Download GTU (Gujarat Technological University Ahmedabad) B.Tech/BE (Bachelor of Technology/ Bachelor of Engineering) 2020 Summer 3rd Sem 3130006 Probability And Statistics Previous Question Paper

Enrolment No.___________

**GUJARAT TECHNOLOGICAL UNIVERSITY**

**BE - SEMESTER? III EXAMINATION ? SUMMER 2020**

**Subject Code: 3130006**

**Date:28/10/2020**

**Subject Name: PROBABILITY AND STATISTICS**

**Time: 02:30 PM TO 05:00 PM**

**Total Marks: 70**

**Instructions:**

**1. Attempt all questions.**

**2. Make suitable assumptions wherever necessary.**

**3. Figures to the right indicate full marks.**

4. Statistical Tables are required.

4. Statistical Tables are required.

**Q.1 (a)**An insurance company obtained the following data for accident claims (in

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thousand rupees) from a particular region. Find its mean, median and Mode.

Amount

1 - 3

3 - 5

5 - 7

7 - 9

9 - 11

11 - 13

Frequency

6

47

75

46

18

8

**(b)**A market survey was conducted in four cities to find out the preference for brand

**04**

soap. The responses are shown below:

Delhi

Kolkata

Chennai

Mumbai

Yes

45

55

60

50

No

35

45

35

45

No opinion

5

5

5

5

( a ) What is the probability that a consumer preferred brand , given that he

was from Chennai?

( b ) Given that a consumer preferred brand , what is the probability that he

was from Mumbai?

**(c)**( i ) The number of monthly breakdowns of a computer is a random variable

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having Poisson distribution with mean 1.8. Find the probability that the

computer will function for a month ( a ) without a breakdown ( b ) with at

least one breakdown.

( ii ) Assume that 5 % of the apples weigh less than 150 and 20 % of the

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apples weigh more than 225 . If the distribution of the weight of the

apples is normal, find the mean and standard deviation of the distribution.

**Q.2 (a)**The probability that one of the ten telephone lines is busy at an instant is 0.2.

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( a ) What is the probability that 5 of the lines are busy?

( b ) What is the probability that all the lines are busy?

**(b)**An auto company claims that the mean petrol consumption of its new six cylinder

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car is 9.5 km

per liter which is lower than the existing auto engine. It was found

that the mean petrol consumption of a sample of 50 of these cars was 10 km per

liter with a standard deviation of 3.5 km per liter. Test the claim at 5 % level of

significance.

**(c)**( i ) The life of batteries manufactured by a battery manufacturer can be

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modelled as a random variable having approximately a normal distribution

with = 50 months and = 6 months. Find the probability that the mean

of a random sample of 36 such batteries will be less than 48 months.

( ii ) If two random variables and have the joint density

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(2 + ), 0 < < 1, 0 < < 1

(, ) = {

,

0,

find and the mean of the conditional density 1( | 0.5) where 1() is

the marginal probability density of .

**OR**

**(c)**( i ) A process for making certain bearings is under control if the diameters of

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1

the bearings have a mean of 0.5000 cm. What can we say about this

process if a sample of 10 of these bearings has a mean diameter of 0.5060

cm and a standard deviation of 0.0040 cm?

( ii ) Three balanced coins are tossed. Let denote the number of heads on the

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first two coins and denote the number of tails on the last two coins. Find

the joint distribution of and .

**Q.3 (a)**Show that and are independent events if () = 0.25, () = 0.40, and

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() = 0.50.

**(b)**Given that = 25, = 125, 2 = 650, = 100, 2 = 460 and

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= 508. Later on it was found that two of the points (8, 12) and (6, 8) were

wrongly entered as (6, 14) and (8, 6). Prove that = 2 3

/ .

**(c)**( i ) The runs scored by two batsmen and in 10 matches are given in the

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following table:

14

13

26

53

17

29

79

36

84

49

37

22

56

52

14

10

37

48

20

4

Who is more consistent?

( ii ) Calculate the first four moments about the mean of the following data:

**04**

5

10

15

20

25

6

10

14

6

4

**OR**

**Q.3 (a)**The number of page requests that arrive at a Web server is a Poisson random

**03**

variable. Its probability distribution is as follows:

Number of

0

1

2

3

4

5

6

requests/sec.

Probability

() 0.368 0.368 0.184 0.061 0.015 0.003 0.001

Find the mean and variance of this probability distribution.

