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B.Sc., DEGREE EXAMINATION
PART III – Mathematics/Mathematics (CA)
Allied A –STATISTICS FOR MATHEMATICS-I
Unit=I
SECTION-A
1. A random variable X has the following probability distribution:
x: 0
1 2 3
p(x): 3k 5k 7k 5k
The value of k is
a)1/8 b) 1/20 c)1/12 d)1/14
2. If X is random variable, then V(aX+b) is equal to
a) V(X) b) V(X) c) a 2 V(X) + b d) a 2 V(X)+b 2
3. If X and Y are two independent random variables then
a) f(x,y) = f(x).g(y) b) f(x,y) > f(x).g(y) c) f(x,y) < f(x).g(y) d) f(x,y) = f(x/y).g(y)
4. If X and Y are independent then the conditional expectation E(X/Y) is
a) E(Y) b) E(Y/X) c) E(XY) d) E(X)
2
5.
represents
3
2
4
a)
d) None of these
6. The expectation of the number on the die when a six faces die is thrown is
a) 7/3 b) 3/7 c) 7/2 d)2/7
10. Stochastic variable is another name of _____
a) Continuous variable b) Discrete variable
c) Random variable d) None of these
11. Which of the following is a discrete random variable
a) Number of road accidents occurs in a day in a city
b) Life time of a mobile phone
c) Height of a randomly selected student from a college
d) All of the above
12. X is a random variable. Then which of the following is a random variable?
a) aX + b, where a and b constants b) X2 c) X3 d) All the above
13. Values taken by a random variable will always be a _____
a) Positive integer b) Positive real number
c) Real number d) Odd number
14. f(x) = P (x = x) denotes ____ of a random variable X.
a) Probability mass function b) Probability density function
c) Distribution function
d) None of these
15. A random variable X has the following probability function.
x
– 2 –1 0 1 2
3
P(x)
0.1 k 0.2 2k 0.3 k
What is the value of k?
a) 0.4 b) 3 c) 0.1 d) 2
𝐾
16. For the random variable X which has the probability function f(x) = 𝑥! (x = 0, 1, 2, ……..)
the
distribution function is given by
𝐾
a) 𝑒 b) e c) ke d) k
17. If X is a r.v having pdf f (x),then E(X) is called .......
a).Arithmetic mean b).Geometric mean
c).Harmonic mean d)First Quartile
18. The expected value of a constant ‘b’ is ____
1
a) b b) 0 c) 1 d) 𝑏
19. For constants C1 and C2 the expected value of c1 X + c2 is equal to
a) E(c1x + c2) = c1x
b) E(c1x + c2) = c2
c) E (c1x + c2) = c1 E(X) + c2
d) E(c1x+c2) = c1x + E(c2)
20. If a and b are two constants, then which one of the following statements is incorrect?
a) E [aX+b] = aE(x)+b b) E[aX+bX] = aE(X)+bE(X)
c) E[aX+b] = a+b
d) E[(a+b) X] = (a+b) E(X)
21. In terms of moments the mean can be expressed as _____
a) μ31
b) μ21
c) μ1 1 d) μ0 1
22. IF random variables X and Y have expected values 32 and 28 respectively then E(x-y) will
be
equal to
a) 60 b) 4
c) 16
d) None of these
23. If X is a random variable variance of X, Var [X] = E [X-E(x)]2= E(x2) – [E(x)]2 provided
a) E(x2) exists b) E(x2) does not exist
c) Existence or non-existence of E(x2) cannot be proved d) None of these
24. If X is a random variable the rth moment of X usually denoted by μr1 is defined as ______
a) μr 1 = rE(x) b) μr 1 = E[r(x)] c) μr 1 = = r+E(x) d) μr1 = E(xr)
25. The relationship between mean μ, variance σ2 and second moment about the origin μ21 is
given by
a) σ2 = μ + μ21 b) σ2 = μ –μ21 c) σ2 = μ21+μ d) σ2= μ21-μ
SECTION-B
1.
2.
3.
4.
5.
6.
7.
8.
Define distribution function and state its properties.
Define p.m..f and pd.f .give examples.
Define Mathematical expectation and sate its properties.
Define moments. Establish the relationship between raw moments and central moments
Evaluate E (2x) for the following pdf.
