R07

Code No: 45040

Set No - 1

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III B.Tech I Semester Regular Examinations,Nov/Dec 2009 PROBABILITY AND STATISTICS Metallurgy And Material Technology Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ?????

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1. (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.

(b) A sample of 105 Iron bars whose mean length is 10 ft is drawn. Is it drawn from a population whose mean is 12 ft and standard deviation 4 ft.? [8+8]

or

2. (a) A class contains 10 men and 20 women of which half the men and half the women have brown eyes. Find the probability that the person is a man or has brown eyes. (b) There are three bags. Bag I contains - 1 White, 2 red and 3 green balls Bag II- contains- 2 White, 3 red and 1 green balls Bag III- contains- 3 White, 1 red and 2 green balls One bag is selected and one ball is drawn. Find the probability that it is from

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i. Bag- I ii. Bag-II iii. Bag-III.

[8+8]

3. (a) Customers in a bank arrive with inter arrival time 2 minutes. The service rate is 4 per minute. Find i. Equipment utilisation ii. Expected number of customers in the system

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(b) A super market has a single cashier. During peak hours, customers arrive at a rate of 20 per day. The average number of customers that can be processed by the cashier is 24 per hour. Calculate

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i. Average number of customers in the system. ii. The average time a customer spends in the system.

[8+8]

4. 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 colled with liquid nitrozen. Super Conductor Failures

Method-I 31 19

Method-II 42 8

Method-III Method-IV 22 25 28 25

Perform a Chi-square test at .o5 level whether there is a significance difference between the proportions. [16] 1

Code No: 45040

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Set No - 1

5. (a) The average income of 100 people of a city is Rs.210 with a standard deviation of 10 Rs. For another sample of 150 persons the average income was Rs.220 with a standard deviation of Rs.12 Test the significance between the difference of two means at 5% level.

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(b) Random samples of 600 men and 900 women in a locality were asked whether they would like to have a bus stop near their residence 350 men and 475 women were in favour of the proposal. Test the significance between the difference of two proportions at 5% level. [8+8] 6. (a) If X is a continuous random variable and Y= aX + b, then prove that

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i. E(Y)= a E(X) + b ii. V(Y) = a2 V(X)

(b) Derive formulae to find the mean and variance of the Binomial distribution. [8+8]

(a) The population mean

or

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

(b) The population Standard deviation

(c) The mean of the sampling distribution of mean

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(d) Standard deviation of the sampling distribution of means

[16]

8. (a) Use recurrence formula to find the probabilities when x=0,1,2,3,4and 5 if x is a Poisson variate with mean 3.

?????

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(b) The marks obained in Statistics in a certain examination are found to be normally distributed. If 15% of the candidates get≥60 marks, 40 % get <30 marks, find the mean and the Standard deviation of marks. [8+8]

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Code No: 45040

R07

Set No - 2

1. A population consists if six numbers. 5, 7, 10, 12, 15, 17.

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(a) Write samples of size 2 drawn with replacement.

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III B.Tech I Semester Regular Examinations,Nov/Dec 2009 PROBABILITY AND STATISTICS Metallurgy And Material Technology Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ?????

(b) Find the mean of the sampling distribution of mean

(c) Standard deviation of the sampling distribution of means

[16]

2. (a) If the probability that an individual suffers a bad reaction from a certain injection is .003. Find the probability that out of 1000 individuals

or

i. Exactly 3 ii. ≥ 3 iii. None suffers from a bad reaction.

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(b) 1000 students appear for an examination. It was found that the marks are normally distributed with mean 50 and stsandard deviation 10. The students who get more than or equal to 60 will be placed in 1st division, who get between 50 and 60 will be placed in 2nd division, who get between 40 and 50 in 3rd division, Who get more than 75 in distinction and who get less than 40 will be failed. Find the number of students who get i. Distinction ii. 2nd division iii. Failed

[8+8]

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3. (a) A discrete random variable has the following distribution. x -3 -2 -1 0 1 2 3 Find P(x) k .1 k .2 2k .4 2k

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i. mean ii. variance

(b) Six cards are drawn from a pack of 52 cards. Getting a red card is a success. Find the probability of getting the success i. At least once ii. 3 times iii. P(x < 3)

[8+8]

4. (a) A random sample of 16 values from a normal population showed a mean of 41.5 inches and the sum of the squares of deviation from mean is 13.5 inches. Find the maximum error with 99% confidence. 3

Code No: 45040

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Set No - 2

(b) A sample of size10was taken from a population whose S.D is .03 and the mean is Construct 95% confidence interval for the mean. (c) It is claimed that a random sample of 100tyres with a mean life of 15269 is drawn from a population of tyres which has a mean life of 15200 km. and a standard deviation of 1248 km. Test the validity of the claim. [5+5+6]

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5. (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.

or

(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]

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6. (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. (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]

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7. Derive differential equations for P (t) for pure birth and death process and solve them. [16] 8. (a) What is the probability of getting two queens, if we draw two cards from a pack of 52 cards.

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i. With replacement ii. Without replacement

(b) In a certain college 25% of the students failed in mathematics, 15% failed in chemistry. A student is selected at random. i. If he failed in Mathematics, what is the probability that he failed in Chemistry ii. If he failed in Chemistry, what is the probability that he failed in Mathematics. [8+8] ?????

4

Code No: 45040

R07

Set No - 3

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III B.Tech I Semester Regular Examinations,Nov/Dec 2009 PROBABILITY AND STATISTICS Metallurgy And Material Technology Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ?????