**(b)**From the following data of the marks obtained by 8 students in Computer

**04**

Networking (CN) and Complier Design (CD) papers, compute rank coefficient

of correlation.

CN

15

20

28

12

40

60

20

80

CD

40

30

50

30

20

10

30

60

**(c)**( i ) Find out mean deviation about median for the following series:

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Size

4

6

8

10

12

14

16

Freq.

1

2

4

5

4

3

1

( ii ) Find the coefficient of skewness based on the Method of Moments for the

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following data:

Class

0 - 10

10 - 20

20 - 30

30 - 40

40 - 50

13

20

30

25

12

**Q.4 (a)**In a certain district, the proportion of highway sections requiring repairs in any

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given year is a random variable having the probability density

12 2(1 - ) , 0 < < 1

() = {

.

0,

Find the distribution function and use it to determine the probability that at least

half of the highways sections will require repairs in any given year.

**(b)**At a checkout counter customers arrive at an average of 1.5 per minute. Find the

**04**

probabilities that ( i ) at most 4 will arrive in any given minute; ( ii ) one

customer will arrive in the first one minute and two customers will arrive in

the next one minute.

**(c)**Fit a parabola = + + 2 to the following data:

**07**

0

1

2

3

4

1

4

10

17

30

**OR**

**Q.4 (a)**The joint probability density of two random variables 1 and 2 is given by

**03**

2

1 (

(

1 + 22), 0 < 1 < 1, 0 < 2 < 2

1, 2) = {5

.

0,

Find the marginal densities of both the random variables and check whether the

two random variables are independent.

**(b)**A safety engineer feels that 30 % of all industrial accidents in her plant are caused

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by failure of employees to follow instructions. If this figure is correct, find

approximately, the probability that among 84 industrialized accidents in this

plant anywhere from 20 to 30 (inclusive) will be due to failure of employees to

follow instructions.

**(c)**The following show the gain in reading speed of 3 students in a speed-reading

**07**

program, and the number of weeks they have been in the program:

No. of weeks

3

5

2

8

6

9

3

4

Speed gain

86 118 49 193 164 232 73 109

Fit a straight line by the method of least squares.

**Q.5 (a)**Suppose that the time it takes to get service in a restaurant follows a gamma

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distribution with mean 8 minutes and variance 32 minutes. Suppose that you

went to this restaurant at 6: 30 p.m. What is the probability that you will receive

service before 6: 36 p.m.?

**(b)**If 57 out of 150 patients suffering from certain disease are cured by allopathy

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and 33 out of 100 patients with the same disease are cured by homeopathy, is

there reason to believe that allopathy is better than homeopathy at 0.05 level of

significance?

**(c)**( i ) If two independent random samples of size 1 = 7 and 2 = 13 are taken

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from a normal population, what is the probability that the variance of the

first sample will be at least three times as large as that of the second

sample?

( ii ) A courier service advertises that its average delivery time is less than 5

**04**

hours for local deliveries. A random sample of 10 for the amount of time

this courier service takes to deliver packages to an addressee across town

produced the following times: 8, 3, 4, 7, 10, 5, 6, 4, 5, 8. Is this evidence

support the claim of the courier service at 5 % level of significance?

**OR**

**Q.5 (a)**A power supply unit for a computer component is assumed to follow an

**03**

exponential distribution with a mean life of 1200 hours. What is the probability

that the component will survive more than 1500 hours?

**(b)**Twenty people were attacked by a disease and only 18 survived. Will you reject

**04**

the hypothesis that the survival rate if attacked by this disease is 85 % in favour

of the hypothesis that it is more at 5 % level.

**(c)**( i ) The mea

n life of a random sample of 10 light bulbs was found to be 1456

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hours with a standard deviation of 423 hours. A second sample of 17

bulbs chosen at random from a different batch showed a mean life of 1280

hours with a standard deviation of 398 hours. Is the difference between

the mean life of the two batches significant at 5 % level of significance?

( ii ) The manager of a theatre complex with four theaters wanted to see

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whether there was difference in popularity of the four movies currently

showing for Saturday afternoon with the following results: 86, 77, 84, 81

custormers viewed movies 1, 2, 3, and 4 respectively. Complete the test to

see whether there is a difference at the 5 % level of significance.

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This post was last modified on 04 March 2021