1
F(x) = 2𝑥 , x= 1, 2,3,…….. and 0 otherwise, write your comme3nts about the expected value
Prove that E[(X-c)2] = V(X)+[E(X)-c]2
A coin is tossed until a head appears. What is the expectation of the number of tosses
required?
9. A random variable X has mean 10 and variance 25. Find for what values of a and b does the
variable Y= aX+b has expectation zero and variance unity.
10. Sate and prove multiplication theorem on expectation.
SECTION-C
11. a) i) State and prove addition theorem on expectation.
ii) A continuous r.v X has the following probability law
f(x) = kx2 , 0 ≤X ≤1
=0
elsewhere
Determine k and compute p(X ≤ .5)
12. i) State and prove multiplication theorem
ii) A r.v has the following distribution function
F(x) = 0,
for x ≤0
= x/2, for 0 ≤x< 1
= 1/2, for 1 ≤x< 2
= x/4, for 2 ≤x< 4
= 1,
for x≤4
Is the distribution function continuous? If so, find its probability density function.
13. Explain properties of variance
14. a. Find the expectation of the number on a die when thrown.
b. Two unbiased dice are thrown. Find the expected values of the sum of numbers of points on
them.
15. The diameter of an electric cable, say X,is assumed to be a continuous random variable with
p.d.f f(x) = 6x(1-x) , 0 ≤ x ≤1.
(i) Check that is p.d.f
(ii) Determine a number b such that P(x<b)=P(X>b)
16. In four tosses of a coin, let X be the number of heads. Tabulate the 16 possible outcomes
with the corresponding values of X. By simple counting, derive the probability distribution of X
and hence calculate the expected value of X.
17. What is the expectation of number of failures preceding the first success in an infinite series
of independent trials with constant probability p of success in each trial ?
18. In a sequence of Bernoulli trials,let X be the length of the run of either successes or failures
starting with the first trial. Find E(X) and V(X).
19. Sate and prove Cauchy-Schwarts Ineqality
20. Sate and prove Variance of a Linear Combination of random variables
Unit-II
SECTION A
1.A moment generating function is that____
a) Which gives a representation of all the moments
∞
b) m(t) = E(etx) = ∫−∞ etx f (x)dx if X is continuous
c) The expected value of etx exists for every value of t in some interval –h < t<h,
h>0
d) All the above
2. m(t) = E(etx) = Σx etxf (x)dx is for ____
a) X is continuous b) X is discrete
c) both a and b d) None of these
3. If X is a random variable, the rth central moment of X about A is defined as __
a) E [(X-A)r] b) E [(A+x)r] c) E [(xr-Ar)] d) E [(Ax)r]
3. 101) The rth raw moment μr1 is the coefficient of ___ in moment generating function Mx(t) of
the
random variable X.
𝑡
[𝑟!]2 b)
𝑡𝑟
𝑟!
𝑡
c)
d) None of these
𝑟!
4. The characteristic function φx (t) of a continuous random variable X is given by
∞
∞
∞
a) ∫−∞ etx f (x)dx b) ∫−∞ 𝑒 !𝑡𝑥 f(x)dx
c) ∑−∞ 𝑒 𝑡𝑥
∞
d) ∑−∞ 𝑒 𝑖𝑡𝑥
5. If X is a random variable having the pdf
F(x) = qx-1 P, x = 1, 2, 3, …. P+q = 1. Find moment generating function.
𝑝𝑒 𝑡
a) 1−𝑞𝑒 𝑡
b)
𝑞𝑒 𝑡
1−𝑝𝑒 𝑡
c)
𝑒𝑡
𝑝−𝑞𝑒 𝑡
d)
𝑝𝑒 𝑡
𝑞−𝑞𝑝
11
6. If the random variable X take on three values –1, 0 and 1 with probabilities 32 ,
respectively, what is P(1) if we transform X taking Y = 2x+1
11
a)32 b)
16
32
c)
5
32
d) None of these
16
32
,
5
32
7.If φx(t) is the characteristic function of X, what is the value of φ(0)?
