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Code No: 07A4BS01

Set No. 2

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JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD II B.TECH IIII SEM–REGULAR/SUPPLEMENTARY EXAMINATIONS MAY – 2010 B.Tech II Semester Regular Examinations,May 2010 PROBABILITY AND STATISTICS Common to CE, ME, CHEM, MECT, MEP, AME Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ????? 1. (a) Derive formulae for

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i. The mean waiting time in the system. ii. Expected waiting time of a customer in a non empty queue

(b) The customers arrive at a fast food centre at an interval of 11 minutes and they are served at the rate of 1/9 per minute..Find

[8+8]

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i. Average length of the queue ii. Average waiting time in the queue

2. Bricks made in 2 kilns have graded as I,II and III. The production of Bricks in a particular period was as follows. Determine whether the manager of kiln A justified in claiming that he produces Bricks of a higher quality that of the other one. Grade I 24 31

Grade II 43 57

Grade III 13 32

[16]

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kiln Kiln A Kiln B

3. (a) A continuous Random variable has the p.d.f f(x) = K ( x - 1 )3 if 1 < x ≤ 3 = 0 else where Determine i. K ii. Find mean iii. Variance

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(b) The mean and variance of a binomial distribution are 2 and 8 / 5. Find n Mode Maximum probability P(x > 2)

[8+8]

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i. ii. iii. iv.

4. (a) If a number is selected at random from all four digit numbers formed from the numbers 1, 2, 3 and 4, without repetition. Find the probability that the number is i. Divided by 5 ii. an even number

(b) In a certain college 25% of boys and 10% of girls are studying Mathematics. The girls constitute 60% of the students. If a student is selected and is found to be studying Mathematics, find the probability that the student is a 1

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i. Girl ii. boy.

[8+8]

5. Apopulation consists if six numbers 4, 8,12,16.20,24.Consider all Samples of size two which can be taken without replacement from this population. Find

(b) The population Standard deviation (c) The mean of the sampling distribution of mean (d) Standard deviation of the sampling distribution of mean

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(a) The population mean

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6. (a) A manufacturer of pins knows that 2% of his product is defective.If he sells pins in boxes of 100 and guarantees that not more than 4 pins will be defective what is the probability that a box will fail to meet the guaranteed quantity?

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i. Distinction ii. Second Division.

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(b) One thousand students appeared for an examination. It was found that the marks were normally distributed with mean 70 and standard deviation 15. The students who will get greater than or equal to 60 will be placed in first division., between 50 and 60 second division, between 40 and 50 third division, who gets less than 40 will be failed and who get more than 75 in distinction. Find the number of students who get [8+8]

7. (a) A random sample of size 81 was taken whose S.D is 4.5 and the mean is 32 Construct 98% confidence interval for the mean. (b) An industrial engineer intends to use the mean of a random sample of size 150 to estimate the average mechanical aptitude of assembly line workers in a large industry. If on the basis of experience the engineer can assume that σ =6.2 for such data, What can be assert with probability .99 about the maximum size of his error?

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(c) An insurance agent has claimed that the average age of policy holders who insure through him is less than the average for all agents which is 53 yrs. A random sample of 200 policy holders who insured through him reveal that the mean and standard deviation [5+5+6]

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8. (a) Comparing the average protein content of two brands of dog food, a consumer testing service finds that fifty 5-pound packages of brand A dog food had an average protein content of 11 ounces per package and a standard deviation of 1ounce, while sixty 5-pound packages of brand B food had an average protein content of 9 ounces per package and a standard deviation of 0.5 ounce. A difference of 0.5 ounces is considered to be not sufficiently important to report as a consumer issue. At the 0.01 level of significance should the testing service report this as issue. (b) A die is thrown 900 times. 1 or 2 was obtained 200 times.Test whether the die is unbiased [8+8] 2

Code No: 07A4BS01

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Set No. 2

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Code No: 07A4BS01

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Set No. 4

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JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B.Tech II Semester Regular Examinations,May 2010MAY – 2010 II B.TECH II II SEM–REGULAR/SUPPLEMENTARY EXAMINATIONS PROBABILITY AND STATISTICS Common to CE, ME, CHEM, MECT, MEP, AME Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ?????

i. P( x = 2 ) ii. P( x ≥ 3) iii. P( 2 < x < 5 ) in a box of 100 items

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1. (a) If 20 % of the items are defective, If x is the number of defective4 items, Find

(b) Eight coins are tossed. Find the probability of getting 3 to 7 heads using normal distribution [8+8]

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2. A typist at an office receives on average 20 letters per day for typing. The typist works 8 hours per day and he takes 15 minutes for typing. (a) What is the average number of letters waiting to be typed?

