200 (A) = 400
(B) = 500
(AB) = 250
4. The following are the data on literacy and criminal tendency in a town.
Total population (in thousands) = 224, Literate criminals (In thousands) = 3
Total literates (in thousands) = 40, Illiterate criminate (in thousands) = 43
Find the association between literacy and criminal tendency and comment on it.
5. In an experiment on immunization of cattle from tuberculosis the following results were obtained:
Affected
Not affected
Inoculated
12
26
Not inoculated
16
06
By calculating Yule's coefficient of association examine the effect of vaccine in controlling the
disease. Calculate coefficient of colligation (Y) & check Q = 2Y / 1+Y2
6. Out of 800 students 320 are married. Among 240 students who failed, 96 were married. Find the
coefficient of association between the attributes marriage and failure of examination. Interpret the
result.
7. The male population of a state is 250 lakh. The number of literate male is 20 lakh and total number
of male criminals is 26 thousand. The number of literate male criminals is 2 thousand. Do you
find any association between literacy and criminality?
8. An investigation was conducted in 1000 business class families in a certain city and the following
data were obtained. Do data confirm the hypothesis ‘that greedy fathers have greedy eldest son?’
Greedy fathers with greedy eldest son
= 408
Greedy fathers with not greedy eldest son
= 204
Not greedy fathers with greedy eldest son
= 106
Not greedy fathers with not greedy eldest son = 282
R.B.MADKHOLKAR MAHAVIDYALAYA, CHANDGAD
Department of Statistics/B.Sc.-I/ Expt. No.: 14
APPLICATIONS OF BINOMIAL & HYPERGEOMETRIC DISTRIBUTION
Name of Student:
Date:
BINOMIAL DISTRBUTION
1. In a computer centre there are 8 computers. The chance of their failure is same during the given
period and is equal to 0.3. Find the probability that during the given period. i) at least one
computer will fail ii) exactly half of the computers will fail; iii) All of them will work efficiently.
2. It is observed that 40% of the students in a class wear glasses. If 5 students of this class are selected at
random, what is the chance that among them,
i) no one wears glasses
ii) at least one wears glasses?
3. The sales manager of an automobile dealer estimates that 90% of the new cars delivered by them have no
defect and so will not be brought back immediately for repair. He sells a fleet of six cars to an important
customer. What is the probability that:
i) no car will be brought back
ii) All cars will be brought back
iii) one or more cars will be brought back for repair?
4. In a precision bombing attack there is 50 % chance that any the bomb will strike the target. Two direct hits
are required to destroy the target completely. How many bombs must be dropped to give a 99% chance
or better of completely destroying the target?
HYPERGEOMETRIC DISTRIBUTION
1. A group of 20 cricket players contains Maharashtrains and remaining non Maharashtrains, An Indian team
of 12 players is to be formed. What is the probability that one fourth of the players in me team are
Maharashtrains?
2. A club has 20 members of which 12 are boys and 8 girls. If a committee of 5is to be selected by drawing
lots, find the probability that the committee will consists of:
I) 3 boys and 2 girls
II) at least 3
boys
III) members of same sex.
3. A box contains 20 balls. 12 of them being white and remaining red. If 6 balls are drawn at random find the
probability that: i) Half of them are whites ii) At most 2 are white iii) At least 2 are white.
4. A box contains 6 white and 4 black balls, 5 balls are drawn at random from the box without replacement.
Find the probability of getting, i) no black ball ii) exactly Two black balls, iii) at least two black balls
iv) at the most two black balls.
5. A box contains 20 computers chips of which 5 are defectives. 4 chips are selected at random. Find the
probability of getting i) no defective chips ii) equal no of good and defective chips. iii) all defective chips.
6. Find the probability of getting i) 4 spade cards ii) At least 4 spade cards when from a well shuffled pack
of cards 6 cards are drawn at random without replacement.
7. An urn contains 7 black and 3 white balls. If 5 balls are drawn, write down the frequency distribution for
the number of black balls obtained. If drawings are made without replacement.
8. A box contains 15 screws which 5 are defective. If 10 screws are drawn at random find the probability of
getting, I) 3 defective screws II) At most 3 defective screws. When the drawings are made without
replacement.
9. As a part of air pollution survey; an investigator decides to inspect the exhaust of 10 trucks out of a fleet of
16 trucks. He suspects that 6 of the trucks emit excessive amounts of pollutants and hence they are deemed
to the bad trucks. Find the probability that the investigator will find
i) exactly 3 bad trucks ii) at
least 3 bad trucks and iii) at most 2 bad Trucks.
R.B.MADKHOLKAR MAHAVIDYALAYA, CHANDGAD
Department of Statistics/B.Sc.-I/ Expt. No.: 15
MULTIPLE REGRESSIONS
Name of Student:
Date:
1. Given the following data,
X1 = 6.8, X2 = 7.0, X3 = 7.4, 1 = 1, 2 = 0.8, 3 = 9.0, r12 = 0.60, r13 = 0.7, r23 = 0.65.
i) Find equation of regression plane of X3 on X1 and X2.
ii) Estimate X3 when X1 = 9, X2 = 4.
