R05220101-probability-and-statistics

Set No. 1

Code No: R05220101

II B.Tech Supplimentary Examinations, Aug/Sep 2008 PROBABILITY AND STATISTICS ( Common to Civil Engineering, Mechanical Engineering, Chemical Engineering, Mechatronics, Production Engineering, Bio-Technology and Automobile Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ⋆⋆⋆⋆⋆ 1. (a) For any three arbitrary events A, B, C prove that P (A′ ∪ B ′ ∪ C) = P (A) + P (B) + P (C) − P (A ∩ B) − P (B ∩ C) − P (C ∩ A) + P (A ∩ B ∩ C) (b) In a certain town 40% have brown hair, 25% have brown eyes and 15% have both brown hair and brown eyes. A person is select at random from the town i. If he has brown hair, what is the probability that he has brown eyes also ii. If he has brown eyes, determine the probability that he does not have brown hair [8+8] 2. (a) A sample of 4 items is selected at random from a box containing 12 items of which 5 are defective. Find the expected number of defective items. (b) Six cards are drawn from a pack of 52 cards. Find the probability that i. at least three are diamonds ii. none is a diamond.

[8+8]

3. (a) Using recurrence formula find the probabilities when x=0, 1, 2, 3, 4 and 5 If the mean of Poisson distribution is 3. (b) If the masses of 300 students are normally distributed with mean 68 kgs and standard deviation 3 kgs how many students have masses. i. Greater then 72 kg ii. Less than or equal to 64 kg iii. Between 65 and 71 kg inclusive

[8+8]

4. Sample of size 2 are collected from the population 3,5,7,9,11 without replacement. Find the (a) Mean of population. (b) Standard deviation of population. (c) Mean of the sampling distribution of means. (d) Standard deviation of sampling distribution of means.

[8+8]

5. (a) A sample of 100 iron bars is said to be drawn from a large number of bars. Whose lengths are normally distributed with mean 4 feet and S.D 0.6ft. If the sample mean is 4.2 ft, can the sample be regarded as a truly random sample? 1 of 2

Set No. 1

Code No: R05220101

(b) A random sample of 500 apples was taken from a large consignment and 60 were found to be bad. Within the 98% confidence limits for the percentage number of bad apples in the consignment. [8+8] 6. The following is the distribution of hourly number of trucks arriving at a company’s warehouse; No.of Trucks 0 1 2 3 4 5 6 7 8 Frequency 52 151 130 102 45 12 5 1 2 Find the mean of this distribution, and using it as parameter λ, fit a Poisson distribution. Test for goodness of fit at the 0.05 level of significance? [16] 7. (a) The following data pertain to the demand for a product ( in thousands of units) and its price ( in cents) charged in five different market areas: Price (x) 20 16 10 11 14 Demand 22 41 120 89 56 Fit a straight line of the form y = ao + a1 x to the above data (b) Fit the model y =axb to the following data: x: 1 2 3 4 5 6 y: 2.98 4.26 5.21 6.10 6.80 7.50

[8+8]

8. Compute the coefficient of correlation and the two lines of regression for the following data. [16] X Y

14 16 17 18 19 20 21 22 23 84 78 70 75 66 67 62 58 60

⋆⋆⋆⋆⋆

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Set No. 2

Code No: R05220101

II B.Tech Supplimentary Examinations, Aug/Sep 2008 PROBABILITY AND STATISTICS ( Common to Civil Engineering, Mechanical Engineering, Chemical Engineering, Mechatronics, Production Engineering, Bio-Technology and Automobile Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ⋆⋆⋆⋆⋆ 1. (a) For any three arbitrary events A, B, C prove that P (A′ ∪ B ′ ∪ C) = P (A) + P (B) + P (C) − P (A ∩ B) − P (B ∩ C) − P (C ∩ A) + P (A ∩ B ∩ C) (b) In a certain town 40% have brown hair, 25% have brown eyes and 15% have both brown hair and brown eyes. A person is select at random from the town i. If he has brown hair, what is the probability that he has brown eyes also ii. If he has brown eyes, determine the probability that he does not have brown hair [8+8] 2. (a) Let x be a discrete random variable having the following probability distribution, then X -2 -1 P(X) 0.1 k

0 1 2 3 0.2 2k 0.3 3k

Find i. K ii. mean iii. variance (b) The probability that the life of a bulb is 100 days is .05. Find the probability that one of 6 bulbs. i. Atleast one ii. greater than 4 iii. none, will be having a life of 100 days.

