First question is compulsory.

Answer any FOUR out of the remaining questions.

Each question carries 14 marks.

1.   (a) An integer is chosen at random from the first 200 Positive integgers what is the probability that the integers chosen an divisible by 6 or by 8.

(b) State and prove addition theorem probabililty for two events.

(c) Define χ2 –test write any two properties of it.

(d) Eight coins are thrown simultaneously. Find the chance of obtaining at least 6 heads.

(e) Which tests is used for small samples write any two properties of it?

(f) Explain null hypothesisa and critical region.

2.     (a) State and prove Baye’s theorem.

(b) A problem in swtatistics is given to five students A,B,C,D and E their chances of solving it are 1/2 ,1/3,1/4,1/5 and 1/6 what is the probability that the problem will be solved?

3.    (a) Prove that Poissom distribution is the limiting case of binomial distribution for very large trial with very small probability.

(b)  among 10,000 random digitas in how many cases do we expect that the digit 3 appears at most 950 times ( the area under standard normal curve for Z+1.667 is 0.4525 approximately).

4.   (a) Define normal distribution.  Write any two properties of it.

(b)   In a certain examination the percentage of candidates passing getting distinction were 45 and 9. Estimate the average marks obtained by the stidents, the minimum pass and distinction marks in 40 and 75 respectively.

5.   (a) If θ is the angel between the two regression lines show that tanθ=1-r2/r σxσyx2y 2    .

(b)  Calculate Karl Person’s coefficient of correlation from the following data and interpret its value.

ROLL No. of Students:    1              2              3              4              5

Marks in Accountancy:  48           35           17           23           47

Marks in Statistics:           45           20           40           25           45

6.  (a) find maximum likelihood estimator for Poission distribution parameter.

(b).  Determine a 95% confidence interval for the variance of the normal population with the sample 145.3,145.1,145.4,146.2.

7.   (a)  the mean life of sample of 10 electric bulbs was found to be 1,456 hours iwht standard deviation of 423 hours.  Asecond sample of 17 bulbs chosen from a different batch showed a mean life of 1,280 hours with standard deviation of 398 hours is there a significant difference between the means of the two batches?

(b)  If in a random sample of 200 persons suffering with ‘headache’ 160 persons gor cured by a drug can we accept the claim of the manufacture that is drig cures 90% of the suffers use 0.01 level of significance.

8.  (a)  Derive the waiting time distribution in the queue for M/M?I model.

(b)  customers arrive at asales counter manned by asingle person according to a Poisson process with a men rawte of 20 per hour.  Rthe time reuired to serve sustomer has on exponential distribution with a mean of 100 seconds.  Find the average waiting time of customer.

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First question is compulsory. Answer any FOUR out of the remaining questions. Each question carries 14 marks. 1.   (a) An integer is chosen at random from the first 200 Positive integgers what is the probability that the integers chosen an divisible by 6 or by 8. (b) State and prove addition theorem...