a) 1 b) 0 c) φ d) None of these
1 3
8. For a given probability distribution f(x) = , 8 [𝒙] x =0, 1, 2, 3 for random variables X, the
moment generating function is ____
1
1
a) e t b) 8 (1+et )3 c) (1+et)2 d) 4et
9. The expected value of X is equal to the expectation of the conditional expectation of X given
Y is-----------a) E(X/Y)
b) E(E(X/Y))
c) E(Y/X)
d) E(E(Y/X))
10. If X is a r.v. with distribution function F(x), then P (X2 ≤ y) is
a) P (-√𝑦 ≤X≤ √𝑦 )
b) P (X ≤√𝑦 ) ) - P (X ≤ -√𝑦 )
c) F (√𝑦 ) () - F (-√𝑦 )
d) All the above
11. If X is a continuous r.v with mean µ and variance 𝜎2 , then for any positive number k,
P{|X- µ|≥ k 𝜎}≤1/k2 is known as:
a) Lyapunov’s inequality
b) Tchebychev’s inequality
c) Weak Law of larege numbers
d) None of the above
12. Moment generating function of a r.v X, then subject to the convergence of the expansion of
logMx(t) in the powers of t,the function: Kx(t)= logMx(t) is known as:
a) M.G.F
b) Characteristic function
c) Cumulate generating function
d) Maclaurin’s expansion function
13. Let X and Y be two r.v. Then for
F(xy)= kxy, 0<x<4 and 1<y<5
0, otherwise to be a joint density function k must be equal to:
a) 1/100
b) 1/96
1/48
none of these
14. If the joint p.d.f of a two-dimensional random variable(x,Y) is given by:
f(xy) = 2, 0<x<1 and 1<y<x
0 otherwise.
a) 2y,0<y<1
Then the conditional density function of X given Y is:
b) 1-2y,0<y<1
c) 1-y,0<y<1
d) none of these
0 otherwise
0 otherwise
0 otherwise
15. If f(x) = e-x, x > 0 find the pdf of y= x1/2
a) y𝑒 −𝑦 2y ≥ 0 , ≥ b) 2 y𝑒 𝑦 2y ≥ 0 c) 2 y𝑒 −𝑦 2y ≥
1
d) 2y𝑒 −𝑦 2y ≥ 0
-
2
2
- 1/k2
- 1/k2
17. The joint cumulative distribution function F(x,y) lies within the values
a) -1 and +1
c) -∞ and 0
b) -1 and 0
d) 0 and 1
18. If x and y are two independent random variables then f(x,y) = ...
a) f(x)+f(y)
c) f(x).f(y)
b) f(x)-f(y)
d) f(x)/f(y)
19. The value of F (-∞+ ∞) = ....
a) 0
b) 1
c)+∞
d) -∞
20. If X and Y are two independent r.v.’s the cumulative distribution function F(x,y)
is equal to
a).F1(x).F2(y)
b).P(X ≤ x , Y ≤ y )
c).both a and b
d).neither a nor b
21. If X and Y are two independent r.v’s then
a).E(XY)=1
b).E(XY) = 0
c).E(XY)=E(X).E(Y)
d).E(XY) = a constant
22. E(Y /X = x) is called ......
a).regression curve of x on y
b).regression curve of y on x
c).both a and b
d).neither a nor b
23. For the joint pdf f(x,y), the marginal distribution of Y given X=x is given as
a).∑x f(xy)
∞
b)∫−∞ 𝑓(𝑥𝑦) dy
∞
c) ∫−∞ 𝑓(𝑥𝑦) dx
∞
∞
d) ∫−∞ ∫−∞ 𝑓(𝑥𝑦) 𝑑𝑥dy
24. If X and Y are independent, the cumulative distribution Fxy(x, y) is equal to
a). FX(x)FY(y)X
c). both a and b
b). P (X ≤ x)P (Y ≤ y)
d). neither a nor b
25. Let (X,Y) be jointly distributed with density function,
f (x, y ) = 𝑒 −𝑥−𝑦 ; 0 < x < ∞, 0 < y <∞ = 0 ; otherwise Then X and Y are
a). Independent
b). Both having the mean unity
c). Both having the variance unity
d). All of the above..
