--- Content provided by FirstRanker.com --- A bag contains 4 white and 3 black balls. Two drawings of 2 balls are made without replacement. What is the probability that the first drawing will give 2 white balls and the second 2 black balls? (2M) Define Marginal and Conditional probability density functions. (3M) If X is a continuous random variable with 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 the value of k. (2M) Define null hypothesis and alternative hypothesis. (3M) Write about Type-I and Type-II errors. (2M) Write properties of good estimator. (3M) If 300 screws are chosen at random from a large batch and 16 screws were defective, find 95% confidence interval for the proportion of defective screws in the batch. (2M) Write about various types of control charts. (3M) Explain about OC curve. (2M)
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PART – B (50 Marks)
(Answer any one full question from each unit)
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UNIT – I
- a) State and prove Baye’s theorem. (5M)
- b) Two dice are thrown. Let X assign to each point (a, b) in S the maximum of its numbers i.e., X(a, b) = max(a, b). Determine the probability distribution of X. (5M)
OR
- a) In a certain town 40% have brown hair, 25% have brown eyes and 15% have both brown hair and brown eyes. A person is selected at random from the town.
- If he has brown hair, what is the probability that he has brown eyes also?
- If he has brown eyes, what is the probability that he has brown hair also?
- What is the probability that he has either brown hair or brown eyes? (5M)
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- b) 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
Determine:- k
- Evaluate P(X<6), P(X≥6)
- P(0<X<5) (5M)
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UNIT – II
- a) Define Binomial distribution and derive mean and variance of Binomial distribution. (5M)
- b) The diameter of an electric cable is assumed to be continuous random variable X with p.d.f f(x) = 6x(1-x), 0 ≤ x ≤ 1. Determine a number ‘b’ such that P(X < b) = P(X > b). (5M)
OR
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- a) Define Normal distribution and derive mean and variance of Normal distribution. (5M)
- b) Out of 800 families with 5 children each, how many would you expect to have
- 3 boys
- at least one boy
- no girl
- 2 or 3 girls (5M)
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UNIT – III
- a) Explain about tests of hypothesis concerning one mean. (5M)
- b) The mean life time of 100 fluorescent light bulbs produced by a company is computed to be 1570 hours with a standard deviation of 120 hours. If µ is the mean lifetime of all the bulbs produced by the company, test the hypothesis µ=1600 hours, against the alternative hypothesis µ≠1600 hours, using a 5% level of significance. (5M)
OR
- a) Explain about tests of hypothesis concerning one proportion. (5M)
- b) Experience shows that 20% of a manufactured product is of top quality. In one day’s production of 400 articles only 70 are of top quality. Test the hypothesis at 0.05 level. (5M)
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UNIT – IV
- a) Explain the terms:
- Parameter
- Statistic
- Sampling distribution
- Standard error (5M)
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- b) A random sample of 200 tins of coconut oil gave an average weight of 4.95 kgs with a standard deviation of 0.21 kgs. Find 95% and 99% confidence limits for the mean weight of the tins from which the sample was taken. (5M)
OR
- a) Write about estimation of one mean and one proportion. (5M)
- b) 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
- The population mean
- The population standard deviation
- The mean of the sampling distribution of means
- The standard deviation of the sampling distribution of means (5M)
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UNIT – V
- a) Explain about control charts for measurements (X and R charts). (5M)
- b) The following data give the number of defects in 10 machines: 3, 6, 4, 2, 5, 7, 8, 2, 5, 3. Draw a control chart for the number of defects. (5M) FirstRanker.com
OR
- a) Explain about control charts for attributes (p and c charts). (5M)
- b) The following table gives the number of missing rivets noticed at final inspection of aircraft:
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Aircraft number 1 2 3 4 5 6 7 8 9 10 Number of missing rivets 4 12 6 8 10 3 15 7 0 9
Draw a control chart for the number of defects and comment on the state of control of the process. (5M) FirstRanker.com