In a television appearance, an MLA from a state asks voters in that state to indicate whether the MLA should vote against or for a particular piece of legislation. The MLA office receives 4100 replies. Since voters from western part tends to be democrats while those from eastern parts tend to e communists, the MLA divides the replies on the basis of their geographical origin. 68% of the replies from the western part are against the legislation and 36% of the replies from the eastern part are against it.

(i) Did the MLA carry out a stratified sampling?

(ii) Did the MLA carry out a stratified random sampling?

(iii) How would you justify the sample design?

(iv) What improvements would you suggest?

The following table gives the hours spent by 50 students in the playground in a week:

Calculate the mean median and standard deviation of the hours required by students.  

The natural angular frequency of an oscillator is 0. It is subjected to a damping force proportional to velocity and the damping force per unit mass per unit velocity is b. A periodic force given by F cos t is applied to it. Establish the differential equation for the oscillator and obtain the condition for velocity resonance.

 Find the probability that at most 5 defective fuses will be found in a box of 200, if experience shows that 20% of such fuses are defective.

i) Two collinear SHMs of amplitudes 3 and 4 units add up to an SHM of amplitude   units. Calculate the phase difference between the two superposing SHMs.

ii) A particle is simultaneously subjected to two mutually perpendicular oscillations given by

x = 3 sin cm, y = 3 cos cm Determine its trajectory of motion.

Consider the case of tossing two dices. Let

A  : be the event of getting an odd total :

B : the event of getting on the first dice, and :

C:  the event of getting a total of seven on two dices.

Then

(i) Find (P A ∩C), (P A ∩ )B and (P B∩ C).

Check whether A,B and C are independent or not. Give reasons for your answers.

The displacement of an object executing SHM is given by:

x = 0.02 cos 4 (t + 0.625)m

Calculate the amplitude, period, maximum velocity, maximum acceleration and initial displacement of the object.

Consider a population of 11 schools from which a sample of size 3 is to be selected. List all the possible samples by circular systematic sampling.

Plating of gold on wrist watches requires a chemical called . AiCI 3 Concentration of the chemical that is added to the solution is an important factor. The objective is to choose that concentration for which plating is uniform. Three concentration of AiCI 3 chosen were 5%, 15% and 20%. These were added in the solution and plating was done and thickness (in units) measured on 5 samples is gives below

Sample ® ¯ Conc. of AiCI3		1	2		3	4	5
5%		3	2		4	2	4
15%		4	4		3	2	4
25%		3	3		4	4	2

Use ANOVA to comment on whether the concentration of AiCI gives same result or 3 not. (Use 05.0 α = )  

Establish the differential equation for simple harmonic motion (SHM) in one dimension. Show that for SHM, the velocity and acceleration of the particle is proportional to and  respectively, where  is the angular frequency of the particle.

In a locality of 18,000 families, a random sample of 840 families was taken. Of these 840 families, 206 families were found to have a monthly income of Rs. 500 or less. Give the confidence interval for the families having income Rs. 500 or less

Two samples of 9 and 8 sizes give the sum of squares of deviations from respective means equal to 160 and 91 inches squares. Test whether these samples have been drawn from same normal population or not? (Use 05.0 α = )

A bicycle shop sells the following number of bicycles from 1990 to 2000.

Compute the first three moving averages of length 3 for the bicycle sales data and place them in line with the corresponding year.  

There are two samples of 1200 and 900 people drawn from populations respectively, which have 30% and 25% of fair-haired people. Test whether the samples drawn from this population maintain difference or not. (Use α = 05.0 )

Two cards are drawn simultaneously or successively without replacement from a well-shuffled deck of 52 cards. Find the probability distribution of number of aces (x) Give a graphical representation of probability distribution.

Following is the distribution of marks (out of 25) obtained by 10 students in Physics and Mathematics.

 Draw a scatter diagram for all the 10 students and calculate the correlation between marks of Physics and Mathematics.

a) The input to a single input neuron is 2, its weight is 2.3 and its bias is - 3 . The neuron output for Linear transfer function is -1 .

b) The SOM is useful for classification.

c) Gradient based optimization methods are used when the objective function is not smooth and one needs efficient local optimization.

d) The -cut of a fuzzy set A in is defined as A . {x ^{_A (x)  ₀ }.}   

e) If W (k₀) = W (k₀ +1 (k₀ + 2) , then perceptron is non-linear separable.