March 2009     .

1 Semester

Time : 3 Hours)                                                                                                           (Max. Marks : 90

Note :  (1)        Attempt any five questions with a minimum of two questions from each Section.

(2)        All questions carry equal marks.

(Section A)

1.         (a)        A survey was conducted in a region to study the radio listening habits and cinema-viewing habits of the inhabitants. The following ' data were obtained. The figures are in thousands:

Number of people who listen to Radio

Programme t regularly                                                                             37,055

Number of people who listen to Radio

Programme. II regularly                                                                          29,272

Number of people who see Cinema regularly                                         52,606

Number of people who listen to both the Radio

Programmes regularly                                                                             15,078

Radio Programme I and Cinema                                                            27,341

Radio Programme 11 and Cinema                                                          26,272

Radio Programme 1, !land Cinema                                                           9,787

Find out how many people do atleast one thing. Check also the consistency of the data.

(b)        The proportion of bulbs failing before x unit of time is represented by F(x), which is assumed to be of the following form :

It is given that F(200) = 0.30; F(300) = 0.45.

(i)         Determine the constants X and a.

(ii)        Is F(x) continuous?

(iii)

2.         (a)        Find the derivative of with respect to x.

(b)        Determine the criticil path far the function :

and classify them as maxima, minima.

(c)        With usual notations determine

3.         (a)        Find the sum S of the infinite series for 1 x 1 < 1.

Where, S = 1 + 2x + 3x2 + 4x3 + ………..

(b)        A depositor puts Rs. 100 per quarter into a saving bank account that pays 5% per annum compounded quarterly for a period of 5 years. How much would need to be deposited at the beginning of the annuity (with the same rate of Interest) in order to accumulate the same amount as the annuity at the end of 5 years ?

(c)        A person saves Rs. 1,000 uniformly every year for 4-years. Compounding is done continuously at 10% per year. What will be his total savings at the end of four years?

4.         (a)        Determine for what value of k the system of equations have (i) unique solution, (ii) an infinite number of solutions :

x+y+z = 6

x + 2y + kz = 10

x +2y + 3z = 10

Hence obtain anyone unique solution for the system of equations.

(b)        If A show that for any positive integer m,

Section B

5.         (a)        The following table shows the distribution of some families according to their expenditure per week. Number of families corresponding to expenditure groups Rs. (10 - 20) and Rs. (30 - 40) are missing from the table. The median and mode are given to be Rs. 25 and 24 respectively. Calculate the missing frequencies and then find the arithmetic mean and standard deviation of the data :

Experliture               0-10            10-20            20-30             30-40         40-50

No. off amilies         14                ?                   27                   ?                15

(b)        What do you mean by Dispersion ? Discuss the various measures of dispersion. How do you compare the variability of two series?

6.         (a)        60% of the employees of the XYZ Corporation are college graduates. Out of these, 10% are in sales. Out of the employees who did not graduate from college, 80% are in sales. What is the probability that an employee selected at random:

(i)         is in sales ?

(ii)        is neither in Sales nor a College graduate ? •

(b)        In 100 sets of ten Losses of an unbiased coin, in how many cases should we expect :

(i)         seven heads and three tails,

7.         (a)        Discuss different measures of seasonal variations in a time series.

(b)        In two sets of variable X and `I with 50 observations each the following data were observed:

= 10, = 3, = 6, = 2 and  (X, Y) = 0.3. But on subsequent verification it was found that one value of X ( =10) and one value of Y ( =6) were inaccurate and hence weeded out. With the remaining 49 pairs of values, how is the original value of  affected?

8.         (a)        A survey of 320 families with 5 children each revealed the following distribution :

No-of boys      5          4          3          2          1          0

No. of girls      0          1          2          3          4          5

No. of amities 14        36        110      88        40        12

Is this result consistent with the null hypothesis that the male and female births are equally probable?

Given for 5 d. f P(X2 > 11.07) = 0.05.

(b)        Write explanatory note on t-test and z-test.