SECTION B
1. Define Moment generating function
2. Define Cumulant Generating function
3. Define Characteristic function
4. Explain Bivariate R.V
5. Define Marginal distribution of X and Y
6. Define Conditional Distribution of X and Y
7. Explain mathematical expectation of bivariate random variable
8. Define Change of variable
9. Define Central limit theorem
10. Define Weak law of large numbers
SECTION C
11. State and prove Tchebychev’ inequality
12. Explain Properties of M.G.F
13. Explain properties Characteristic Function
14. Define Marginal p.m.f and p.d.f
1
15. The joint probability distribution of two r.v X and Y is given by: P(X= 0, Y= 1) =3 ,
1
1
P(X= 1, Y= -1)= 3 and P(X= 1,Y= 1) = 3. Find (i) Marginal distribution of X and Y
(ii) the conditional distribution of X given Y=1.
16. Explain conditional expectation and variance.
17. Let X and Y be two r.v each taking three values -1,0, and 1 having the joint probability
distribution
X
-1
0
1
Total
-1
0
.2
0
.2
0
.1
.2
.1
.4
1
.1
.2
.1
.4
Total
.2
.6
.2
1.0
(i) Show that X and Y have different expectations.
(ii) Prove that X and Y are uncorrelated
(iii) Find Var X and Var Y
(iv) Given that Y=0, what is the conditional probability distribution of X.
(v) Find V(Y|X=-1)
18. Let
1
f(x) = 4 x= -1
1
4
1
2
x=0
x=1
Find the change of variable y=x2
19. Explain Jacobian transformation
20. Stae and prove C.L.T
Unit-III
SECTION A
1. If X B (n, p), the distribution of Y = n - X is .......
a).B(n,1) c).B(n,p)
b).B(n,x) d).B(n,q)
2. A family of parametric distribution in which mean =variance is
a).Binomial distribution
b).Gamma distribution
c).Normal distribution
d) Geometric distribution
3.If X ~B (3, 1/2) and Y~B(5, 1/2), the probability of P(X+Y=3) is ....
a).7/16
b).11/16
c).7/32
d).None of the above.d).Poisson distribution
4. If X ~B (n, p) , mean = 4, variance = 4/3, then P (X = 5)=......
2
a)[.3]6
1
2
b) [.3]6
1
c)[.3]5[.3]
2
d)4)[.3]6
5. If X ~N (μ, 𝜎2) ,the points of in enflexion of normal distribution curve are
a). ±μ
c).μ ± 𝜎
b). 𝜎 ± μ
d). ± 𝜎
6. An approximate relation between QD and SD of normal distribution is
a).5QD = 4SD b).2QD = 3SD
c).4QD = 5SD d).3QD = 2SD
7. An approximate relation between MD about mean and SD of a normal
distribution is
a).5MD = 4SD b).3MD = 3SD
c).4MD = 5SD d).3MD = 2SD
8. The area under the standard normal curve beyond the lines z = ± 1.96 is
a).95%
b).90%
c).5%
d).10%
9. Let X is a binomial variate with parameters n and p. If n=1, the distribution of X
reduces to
a).Poisson distribution
b).Binomial distribution
c).Bernoulli
d).Discrete Uniform distribution
10. If X is a normal variate with mean 20 and variance 64, the probability that X lies
between 12 and 32 is
a).0.4332
b).0.7475
c).0.1189
d).0.5
11. If Z is a standard normal variate, the proportion of items lying between Z=-0.5
and Z=-3.0 is
a).0.4987
c).0.3072
b).0.1915
d).0.3098
12. If X is a normal variate representing the income in Rs.per day with mean =50 and
SD=10. If the number of workers in a factory is 1200,then the number of workers
having income more than Rs.62 per day is
a).462
b).138
c).738
d).None of these
13. Assuming that the height of students is distributed as N(μ, 𝜎 2).Out of a large
number of students, 5 % are above 72 inches and 10% are below 60 inches. The mean and SD of
the normal distribution are
a).μ = 0, 𝜎 = 1 c).μ = 66, 𝜎 = 4
b).μ = 65, 𝜎 = 5 d).μ = 65, 𝜎 = 4
14. The mgf of binomial distribution is
a) (q + p𝑒 𝑡 )n b) (p +q𝑒 𝑡 )n
c) (q + p )n
d).(q +𝑒 𝑡 )n
15. Binomial distribution with parameters n and p is said to be symmetric if
a).q < p
b).q > p
c).q = p
d).q ≠ p
16. Mean of a chi-square distribution
a) n
b) 2n
c)n2
d)2n+1
17. Which of the following is a sampling distribution:
(i) Binomial
(ii) Poisson
(iii) Chi-square
(iv) None of these
18. If X follow standard normal distribution, then Y =X 2 follows,
(a) Normal
(b) Chi-square with 2 d.f.
