B.Sc., DEGREE EXAMINATIONPART 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 3p(x): 3k 5k 7k 5kThe value of k isa)1/8 b) 1/20 c)1/12 d)1/142. 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 thena) 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)5. If X is r.v with mean X then the expression E(X-X)2 represents

a) 3 b) 2 c) 4 d) None of these6. 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)2/710. Stochastic variable is another name of _____a) Continuous variable b) Discrete variablec) Random variable d) None of these11. Which of the following is a discrete random variable

a) Number of road accidents occurs in a day in a cityb) Life time of a mobile phonec) Height of a randomly selected student from a college

d) All of the above12. 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 above13. Values taken by a random variable will always be a _____a) Positive integer b) Positive real numberc) Real number d) Odd number14. f(x) = P (x = x) denotes ____ of a random variable X.a) Probability mass function b) Probability density functionc) Distribution function d) None of these15. A random variable X has the following probability function.x – 2 –1 0 1 2 3P(x) 0.1 k 0.2 2k 0.3 kWhat is the value of k?a) 0.4 b) 3 c) 0.1 d) 2

16. For the random variable X which has the probability function f(x) = Kx! (x = 0, 1, 2,

……..) thedistribution function is given by

a)Ke 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 meanc).Harmonic mean d)First Quartile18. The expected value of a constant ‘b’ is ____

a) b b) 0 c) 1 d) 1b

19. For constants C1 and C2 the expected value of c1 X + c2 is equal toa) E(c1x + c2) = c1x b) E(c1x + c2) = c2c) 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) μ3

1 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 beequal toa) 60 b) 4 c) 16 d) None of these23. If X is a random variable variance of X, Var [X] = E [X-E(x)]2= E(x2) – [E(x)]2 provideda) E(x2) exists b) E(x2) does not existc) Existence or non-existence of E(x2) cannot be proved d) None of these24. 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 bya) σ2 = μ + μ21 b) σ2 = μ –μ21 c) σ2 = μ21+μ d) σ2= μ2

1-μ

SECTION-B

1. Define distribution function and state its properties.2. Define p.m..f and pd.f .give examples.3. Define Mathematical expectation and sate its properties.4. Define moments. Establish the relationship between raw moments and central moments5. Evaluate E (2x) for the following pdf.

6. F(x) = 12x , x= 1, 2,3,…….. and 0 otherwise, write your comme3nts about the expected value

7. Prove that E[(X-c)2] = V(X)+[E(X)-c]2

8. 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 lawf(x) = kx2 , 0 ≤X ≤1= 0 elsewhereDetermine k and compute p(X ≤ .5)

12. i) State and prove multiplication theoremii) A r.v has the following distribution functionF(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 variance14. 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 ! c)

tr ! d) None of these

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) ∑−∞

∞

etx d) ∑−∞

∞

eitx

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.

a) p et

1−q et b) q et

1−pet c) et

p−q et d) pet

q−qp

6. If the random variable X take on three values –1, 0 and 1 with probabilities 1132 ,

1632 ,

532

respectively, what is P(1) if we transform X taking Y = 2x+1

a)1132 b)

1632 c)

532 d) None of these

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

8. For a given probability distribution f(x) = , 18¿] x =0, 1, 2, 3 for random variables X, the

moment generating function is ____

a) e t b) 18 (1+et )3 c) (1+et)2 d)

14 et

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) isa) P (-√ y≤X≤√ y ¿¿b) P (X ≤√ y ¿¿ ) - P (X ≤ -√ y ¿¿c) F ¿ () - F (-√ y ¿¿d) All the above

11. If X is a continuous r.v with mean µ and variance σ 2 , then for any positive number k,

P{|X- µ|≥ kσ}≤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. Then the conditional density function of X given Y is:

a) 2y,0<y<1 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− y2y≥ 0 , ≥ b) 2 ye y2y≥ 0 c) 2 ye− y2y≥ d) 12ye− y2y≥ 0

16. If X is a r.v and k is any positive number then p(X- < k )

a) 1/k2 b) 1/k2 c) ) 1- 1/k2 d) 1- 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

15. The joint probability distribution of two r.v X and Y is given by: P(X= 0, Y= 1) =13 ,

P(X= 1, Y= -1)= 13 and P(X= 1,Y= 1) =

13 . 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-101

0.20

.1

.2

.1

.1

.2

.1

.2

.6

.2Total .2 .4 .4 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

f(x) = 14 x= -1

14 x=0

12 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 distributionb).Gamma distributionc).Normal distributiond) Geometric distribution3.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)=......

a)[.23]6 b) [.