2. In a study of a random sample of 120 students the following results are obtained
Means: X1 = 68, X2 = 70, X3 = 74
S. D.s: 1 = 10, 2 = 25, 3 = 81.
Sample correlation coefficients r12 = 0.60, r13 = 0.7, r23 = 0.65. Where X1, X2 and X3
denote percentage of marks obtained by a students in test I, II and the final examination
respectively.
i) Obtain the least square regression equation of X3 on X1 and X2.
ii) Estimate the percentage of marks in the final examination if he gates 60% and
70% in test I and II respectively.
3. The neoguts (X1) in pound the heights (X2) in inch and the age (X3) in year of 10
boys is as follows:
X1 : 64 71 53 67 55 58 77 57
56 76
X2 : 57 59 49 62 51 50 55 48
52 61
X3 : 8
10
6 11
8
7 10
9 10
6
Find the plane of regression of X1 on X2 and X3. Hence estimate the value of X1
when X2=60 inches and X3=12.
4. The following results were obtained from measurement on length in mm. (X1)
Volume cc (X2) and breadth in mm. (X3) of 250 eggs.
X1 = 55.95, X2 = 51.48, X3 = 56.03, 1 = 2.26, 2 = 4.40, 3 = 4.5,
r12 = 0.56, r13 = 0.97, r23 = 0.58.
Find the equation of the line of regression plane of X2 on X1 and X3.
5. The data below relate to the three variables X1, X2, and X3,
N=12, X1= 753, X2= 643, X3 = 106, X1 2= 48139, X22 = 34843, X3 2 = 1976,
X1X2= 40830, X1X3= 6796, X2X3= 5776
Obtain the equation of the plain of regression of X 3 on X1 and X2.
6.If Z1=X1+X2, Z2=X2+X3, Z3=X3+X1 where X1, X2, X3 are mutually uncorrelated variables
each with mean zero and variance one, show that partial regression coefficients of Z 1 on Z2 with Z3
respect is 1/3
R.B.MADKHOLKAR MAHAVIDYALAYA, CHANDGAD
Department of Statistics/B.Sc.-I/ Expt. No.: 16
MULTIPLE AND PARTIAL CORRELATION
Name of Student:
Date:
1.
Data given bellow relates the score in three subjects of a group of 10 students.
Subject X : 68 72 36 53 47 74 65 73 52 70
Subject Y : 53 65 52 50 48 42 61 66 43 50
Subject Z
: 48 49 37 39 35 50 47 40 31 47
a) Find Rx.yz and rxy.z
b) If X1 = X – Y, X2 = Y - Z and X3 = Z - X find 21.23, R1.23 and r12.3.
2.
X1, X2, X3 are three variables measure from their means, with
N= 10, X1X2 = 60, X1X3 = 40, X2X3 = 60, X12 = 90, X22 = 160, X32 = 40.
Find i) Variance of the residual X1.23
ii) Multiple correlation coefficient R1.23
iii) Partial correlation coefficient r12..3
3.
The following constants are obtained from measurements on length in mm. (X1)
volume in cc (X2) and weight in gm (X3) of 300 eggs.
2
2
2
1 = 5.1076, 2 = 19.272, 3 = 19.4481, r12 = 0.578, r23 = 0.974.
Obtain
i) variance of the residual X1.23
ii) multiple correlation coefficient R1.23
iii) partial correlation coefficient r12..3
4.
From large group of students in Statistics.
X1 = score in theory, X2 = score in Statistical methods,
X3 = score in practical Statistics,
Which are approximately normally distributed with, X1 = 50.4, X 2 = 45.1, X 3 = 55.3,
= 6.9, 2 = 6.4, 3 = 6.8, r12 = 0.69, r13 = 0.45, r23 = 0.58.
Obtain i) R1.23 and r12.3
& Show that ii) R1.23
r 12.3
1
5.
r 13.2
Following data relates the study factors which influences the academic success.
x1 = honors point
x2 = general intelligence
x3 = hours of study
(x1 - X1 ) (x2 – X 2 ) = 455,
(x1 - X1 ) 2 = 176,
Find
(x1 - x1) (x3 – X 3 ) = 22,
(x2 – X 2 ) 2 = 2564,
i) R1.23, R2.13, R3.12,
(x2 – X 2 ) (x3 – X 3 ) = -258,
(x3 – X 3 ) 2 = 166 with n = 15
ii) r12.3, r23.1, r32.1.
6.
Five thousand candidates were examined in three subjects a, b and c and following
constants are obtained.
a = 6.2, b = 9.4, c = 8.7, rab = 0.47, rbc = 0.38, rac = 0.29.
Find (1) 2a.bc , (2) Ra.bc, (3) rab.c.
7.