(b) The probability that the letter arriving at the typist will have to wait. (c) What is the average waiting time needed to have a letter to be typed? [16]

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(d) The busy tlme.

3. (a) A continuous Random variable has the p.d.f f(x)=Ke−λx If x≥0,=0 otherwise Determine i. K ii. The mean iii. variance

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(b) A summing that half of the population are consumer of rice. If 8 individuals are taken find the probability that i. only two ii. At least two iii. 1 ≤ x ≤ 4 are consumer of rice

[8+8]

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4. (a) Five measurements of the tar content of a certain kind of cigarette yielded 15.5, 15.6,16,14.9 and 16.2 Construct 99% confidence interval for the mean. (b) It was found that the average income of 36 workers is 12000 per annum. .Is this sample has been taken from a large population whose mean of annual income is 11,800 and S.D 800. [8+8]

5. (a) A sample of 100 electric bulbs produced by a manufacturer A showed a mean life time of 1190hrs and a standard deviation of 90 hrs. Another sample of75 bulbs produced by a manufacturer B showed a mean life of 1230hrs with a S.D of 120hrs. Test the significance of the difference between the means at 99% level. 4

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Code No: 07A4BS01

Set No. 4

(b) In a random sample of 400 industrial accidents , it was found that 231 were due to unsafe working conditions Find i. The maximum error ii. Construct 95% confidence interval for the proportion.

[8+8]

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6. (a) If A and B are mutually exclusive events such that P(A) = 4 P(B) and A∪B = S. Find i. P(A∩ BC ) ii. P( A∩ B) iii. P( AC ∪ B)

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(b) Three machines I, II and III produce 40%, 30% and 30% of the total number of items of a factory. The percentages of defective items of these machines are 4%, 2% and 3%. An item is selected at random and found to be defective. Find the probability that it is from

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i. Machine- I ii. Machine-II iii. Machine-III. ‘

[8+8]

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7. A population consists if five numbers 12,32,40.51,60.Consider all Samples of size two which can be taken without replacement from this population. Find (a) The population mean

(b) The population Standard deviation

(c) The mean of the sampling distribution of mean (d) Standard deviation of the sampling distribution of means

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8. Fit a poisson distribution to the following data and test the Goodness of Fit at .05 level of significance. 5 6 2 1

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x 0 1 2 3 4 f 275 72 30 7 5

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Code No: 07A4BS01

Set No. 1

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1. (a) If A is any event , prove that P(AC ) = 1- P(A)

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JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD II B.TECH IIIISEM–REGULAR/SUPPLEMENTARY EXAMINATIONS MAY – 2010 B.Tech II Semester Regular Examinations,May 2010 PROBABILITY AND STATISTICS Common to CE, ME, CHEM, MECT, MEP, AME Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ?????

(b) If A and B are two events prove that, if A⊆B, then P(A) ≤ P(B)

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(c) What is the probability of getting two queens, if we draw two cards from a pack of 52 cards i. With replacement ii. Without replacement.

[5+5+6]

i. ii. iii. iv.

Expected queue length Waiting time Busy period Mean arrival rate

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2. (a) Define the terms

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(b) In a colour T.V manufacturing plant, a loading unit takes exactly 10 minutes to load 2 T.V. sets into a wagon and again comes back to the position to load another set of T.V. If the arrival rate is 2 T.V sets per 20 minutes. Calculate the average time of T.V sets in a stationary state. [8+8] 3. (a) Find 95% confidence interval for the mean of a normally distributed population from which the following sample was taken 15,17,10,18,16,9,7,11,13 and 14

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(b) A random sample of size 225 is taken whose mean is 80. Can this be regarded as a sample from a population with mean weight 82 and standard deviation15. [8+8]

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4. (a) Two horses A,B were tested according to the time (in seconds) to run a particular track with the following results. Horse A 28 30 32 33 33 29 34 Horse B 29 30 30 24 27 29 Test whether the two horses have the same running capacity at 95% level. (b) The daily wages in rupees of skilled workers in two cities are as follows. City Size of the sample of workers S.D. of wages in the sample City A 16 25 City B 13 32 Test at the 0.01 level whether the variances of the wages in two cities are equal. [8+8] 6

Code No: 07A4BS01

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Set No. 1

5. (a) In a city A 20% of a random sample of 900 school boys had a certain slight physical defect. In another city B 18.5% of a random sample of 1600 school boys had the same defect. Is the difference between the proportions is significant at .05 level of significance

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(b) Samples of students were drawn from two universities and from their weights in kgm and deviations are calculated. Make a large sample test to test the significance of the difference between the means [8+8] 6. (a) Prove that the Poisson distribution can be approximated by the binomial distribution.