(c) Chi-square with 1 d.f.
(d) Nome of these
19. The range of a chi-square variable is,
(a) 0 to n
(b) 0 to ∞
(c) -∞ to ∞
20. For random variable following chi-square distribution,
(a) mean = 2(variance)
(b) 2(mean) = variance
(c) Mean = variance
(d) None of these
21. Variance of a chi-square random variable with ‘n’ d.f. is,
(a) 2n
(b) n+2
(c) n
(d) None of these
22. If X and Y are two independent ch-square variables with degrees of freedom 3 and 4
respectively, then Z=X+Y follows,
(a) Chi-square with 7 d.f.
(b) Chi-square with 12 d.f.
(c) Chi-square with 1 d.f.
(d) None of these
23. The probability distribution of the sum of squares of ‘n’ independent standard normal
random variables is,
(a) Normal
(b) Chi-square
(c) t
(d) None of these
24. ‘student’ is the penname of,
(a) Newton
(b) Chebychev
\
25. The range of a t variable is,
(i) 0 to n
(ii) 0 to ∞
(iii) -∞ to ∞
(c) Laplace
(iv) None of these
(d) Gosset
SECTION B
1. Define Binomial distribution.Mension their properties
2. Define poisson distribution
3. Define Normal distribution
4. Find mean of binomial distribution
5. Find variance of binomial distribution
6. Find mean of poisson distribution
7. Find variance of poisson distribution
8. Find mean of Normal Distribution
9. Explain point of inflexation in Normal distribution
10. Explain properties of Normal distribution
SECTION C
11. Find m.g.f of Binomial distribution. Hence find variance
12. Describe limiting case of binomial distribution
13. Find m.g.f of Normal distribution
14. Find Mean of chi-square distribution
15. Explain relationship between chi-square,t and F
16. Find mean, median, mode of Normal distribution
17. Find mean deviation about mean of Normal distribution
18. Explain F distribution their properties
19. Find m.g.f of chi-square distribution
20. Explain additive property of Normal distribution.
Uni1-IV
SECTION A
1. If the two lines of regression are perpendicular to each other, the
relation between the regression coefficient is
a) bxy = byx
b) bxy .byx = 1
c) bxy + byx = 1
d) bxy + byx = 0
2. If x and y are independent random variables with zero mean and unit
variance, then correlation coefficient (cc) between X + Y and X – Y is
a)
c)-
1
2
1
√2
b) 0
d)
1
√2
3.On the basis of 3 pairs of observations (-1, 1), (0, 0) (1, 1) a statistician
obtains the linear regression of Y on X by the method of least squares.
Which of the following best describes the line of regression.
School of Distance Education
a) A straight line parallel to but not identical with the horizontal axis
b) A straight line identical with the vertical axis
c) A straight line identical with the horizontal axis
d) A straight line parallel to but not identical with the horizontal axis
4. If we get a straight line parallel to the x-axis when the bivariate data
were plotted on a scatter diagram, the correlation between the variable
is
a) 0
b) +1
c) -1
d) none of these
5. Let ‘n’ pairs of observations are collected from a bivariate distribution
(X, Y) with Correlation Coefficient 0.75. Suppose that each x values be
increased by 5 and each Y values decreased by 5. Then the new
correlation will be
a) > 0.75
b) < 0.75
c) 0.75
d) None of these
6. Suppose Correlation Coefficient between X and Y is 0.65. Suppose that
each Y values are divided by -5 then the new correlation will be.
a) > 0.65
b) 0.65
c) – 0.65
d) 0
7. (X, Y) is a bivariate distribution connected by relation 2x – 3y + 5 = 0.