Three variables X1, X2 and X3 are measured from their means
1 = 2.7, 2 = 2.4, 3 = 2.7, r12 = 0.28, r13 = 0.51, r23 = 0.49.
Calculate r13.2, If Y1 = X1 + X2, Y2 = X1 + X2 + X3 and Y3 = X1 + X3,
Find a) multiple correlation coefficient of Y1 on Y2 and Y3,
b) partial correlation coefficient Y1 and Y2 on Y3.
************************
R.B.MADKHOLKAR MAHAVIDYALAYA, CHANDGAD
Department of Statistics/B.Sc.-I/ Expt. No.: 17
Bivariate Discrete Distributions- I
(Marginal & Conditional distribution, Computation of probabilities of events)
Name of Student:
Date:
1.
Let the joint p.m.f.of X and Y be
P(x, y) = (x + y) / 21; x = 1, 2, 3; y = 1, 2.
=
0
elsewhere
i) Are X and Y independent? ii) Find P (XY is odd), iii) Find the conditional probability
distribution of X given Y= 2, iv) Find marginal p.m.f. of X and marginal p.m.f. of Y.
2.
The joint probability distribution of two discrete random variable and Y given by
P(x, y) = (2x + 3y) / 72 ;
x = 0, 1, 2;
y = 1, 2, 3.
=
0
elsewhere
i)
Find the marginal probability distribution of X and Y.
ii)
Find conditional probability distribution of X given Y = 2.
3.
The joint probability distribution of discrete random variable X and Y given bellow.
X\y
1
2
3
1
1/12
0
1/8
2
1/6
1/6
0
3
0
1/4
5/24
Find i) the marginal probability distribution of X and Y ii) conditional probability
distribution of X given Y = 2.
iii) Conditional probability distribution of Y given X = 3.
4.
A bag contains 2 white, 3 red and 5 green balls. Three balls are drawn at random from this
without replacement. Obtain the joint probability distribution of the number of white balls (X)
and the number of red balls (Y).
5.
The joint probability distribution of X and Y given below.
y\x
1
2
3
4
0
1/24
1/12
1/12
1/24
1
1/12
1/6
1/6
1/12
2
1/24
1/12
1/12
1/24
i) Compute P (X + Y = 3)
ii) Evaluate P (X 2 / Y 1)
iii) Prove that P (XY = 4) = 1 / 6 and P (XY = 2) = 5 / 24.
6.
The joint probability distribution of X and Y is given below:
Y\X
1
1
2
-1
1/15
1/5
2/15
0
2/15
2/15
2/158
i) Find C , ii) P ( X = Y ), P ( X + Y
iv) Are X and Y independent?
2
1/15
1/15
2/15
1 ), iii) P ( X
1,Y
2 ),
R.B.MADKHOLKAR MAHAVIDYALAYA, CHANDGAD
Department of Statistics/B.Sc.-I/ Expt. No.: 18
Bivariate Discrete Distributions- II
(Expectations/ Conditional expectations/ Variances/Conditional Variance/ Covariance/ Correlation coefficient)
Name of Student:
Date:
1. Consider the following joint probability distribution.
y\x
0
1
-1
0
1/3
0
1/3
0
1
0
1/3
i) Show that E (XY) = E(X).E(Y) ii) Are X and Y independent?
iii) Find E(Y/X = 1)
iv) Find conditional variance of Y given X = 1
2. The joint p. m. f. of (X, Y) is given by
P(x, y) = c (x2 + y2);
x = -1, 1
y = -2, 2
= 0
otherwise
i) Find c, ii) The marginal p.m.f. of X and Y, iii) The conditional p.m.f. of X given Y = 2
iv) E (X|Y = 2), V (X|Y = 2), v) E (XY) = E(X).E(Y) vi) Are X and Y uncorrelated
vii) Are X and Y independent?
3. The probability distribution of (X, Y) is given by
x\y
1
1
-1
1/8
1/2
1
1/4
1/8
Find: i) Marginal distribution of X and Y
ii)
Corr (X, Y),
iii) E (X|Y = 1).
4. The joint probability distribution of (X1, X2) is
(X1, X2) :
(1, 1)
(1, 2)
(2, 1)
(2, 2)
P (X1, X2) :
1/6
2/7
3 / 14
1/3
Find the conditional mean and conditional variance of X 2 given X1 = 2.
5.
The joint probability distribution of (X, Y) is given by
(X, Y) :
(0, 1)
(0, 2)
(1, 1)
(1, 2)
P (X, Y) :
1/6
1/3
1 / 12
5 / 12
Find E(X + Y), E (X.), E(X2 Y2), E (XY+X2Y2).
8. Below is given the joint probability distribution of X and Y.
y\x
-1
0
1
Find:
-1
1/8
1/8
1/8
0
1/8
0
1/8
i) Corr (4/3X+2, 5Y-7),
1
1/8
1/8
1/8
ii) Corr (3X-5, 4-2Y)