[8+8]

3. (a) The probabilities of a poisson variate taking the values 1 and 2 are equal. Find i. µ ii. p(x ≥ 1) iii. p(1 < x < 4) (b) In a sample of 1000 cases, the mean of a certain test is 14 and standard deviation is 2.5. Assuming the distribution to be normal, find i. how many students score between 12 and 15 ii. how many score above 18 1 of 2

Set No. 2

Code No: R05220101 iii. how many score below 8

[8+8]

4. Sample of size 2 are taken from the population 1,2,3,4,5,6 without replacement Find (a) (b) (c) (d)

The The The The

mean of the population. standard deviation of the population. mean of sampling distribution of means. standard deviation of sampling distribution of means.

[4×4]

5. (a) A die is thrown 256 times an even digit turns up 150 times. Can we say that the die is unbiased. (b) If we can assert with 95% that the maximum error is 0.05 and p=0.2, find the sample size. (c) Write about null hypothesis and testing of null hypothesis . [5+5+6] 6. (a) To examine the hypothesis that the husbands are more intelligent than the wives, an investigator took a sample of 10 couples and administered them a test which measures the IQ as follows: Test the hypothesis with a reasonable test at the level of significance of 0.05? Husbands: 117 105 97 105 123 109 86 78 103 107 Wives 106 98 87 104 116 95 90 69 108 85 (b) In an investigation on the machine performance the following results were obtained: No.of Units inspected No. of defectives Machine 1 375 17 Machine 2 450 22 Test whether there is any significant performance of two machines at α=0.05 [8+8] 7. (a) Derive normal equations to fit the curve of the form y = a + bx2 (b) Fit a curve of the form y = a + bx for the following data and find y when x = 12 [6+10] x 1 2 3 4 5 6 7 8 9 y 9 8 10 12 11 13 14 16 15 8. Observations on price x and supply y the following data was obtained. X X X X X x = 130, y = 220, x2 = 2288, y 2 = 5506 and xy = 3467 Find (a) coefficient of correlation (b) The line of regression of y or x (c) The standard error of estimate. ⋆⋆⋆⋆⋆ 2 of 2

[6+5+6]

Set No. 3

Code No: R05220101

II B.Tech Supplimentary Examinations, Aug/Sep 2008 PROBABILITY AND STATISTICS ( Common to Civil Engineering, Mechanical Engineering, Chemical Engineering, Mechatronics, Production Engineering, Bio-Technology and Automobile Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ⋆⋆⋆⋆⋆ 1. (a) For any three arbitrary events A, B, C prove that P (A′ ∪ B ′ ∪ C) = P (A) + P (B) + P (C) − P (A ∩ B) − P (B ∩ C) − P (C ∩ A) + P (A ∩ B ∩ C) (b) In a certain town 40% have brown hair, 25% have brown eyes and 15% have both brown hair and brown eyes. A person is select at random from the town i. If he has brown hair, what is the probability that he has brown eyes also ii. If he has brown eyes, determine the probability that he does not have brown hair [8+8] 2. (a) Probability density function of a random variable X = 1/2 sin x in 0 ≤ x ≤ π = 0 else where. Find the mean, mode and median for the distribution and also find the probability between 0 and π2 (b) Two dice are thrown 5 times. If getting a double is a success. Find the probability that getting the success i. atleast once ii. Two times.

[8+8]

3. (a) Prove that Poisson distribution is the limiting case of Binomial distribution for very large number of trials with very small probability. (b) Assume that the reduction of a person’s oxygen consumption during a period of transcendental meditation ( TM) is a continuous random variable X with mean 37.6 cc/mt and standard deviation 4.6 cc/mt. Determine the probability that during a period of TM a person’s oxygen consumption will be reduced by i. atleast 44.5 cc/mt ii. atmost 35.0 cc/mt

[8+8]

4. (a) A random sample of size 81 is taken from an infinite population having the mean 65 and standard deviation 10. What is the probability that x will be between 66 and 68? (b) Write about i. Critical region ii. Two tailed test.

[8+8] 1 of 2

Set No. 3

Code No: R05220101

5. (a) A die is thrown 256 times an even digit turns up 150 times. Can we say that the die is unbiased. (b) If we can assert with 95% that the maximum error is 0.05 and p=0.2, find the sample size. (c) Write about null hypothesis and testing of null hypothesis .

