Download AKTU B-Tech 3rd Sem 2016-2017 NCS 303 Computer Based Numerical And Statistical Techniques Question Paper

II B.Tech I Semester Regular Examinations, November 2018

PROBABILITY AND STATISTICS

(Com. to CE, CSE, IT)

Time: 3 hours Max. Marks: 70

Note: 1. Question Paper consists of two parts (Part-A and Part-B)

2. Answer ALL the question in Part-A

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3. Answer any FOUR questions from Part-B

PART –A

(14 Marks)

1. 1. Define Random variable and mathematical expectation. (2M)
   2. Write properties of normal distribution. (2M)
   3. Define null hypothesis and alternative hypothesis. (2M)

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   4. Write the control limits for R-chart. (2M)
   5. Define Markov chain and stochastic matrix. (3M)
   6. Write about classification of states of a Markov chain. (3M)

PART -B

(56 Marks)

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2. A random variable X has the following probability function:

   x	0	1	2	3	4	5	6	7
   P(x)	0	k	2k	2k	3k	k2	2k2	7k2+k

   1. Find k
   2. Evaluate P(X<6), P(X=6) and P(0<X<5)

   (14M)

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3. If 20% of the bolts produced by a machine are defective, determine the probability that out of 4 bolts chosen at random

   1. none is defective
   2. 1 is defective
   3. at most 2 are defective

   (14M)

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4. The continuous random variable X has probability density function given by $$f(x) = \begin{cases} kx, & 0 \le x \le 2 \\ 2k, & 2 \le x \le 4 \\ -kx + 6k, & 4 \le x \le 6 \end{cases}$$ Find k and mean value of X. (14M)

5. A population consists of the four numbers 3, 7, 11, 15. Consider all possible samples of size 2 which can be drawn without replacement from the population. Find

   1. the population mean

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   2. the population standard deviation
   3. the mean of the sampling distribution of means
   4. the standard deviation of the sampling distribution of means. (14M)

6. The nicotine content in milligrams in two samples of tobacco were found to be

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   Sample 1	21	24	25	26	27
   Sample 2	22	27	28	29	31	36

   Can it be concluded that the two samples came from the same normal population? (14M)

7. The following are the figures of defectives in 20 samples each containing 200 items

   4, 5, 14, 4, 22, 16, 6, 16, 9, 6, 8, 10, 1, 13, 17, 18, 20, 9, 7, 8

   Calculate the control limits for p-chart and state whether the process is under control. (14M)

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8. A training process is considered as a two state Markov chain. The transition probability of an employee that is in state I (Needs Training) or state II (Is Well-Trained) is given by the following matrix

   $$ \begin{bmatrix} 0.6 & 0.4 \\ 0.05 & 0.95 \end{bmatrix} $$

   Determine the probability that an employee remains in the trained state after 2 transitions, given that he was in state I initially. Also find the steady state probabilities. (14M)

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