**(Time: 2 ½ Hours)**

**[Total Marks: 75]**

**N.B. 1) All questions are compulsory.**

**2) Figures to the right indicate marks.**

**3) Illustrations, in-depth answers, and diagrams will be appreciated.**

**4) Mixing of sub-questions is not allowed.**

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Q. 1) Attempt All (Each of 5 Marks) **(15M)**

a) Multiple Choice Questions.

i) If βyx < 1, then βxy is ………………

a) Less than 1|

b) Greater than 1

c) Equal to 1

d) Equal 0

ii) If the correlation coefficient will have a positive sign then …………..

a) X is increasing and Y is decreasing

b) Both X and Y are increasing

c) X is increasing and Y is decreasing

d) None of the above

iii) For two independent events A and B, P(A) = 0.3 and P(B) = 0.4 than P(A ∩ B)= ……………..

a) 0.12

b) 0.3

c) 0.4

d) 0.2

iv). If the lower and upper limits of a class are 10 and 40 then the class-mark (midpoint ) of the class is …………..

a) 25.0

b) 12.5

c) 15.0

d) 30.0

v) The measure of central value which can not be calculated with open-end classes in case of grouped frequency distribution is ……………..

a) Median

b) Mean

c) Mode

d) Third quartile

B) Fill in the blanks

i) Median is same as ………… quartile.

ii) More than cumulative frequency is …………….. in nature.

iii) The average of the upper and lower class boundaries is called as ………………..

iv) If the correlation coefficient between X and Y is perfect then regression lines of X on Y and Y on X are …………………..

v) P(A ∪ A’) ………………………

C) Short Answers.

i) Define mutually exclusive events.

ii) State the probability of the union of two events when they are independent.

iii) State relation between mean, median, and mode when frequency distribution is positively skewed

iv) State range of correlation coefficient.

v) State relation between probabilities of two events A and B when B is a subset of A.

Q. 2) Attempt the following (Any THREE) (Each of 5 Marks) **(15M)**

a) Define mean, median, and mode. Explain how to calculate them for continuous frequency distribution.

b) Explain the procedure of drawing less than an ogive curve for continuous frequency distribution.

c) Prepare frequency distribution for the following data on the number of mangoes; 3,0,0,1,3,2,1,0,4,2,3,3,0,1,2,1,4,3,2,0,1,4,2,1,1,1,3,2,2.

d) Represent the following information using Histogram.

Monthly income |
50-100 | 100-150 | 150-200 | 200-250 | 250-300 |

Number of Employees |
30 | 50 | 100 | 40 | 30 |

e) Explain the concepts of discrete and continuous variables using illustrations.

f) Find, mean, variance, and standard deviation for the following data.

90,99,70,32,76,68,75,31,39,89,40,66,42,93,53,97,43,92,95,36,67,55,47,37

Q 3) Attempt the following (Any Three) ( Each of 5 Marks ) **(15M)**

a) Define first four raw moments about zero and first four central moments. Write down the relations between raw and central moments.

b) What do you understand by kurtosis? Distinguish clearly by drawing figures between leptokurtic and platykurtic.

c) For the following frequency distribution obtain a coefficient of skewness based on quartiles.

Marks |
20-30 | 30-40 | 40-50 | 50-60 | 60-70 | 70-80 |

Number of students |
5 | 20 | 14 | 10 | 8 | 5 |

d) Represent the positive and negative correlation coefficient by scattering diagram.

e) Explain the concept of correlation and regression. How regression is different than correlation?

f) for the following data obtain the coefficient of the regression line of X on Y.

X |
45 | 44 | 50 | 53 | 66 | 30 | 48 |

Y |
42 | 40 | 41 | 42 | 56 | 30 | 43 |

Q. 4) Attempt the following (Any THREE) (Each of 5 Marks) **(15M)**

a) Explain the following concepts;

i) Sample space

ii) Independent events.

b) Define conditional probability and state Baye’s theorem

c) The probability that a student passes a Physics test is 2/3 and the probability that he passes both the Physics test and English test is 14/45. the probability that he passes at least one test is 4/5. What is the probability that he passes the English test?

d) A box contains 6 red 4 white and 5 back balls. A person draws 4 balls from the box at random. Find the probability that among the balls drawn there is at least one ball of each color.

e) In 2002 there will be three candidates for the position of principal, Dr. X, Dr. Y, and Dr. Z – whose chances of getting the appointment are in the proportion 4:2:3 respectively. the probability that Dr. X if selected would introduce co-education in the college is 0.3. The probabilities of Dr. Y and Dr. Z doing the same are respectively 0.5 and 0.8

i) What is the probability that there will be co-education in the college in 2003?

ii) If there is co-education in the college in 2003, what is the probability that Dr. Z is the principal?

f) Bag I contain 6 blue and 4 red balls. Bag II contains 2 blue and 6 red balls. Bag III contains 1 blue and 8 red balls. A bag is chosen at random and two balls are drawn without replacement from this bag. Bothe the balls were blue. Find the probability that bag II was chosen.

Q. 5) Attempt the following (Any THREE) (Each of 5 Marks) **(15M)**

a) Define variance, standard deviation, and coefficient of variation. Explain to calculate them for raw data.

b) Explain the regression model and write the properties of the regression coefficient.

c) An MBA applies for a job in two firms X and Y. The probability of his being selected in firm X is 0.7 and being rejected at Y is 0.5. The probability of at least one of his applications being rejected is 0.6. What is the probability that he will be selected in one of the firms.

d) The probabilities of X, Y, and Z becoming managers are 4/9, 2/9, and 1/3 respectively. the probabilities that the bonus scene will be introduced if X,Y, and Z becomes managers are 3/10, 1/2, 4/5 respectively.

i)What is the probability that the bonus scheme will be introduced and

ii) If the bonus scheme has been introduced, what is the probability that the manager appointed was X?

e) Represent the following data by Stem and Leaf diagram.

86,46,44,68,47,81,77,48,50,87,41,88,59,80,52,85,56,61,58,72,69,82,78,60,54,71.

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