[5+5+6]

6. Four methods are under development for making discs of a super conducting material. Fifty discs are made by each method and they are checked for super conductivity when cooled with liquid. 1st Method 2nd Method 3rd Method 4th Method Super Conductors 31 42 22 25 Failures 19 8 28 25 Test the significant difference between the proportions of Superconductors at .05 level. [16] 7. (a) The following data pertain to the demand for a product ( in thousands of units) and its price ( in cents) charged in five different market areas: Price (x) 20 16 10 11 14 Demand 22 41 120 89 56 Fit a straight line of the form y = ao + a1 x to the above data (b) Fit the model y =axb to the following data: x: 1 2 3 4 5 6 y: 2.98 4.26 5.21 6.10 6.80 7.50

[8+8]

8. (a) The following table gives experimental values of the three variates X,Y and Z. Fit a multiple regression of the type Z = α X + βY. X 1 2 3 5 Y 1 3 4 2 Z 7 18 25 23 (b) The following are the marks obtained by 12 students in Economics and Statistics: Economics(x) 78 56 36 66 25 75 82 62 Statistics(y) 84 44 51 58 60 68 62 58 Compute the Spearman rank correlation coefficient between x and y. [8+8] ⋆⋆⋆⋆⋆

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Set No. 4

Code No: R05220101

II B.Tech Supplimentary Examinations, Aug/Sep 2008 PROBABILITY AND STATISTICS ( Common to Civil Engineering, Mechanical Engineering, Chemical Engineering, Mechatronics, Production Engineering, Bio-Technology and Automobile Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ⋆⋆⋆⋆⋆ 1. (a) Two cards are selected at random from 10 cards numbered 1 to 10. Find the probability that the sum is even if i. the two cards are drawn together ii. the two cards are drawn one after the other with replacement. (b) State and prove Baye’s theorem. 3 2 5 , 10 , 10 . The (c) The probabilities of A,B,C to become M.D’S of a factory are 10 probabilities that bonus scheme will be introduced if they become M.D’s are .02, 03 and .04. Find the probabilities A,B,C to be become M.D’s if bonus scheme introduced. [5+5+6]

2. (a) Define random variable, discrete probability distribution, continuous probability distribution and cumulative distribution. Give an example of each. (b) The mean of Binomial distribution is 3 and the variance is 94 . Find i. The value of n ii. p(x ≥ 7) iii. p(1 ≤ x < 6).

[8+8]

3. (a) A Poisson distribution has a double mode at x = 2 and x = 3, find the maximum probability and also find p(x≥2). (b) The weekly wages of 1000 workers are normally distributed around a mean of Rs.70 and S.D of Rs.5/- Estimate the number of workers whose weekly wages will be i. between Rs.70 and Rs.72 ii. between 69 and 72

[8+8]

4. Sample of size 2 are taken from the population 1,2,3,4,5,6 without replacement Find (a) The mean of the population. (b) The standard deviation of the population. (c) The mean of sampling distribution of means. (d) The standard deviation of sampling distribution of means.

[4×4]

5. (a) 400 articles from a factory are examined and 3% are found to be defective. Construct 95% confidence interval. 1 of 2

Set No. 4

Code No: R05220101

(b) A sample of 64 students have a mean weight of 70 kgs. Can this be regarded as a sample from a population with mean weight 65 kgs and standard deviation 25 kgs. (c) Find the size of the sample if the S.D of the population is 9 and there should be 99 confidence that the error of estimate will not exceed 3. [5+5+6] 6. Two independent samples of sound 7 items respectively had the following values. Sample I 11 11 13 11 15 9 12 14 Sample II 9 11 10 13 9 8 10 Is the difference between the means of samples significant?

[16]

7. (a) Fit a curve of the form y = axb for the following data x 61 26 7 2.6 y 350 400 500 600 (b) Derive normal equations to fit the straight line y = a0 + a1 x

[10+6]

8. Two independent variables x and y have means 5 and 10 and variances 4 and 9 respectively. Find the coefficient of correlation between u and v where (a) u = 3x+4y , v =3x-y (b) If x and y are not independent and r=0.5 , u = x+y , v= x-y ⋆⋆⋆⋆⋆

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[8+8]