13 ]6 c)[.

23]5[.

13 ] d)4)[.

23 ]6

5. If X ~N (μ, σ 2) ,the points of in enflexion of normal distribution curve are

a). ±μ b). σ ± μ

c).μ ± σ 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 normaldistribution 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 Xreduces toa).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 liesbetween 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=-0.5and 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 andSD=10. If the number of workers in a factory is 1200,then the number of workershaving income more than Rs.62 per day is

a).462 c).738b).138 d).None of these

13. Assuming that the height of students is distributed as N(μ,σ 2).Out of a largenumber 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 ifa).q < p b).q > pc).q = p d).q ≠ p16. Mean of a chi-square distributiona) n b) 2n c)n2 d)2n+117. 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 ∞ (d) None of these

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 4respectively, 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 normalrandom variables is,

(a) Normal (b) Chi-square (c) t (d) None of these

24. ‘student’ is the penname of,

(a) Newton (b) Chebychev (c) Laplace (d) Gosset\25. The range of a t variable is,

(i) 0 to n (ii) 0 to ∞ (iii) -∞ to ∞ (iv) None of these

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, therelation 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 unitvariance, then correlation coefficient (cc) between X + Y and X – Y is

a) 12 b) 0

c)- 1√2

d) 1√2

3.On the basis of 3 pairs of observations (-1, 1), (0, 0) (1, 1) a statisticianobtains 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 axisb) A straight line identical with the vertical axisc) A straight line identical with the horizontal axisd) 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 datawere plotted on a scatter diagram, the correlation between the variableis

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 beincreased by 5 and each Y values decreased by 5. Then the newcorrelation 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 thateach 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

c) 12 d) -

12

8. Let β be the Regression Coefficient and r be the Correlation CoefficientThen

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 b) Greater than 1

c) Equal to 1 d) Equal to 0

14. If x and y are independent, the value of regression coefficient byx isequal to

a) 1 b) 0 c) ∞ d) Any positive value

15. The lines of regression intersect at the point

a) (X, Y) b) (X , Y )

c) (0, 0) d) (1, 1)

16. The co-ordinates (X , Y ) 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 lines17. 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 thenew 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 eachvalue 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 CorrelationCoefficient 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 125. 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.

13.Prove that Spearman’s rank correlation is given by 1-6∑ d2

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 ratiosσx2:σy2: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. Theremaining 20 students has get 5 marks for English and 25 marks forMaths. Then the coefficient of correlation between their marks inEnglish 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 beincreased by 5 and each Y values decreased by 5. Then the newcorrelation 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 thateach 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, theregression coefficient is:

a) reduced by 50 b)15 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)= 12the value of V(y) is

a) 16 b) 4 c) 34 d)

43

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 between2x-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 andY is

a) 1 b) ½ c) ¼ d) 0

10. In rank correlation, let di = xi – yi then ∑i=1

n

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 3 4 6 8

Y: 2.4 3 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 3 4 5

Y: 16 18 19 20 24

6. Fit a straight line equation to the following data:

X: 2 6 4 8 9 3

Y: 1.2 1.3 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 4 6 8 10 12 14 16

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= 493 .

(i) the coefficient of correlation

(ii) the ratios σx2:σy2: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 threeobservations:Temperature(F) 99 97 104 97 2 0 7

Pressure (mg) 100 80 150 80 20 70 0 The correlation coefficient is closest to

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 4 6 8 10

Y: 5.012 10 31.62 28 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 75 82 81 90 80 88 85 92 96

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,∑ x2=26000, ∑ y2=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 100-120 120-140 140-160 160-18030-40 4 3 1 040-50 3 5 4 150-60 2 4 4 360-70 0 2 3 1

27. In two sets of variables x and y with 10 observations each the following data were observed.

x= 12, S.D. of x=3

y= 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,∑ x2=2288, ∑ y2=5506

Obtain the line of regression of Yon X and estimate the supply when the price is 16 units.