Then Correlation Coefficient is
a) +1
b) -1
1
1
c) 2
d) - 2
8. Let β be the Regression Coefficient and r be the Correlation Coefficient
Then
a) β > r
b) β < r
c) -1 ≤ βr ≤ +1 d) βr ≥ 0
9. If X and Y are two variables, there can be at most
a) One regression line
b) Two regression lines
c) Three regression lines
d) An infinite no. of regression linesLet β be the
10. In a regression line of Y on X, the variable X is known as
a) Independent variable
b) Dependent variable
c) Sometimes independent and some times dependent variable
d) None of these
11. In the regression line Y = a + bx, b is called the
a) Slope of the line
b) Intercept of the line
c) Neither (a) nor (b) d) both (a) and (b)
12. If byx and bxy are two regression coefficients, they have
a) Same sign
b) Opposite sign
c) Either same or opposite signs
d) Nothing can be said
13.. If byx>1 then bxy is
a) Less than 1
c) Equal to 1
b) Greater than 1
d) Equal to 0
14. If x and y are independent, the value of regression coefficient byx is
equal to
a) 1
c) ∞
b) 0
d) Any positive value
15. The lines of regression intersect at the point
a) (X, Y)
b) (𝑋̅, 𝑌̅)
c) (0, 0)
d) (1, 1)
16. The co-ordinates (𝑋̅, 𝑌̅ ) satisfy the line of regression of
a) X on Y
b) Y on X
c) Both X on Y and Y on X
d) None of the two regression lines
17. If r = ± 1, the two lines of regression are:
a) Coincident
b) Parallel
c) Perpendicular to each other
d) None of the above
18. If r = 0 the lines of regression are
a) Coincident
b) Parallel
c) Perpendicular to each other d) None of the above
19. Regression coefficient is independent of:
a) Origin
b) Scale
c) Both origin and scale
d) Neither origin nor scale
20. If each value of X is divided by 2 and of Y is multiplied by 2. Then the
new byx is
a) Same as byx
b) twice of byx
c) four time of byx
d) eight times of byx
21. If from each value of X and Y, constant 25 is subtracted and then each
value is divided by 10, then new byx is
a) Same as byx
b) 2 ½ times of byx
c) 25 times of byx
d) 10 times of byx
22. If Correlation Coefficient between X and Y is r, the Correlation
Coefficient between X2 and Y2 is
a) r
b) r2
c) 0
d) 1
23. The unit of Correlation Coefficient is
a) kg/cc
b) per cent
c) non-existing
d) none of the above
24. The range of simple correlation coefficient is
a) 0 to ∞
b) - ∞ to +∞
c) 0 to 1
d) -1 to 1
25. The range of multiple correlation coefficient R is
a) 0 to 1 b) 0 to ∞
c) -1 to 1
d) -∞ to ∞
SECTION B
1. Explain principle of least squares
2. Explain curve fitting
3. Define correlation coefficient
4. Define Rank correlation
5. Derive Karl Pearson correlation
6. Derive Second degree curve
7. Explain Regression equations
8. Write a properties of regression coefficient
9. Explain two regression lines
10. Define multiple correlation and partial correlation
SECTION C
11. Prove that Correlation coefficient is independent of change of origin and scale
12. How can you use scatter diagram to obtain an idea of the extent and nature(direction) of the
correlation coefficient.
6 ∑ 𝑑2
13.Prove that Spearman’s rank correlation is given by 1- 𝑛3 −𝑛 , where di denotes the difference
between the ranks of the ith individual.
14. Explainthe difference between product moment correlation coefficient and rank correlation
coefficient.
15. Find the correlation coefficient between X and a-X is any random variable and a is constant
16. Derive angle between two lines of Regression
17. The lines of regression in a bivariate distribution are: X+9Y= 7 and Y+4X=49/3.
Find (i) the coefficient of correlation
(ii) the ratios 𝜎𝑥 2 : 𝜎𝑦 2:Cov(X,Y)
18. Find correlation coefficient of following data
X: 1 2
3
4 5
Y: 6 7
8
9
10 Comment the nature of correlation
19. Find the correlation of n natural numbers
20. Explain what are regression lines. Why are there two such lines? Also derive their equations.
Unit-V
SECTION A
1. First 5 students get to marks for English and 20 marks for Maths. The
remaining 20 students has get 5 marks for English and 25 marks for
Maths. Then the coefficient of correlation between their marks in
English and Maths will be.