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(b) Suppose that the weights of 800 male students are normally distributed with mean µ = 140 pounds with a standard deviation of 10 pounds.Find the number of students whose weight are i. Between 138 and 148 ii. More than 152

[8+8]

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7. (a) A sample of 5 items is selected at random from a box containing 15 items of which 8 are defective find i. Mean ii. variance of defective items.

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(b) 15 % of items from a factory are defective. Find the probability that in a sample of 12 i. Exactly 9 ii. At least four iii. P(2 < x < 7) are defective

[8+8]

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8. Take 30 slips of paper and label 5 each -4 and 4, four each -3 and 3, three each -2 and 2, two each -1, 1 and If each slip of paper has the same probability of being drawn find the mean and variance of the distribution. [16]

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Code No: 07A4BS01

Set No. 3

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JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD II B.TECHII II B.Tech SEM–REGULAR/SUPPLEMENTARY EXAMINATIONS II Semester Regular Examinations,May 2010MAY – 2010 PROBABILITY AND STATISTICS Common to CE, ME, CHEM, MECT, MEP, AME Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ????? 1. A typist at an office receives on average 30 letters per day for typing. The typist works 8 hours per day and he takes 15 minutes for typing

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(a) What is the average number of letters waiting to be typed?

(b) The probability that more than or equal to 4 letters are being waiting to be typed. (c) What is the average waiting time needed to have a letter to be typed?

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(d) The time that the typist is idle per day

2. (a) For the discrete probability distribution x 0 1 2 3 4 5 6 7 2 2 2 f 0 K 2K 2K 3K k 2K 7K +K Determine

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i. K ii. mean iii. variance.

(b) Two dice are thrown 4 times. Getting a sum of 7 on the faces is a success. Find the probability that the sum gets i. Twice ii. only once.

[8+8]

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3. Six coins are tossed 320 times. P is the probability of getting 6 heads Find (a) µ

(b) P(x≤2)

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(c) P( 1≤ x < 6)

4. Apopulation consists if five numbers 5,8,15,24,32,.Consider all Samples of size two which can be taken without replacement from this population.Find (a) The population mean

(b) The population Standard deviation (c) The mean of the sampling distribution of mean (d) Standard deviation of the sampling distribution of mean

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[16]

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Code No: 07A4BS01

Set No. 3

5. (a) If A,B and C are three events 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) There are Three boxes I ,II and III. Box I contains 4 Red 5 Blue and 6 White balls. BoxII contains 3 Red 4 Blue and 5 White balls. BoxIII contains 5Red 10 Blue and 5 White balls. One box is chosen and one ball is drawn from it. What is the probability that

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i. Red ball is chosen ii. Blue ball is chosen iii. White ball is chosen.

[8+8]

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6. (a) Construct 95% confidence interval for the mean of a normally distributed population from which the following sample was taken 15,17,10,18,16,9,7,11,13 and 14

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(b) A lady stenographer claims that she can take the dictation at the rate of 120 words per minute. Can we reject our claim on the basis of 100 trails in which she demonstrates a mean of 116 words with a S.D of 15 words. [8+8] 7. Fit a binomial distribution to the following data and test the Goodness of Fit at .05 level of significance. 29 60 63

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Poor 23 60 Average 28 79 Very good 9 49

x 0 1 2 3 4 f 38 144 342 287 164

5 25

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8. (a) In a certain factory there are two independent processes for manufacturing the same item. The average weight in a sample of 400 items produced from one process is found to be 150 gms with a standard deviation of 20 gms while the corresponding figures in a sample of 500 items from the other process are 190 and 24. Is there significant difference between the mean (test at 95% level).

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(b) Among 100 fish caught in a large take ,18 were inedible due to the pollution of the environment. With what confidence can we assert that the error tof this estimate is at most .065? [8+8] ?????

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