(a) The population mean (b) The population Standard deviation

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1. Apopulation consists if five numbers 2,3,6,8,11,. Consider all Samples of size two which can be taken without replacement from this population. Find

(c) The mean of the sampling distribution of mean

(d) Standard deviation of the sampling distribution of means

or

2. (a) There are 9 items of which 5 are defective

[16]

i. Write the distribution of defective items ii. Find mean iii. The variance

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(b) If 15 dice are thrown. The probability of getting 2 or 5 on the face is a success. Find i. P(X >2) ii. P(1 < x < 5) iii. P(x = 5)

[8+8]

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3. (a) In a certain factory there are two independent processes for manufacturing the same item. The average weight in a sample of 700 items produced from one process is found to be250 gms with a standard deviation of 30 gms while the corresponding figures in a sample of items from the other process are 300 and 40. Is there significant difference between the mean at 1 % level.

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(b) Among the items produced by a factory out of 500, 15 were defective in another sample out of 400, 20 were defective. Test the significance between the difference of two proportions at 5% level. [8+8]

4. (a) The odds that a book will be reviewed favourably by three independent critics are 5 to 2, 4 to 3 and 3 to 4. What is the probability that of the three reviews a majority will be favourable. (b) A purse contains 2 silver and 4 copper coins and a second purse contains 4 silver and 4 copper coins. A coin is selected from one of the purses. Find the probability that it is i. From purse I ii. From purse - II.

[8+8] 5

Code No: 45040

R07

Set No - 3

5. Problems arrive at a computer centre in Poisson fashion with a mean arrival rate of 25 per hour The average computing job requires 2 minutes of terminal time. Calculate the following. (a) Average number of problems waiting for the queue. (b) The probability that queue length is greater than or equal to five.

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(c) The idle time

(d) The probability that a problem will have to wait for more than 10 minutes. [16]

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6. (a) A Random sample of 100 items is taken from a population whose standard deviation is5.1 The mean of the sample is 21.6.construct 95% confidence interval for the mean. (b) Maximum error =.7. The standard deviation =5 with 95% confidence what is the sample size?

or

(c) A lady stenographer claims that she can take the dictation at the rate of 118 words per minute. Can we reject our claim on the basis of 100 trails in which she demonstrates a mean of 112 words with a S.D of 10 words. [5+5+6] 7. (a) Use recurrence formula to find the probabilities when x=0,1,2,3,4and 5 if x is a poisson variate with mean 2.5.

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(b) A sales tax officer has reported that the average sales of the 500 business that he has to deal with during a year is Rs. 36, 000, with a standard deviation of 10,000. Assuming that the sales in these business are normally distributed find i. The number of business as the sales of which are 40,000 ii. The percentage of business the sales of which are likely to range between Rs.30,000 and 40,000. [8+8]

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8. A pair of dice are thrown 360 times and the frequency of each sum is indicated below: sum 2 3 4 5 6 7 8 9 10 11 12 Frequency 8 24 35 37 44 65 51 42 26 14 14

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Would you say that the die is fair on the basis of the chi square test at .05 level of significance. And test the significance at .05 level [16] ?????

6

Code No: 45040

R07

Set No - 4

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III B.Tech I Semester Regular Examinations,Nov/Dec 2009 PROBABILITY AND STATISTICS Metallurgy And Material Technology Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ????? 1. (a) Three light bulbs are chosen at random from 12 bulbs of which 5 are defective. Find the probability that

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i. All are defective ii. One is defective iii. Two are defective.

or

(b) The probability of A,B and C to become M.D’s of a factory are 1/4, 1/3 and 5/1 2. The probability that bonus scheme will be introduced if they become M.Ds are .02, .03 and .04 Find the probability of becoming M.D’s, if scheme will be introduced. [8+8] 2. A population consists if five numbers. 21, 32, 45, 52 and 60 (a) write all Samples of size three without replacement.

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(b) The population Standard deviation

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

[16]

3. (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.

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(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]

4. (a) If X is a poisson variate such that P(x = 1) = 24P(x = 3) Find i. P(x = 0) ii. P(x < 3) iii. P(2 < x < 6)

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Code No: 45040

Set No - 4

(b) 1000 students appear for an examination. It was found that the marks are normally distributed with mean 35 and standard deviation 5. Find the number of students who get i. Marks between 25 and 40 ii. Marks below 20 iii. more than 50

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[8+8]

5. (a) A random sample of 10 boys has the following I.Q 70, 120, 110,1 01, 88, 83, 95, 98, 107 and 100. Construct 95%confidence interval for the mean.

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(b) If the average time taken to a mechanic to rotate the tyres of a car is 39.5 minutes.If the sample size is 40 and the standard deviation is 1.6. What confidence can be assert that the sample mean does not differ from the true mean by more than .5 minutes. [8+8] 6. Two independent samples from two normal populations respectively had the following values of the variables. 16 26 27 23 24 33 42 35 32 28

22 31

or

Sample I Sample II

Do the estimates of variances differ significantly?

[16]

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7. A ticket issuing office is being manned by a single server. Customer arrive to purchase tickets according to a Poisson distribution with a mean rate of 30 per hour. The time required to serve a customer has an exponential distribution with a mean of 90 seconds. Find (a) Average number of customers in the system. (b) Average number of customers in the queue. (c) Average time a customer spending in the system (d) The variance of queue length.

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(e) The probability that a customer will have to wait for more than 10 minutes (f) The probability that the number of customers in the queue exceeds 5

[16]

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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] ?????

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