a) 0
b) +1 c) -1
d) < 0
2. Let ‘n’ pairs of observations are collected from a bivariate distribution
(X, Y) with Correlation Coefficient 0.75. Suppose that each x values be
increased by 5 and each Y values decreased by 5. Then the new
correlation will be
a) > 0.75
b) < 0.75
c) 0.75 d) None of these
3. Suppose Correlation Coefficient between X and Y is 0.65. Suppose that
each Y values are divided by -5 then the new correlation will be.
a) > 0.65
b) 0.65 c) – 0.65
d) 0
4. If a constant 50 is subtracted from each of the value of X and Y, the
regression coefficient is:
1
a) reduced by 50
b)5 th of the original regression coefficient
c) increased by 50
d) not changed
5. Give the two regression lines x + 2y – 5 = 0, 2x + 3y – 8 = 0 and V(x)= 12
the value of V(y) is
3
4
a) 16
b) 4
c) 4
d) 3
6. The correlation between the five paired measurements (3, 6), (½, 1),
(2, 4), (1, 2), (4, 8) for the variables X and Y is equal to:
a) 0
b) -1
c)1/2
d) 1
7. If the correlation between X and Y is 0.5, then the correlation between
2x-4 and 3-2y is
a) 1
b) 0.5
c) -0.5 d) 0
8. If Var(X + Y) = Var(X) + Var(Y), then the value of correlation coefficient
a) 0
b) 1
c) -1
d) 0.5
9. If Var(X + Y) = Var(X – Y) then the correlation coefficient between X and
Y is
a) 1
b) ½
c) ¼
d) 0
10. In rank correlation, let di = xi – yi then ∑𝑛𝑖=1 𝑑 is
a) 0
b) 1
c) -1
d) none of the above
SECTION B
1. Fit a straight line to the following data
X: 1
2
Y: 2.4 3
3
4
6
8
3.6
4
5
6
2. The following are yield in kg/plot Nitrogen level (N) applied to those plots of an experiment
on paddy
N-level: 0
30
60
90
120
Yield: 15
21
28
27
24
Fit a quadratic curve y=a+bN+cN2 to the above data obtain the level of N which it maximizes
yield. Obtain the maximum yield one can obtain from such trials.
3. Fit an parabola curve of the form Y= a+bX+cX2 to the following data
X:1
1
2
2
3
3
4
5
6
7
Y:2
7
7
10
8
12
10
14
11
14
4. Fit an exponential curve of the form Y= abX to the following data
X: 1
2
3
4
5
6
7
8
Y: 1
1.2
1.8
2.5
3.6
4.7
6.6
9.1
5. Fit a second degree parabola to the following data:
X: 1
2
Y: 16 18
3
4
5
19
20
24
6. Fit a straight line equation to the following data:
X: 2
6
Y: 1.2 1.3
4
8
9
3
1.4
1.5
1.6
1.7
7. Fit a straight line equation to the following date:
X: 12 13
14
15
16
17
18
19
Y: 2
6
8
10
12
14
16
4
8. Find a correlation between two variables
X: 1
2
3
4
5
Y: 6
7
8
9
10
9. Find a correlation coefficient between two variables
X: 45 46
58
60
55
62
Y: 48 56
53
47
43
56
10. Find a correlation coefficient between two random variables
X: 100 108
105
106
107
103
108
Y: 112 119
108
107
120
115
113
11. Obtin the equations of two lines of regression for the following data.Also obtain the estimate
of X for Y=70
X: 65 66
67
67
68
69
70
72
Y: 67 68
65
68
72
72
69
71
12. In a partially destroyed laboratory , record of an analysis of correlation data,the following
results only are legible:
V(X)= 9, Regression equations : 8X-10Y=66=0, 40X-18Y=214.
(i) the mean values X and Y , (ii) the correlation coefficient between X and Y and standard
deviation of Y ?
13. Suppose the observations on X and Y are given as :
X: 59 65
45
52
60
62
70
55
45
49
Y: 75 70
55
65
60
69
80
65
59
61
Where N=10 students, and Y= Marks in Maths, X= Marks in Economics. Compute the least
square regression equations of Y on X and X on ?Y.
14. The lines of regression in a bivariate distribution are : X+9Y=7 and
Y+4X=
49
3
.
(i) the coefficient of correlation
(ii) the ratios 𝜎𝑥 2 : 𝜎𝑦 2:Cov(X,Y)
15. The marks obtained by 10 students in Mathematics(X) and Statistics(Y) are given below.
Find the coefficient of correlation between X and Y
X: 75 30
60
80
53
35
15
40
38
48
Y: 85 45
54
91
58
63
35
43
45
44
16. A study is done of the impact of a drug on body temperature and blood pressure. We have
three
observations:
Temperature(F) 99
97
104
97
Pressure (mg) 100
80
150
80
The correlation coefficient is closest to
2
0
7
20
70
0
17. Five children aged 2, 3, 5, 7 and 8 years old weight 14, 20, 32, 42 and 44 kilograms
respectively.
1 Find the equation of the regression line of age on weight.
2 Based on this data, what is the approximate weight of a six year old child?
18. Fit y= ax+b for the following data
X: 0
2
Y: 5.012
4
6
8
10
31.62 28
10
31
19. Fit a curve of the form Y= a+bX+cX2 to the following data
X: 2
3
4
5
6
Y: 144
172.8
207.4
248.8
298.6
20. Fit a curve of the form y= ax+b for the following data
X: 1
2
3
4
5
6
Y: 2.98
4.26
5.12
6.10
6.80
7.50
21. From the following data, compute the coefficient of correlation between X and Y
X-series
Y-series
No. of items
15
15
Arithmetic mean
25
18
Sum of squares of deviations from mean
136
138
Summation of product of deviations of X and Y series from respective arithmetic mean=122
22. The following are the data on the average height if the plants and weight of yield per plot
recorded from 10 plots rice crop.
Height X: 28 26
32
31
37
29
36
34
39
40
Yield Y: 75
82
81
90
80
88
85
92
96
75
Find i. The correlation coefficient between x and y.
ii. The regression coefficient and write down two regression equations.
iii. The probable value of the yield of that plot having an average plant height og 98 cm.
23. The two regression lines are x+2y-5=0 and 2x+3y-8=0. Obtain the mean and value of x and y
of the variance of x= 12. Determine the variance of y and the correlation coefficient.
24. Given the following summary data and obtain the regression equations and estimate y when
x= 16 and x when y= 95, N= 100. ∑ 𝑥𝑦 = 516000, ∑ 𝑥 = 5000, ∑ 𝑦 = 10000, ∑ 𝑥 2 = 26000,
∑ 𝑦 2 = 1040000
25. Find rank correlation coefficient between marks in Maths and English by 10 students.
Maths:
46 54
35 82 54 60 71
82 54 78
English:
86 70 41 79 62 68 66 79
75 76
26. Find the correlation coefficient and regression equation of y on x from the following
bivariate frequency distribution.
x/y
30-40
40-50
50-60
60-70
100-120
4
3
2
0
120-140
3
5
4
2
140-160
1
4
4
3
160-180
0
1
3
1
27. In two sets of variables x and y with 10 observations each the following data were observed.
𝑥̅ = 12,
S.D. of x=3
𝑦̅= 15,
S.D. of y=4
Correlation coefficient between x and y is 0.5. However on subsequent verification it was found
that one value of x= 15 and y= 13 were wrongly taken as 16 and 18. Find the correct value of
correlation coefficient.
28. The following data pertain to the marks in subjects A and B in a certain examination:
Mean marks in A= 39.5
Mean marks in B= 47.5
S.D of marks in A= 10.8
S.D of marks in B= 16.8
rAB= 0.42
Draw the two lines of regression and explain why there are two regression equations. Given the
estimate of marks in B for candidates who secured 50 marks in A
29. You are given the following information about advertising expenditure and sales:
Advertising Expenditure(X)
Sales(Y)
Mean
10
90
S.D
3
12
Correlation coefficient = 0.8
What should be the advertising budget if the company wants to attain sales target of Rs. 120
lakhs?
30. You are given 10 observations on price (X) and supply (Y) the following data were obtained
∑ 𝑥𝑦 = 3467, ∑ 𝑥 = 130, ∑ 𝑦 = 220, ∑ 𝑥 2 = 2288, ∑ 𝑦 2 = 5506
Obtain the line of regression of Yon X and estimate the supply when the price is 16 units.