Download Code No: X0521 R07 SET - 1 II B. Tech I Semester Supplementary

R07
Code No: X0521
SET - 1
II B. Tech I Semester Supplementary Examinations Dec - 2013
PROBABILITY AND STATISTICS
(Com. to CSE, IT)
Time: 3 hours
Max. Marks: 80
Answer any FIVE Questions
All Questions carry Equal Marks
1.
a) If A and B are two events , then prove that P( A B) P( A) P(B) P( A B).
b) Determine the probability of the following event: The sum 8 appears in a single toss of pair
of fair dice.
c) Police plan to enforce speed limits by using radar traps at 4 different locations within the city
limits. The radar traps at each of these locations L , L , L , L are operated for 40%, 30%,
1
2
3
4
20% and 30% of the time. If a person who is speeding on his way to work has probabilities
of 0.2, 0.1, 0.5 and 0.2 respectively of passing through these locations, what is the
probability that he will be fined?
2.
a) Check whether the following can define probability distribution and explain your answer :
x
f ( x ) 15 for x 0,1,2,3,4,5

b) Consider the density function
f(x 
)
k x ,0 x 1
0 , elsewhere
i) Evaluate k ii) Find the distribution function
F (x) and use it to evaluate P(0.3  X 0.6).
3.
a) Find the mean and variance of Binomial distribution.
b) Calculate the mean and standard deviation of a normal distribution in which 31% are under
45 and 8% are over 64.
4.
Let U be a variable that stands for any of the elements of the population 2, 7, 9 and
1
variable that stands for any of the elements of the population 3, 8. Compute:
a) 
b) 
c) 
d) 
U1
U2
U 1 U 2
U 1 U2
f)  U
g)  U
2
U
Verify that i) 
h) 
UU
2
1
2
1

U U  U
1
2
1

U
2
2
2
ii) 
 
U U  U
U
1
2
1
2
1 of 2
U be a
2
e) 
U
1
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Code No: X0521
R07
SET - 1
5.
a) A random sample of 200 measurements from a large population gave a mean value of 50 and
a S.D. of 9. Determine 95% confidence interval for the mean of population.
b) The values of a normal random variable X obtained in random samples are
55,65,82,48,55,75,70,62 . Find a 95% confidence interval for the mean  of X .
6.
a) A normal population has a mean of 6.8 and standard deviation of 1.5.A sample of 400
members gave a mean of 6.75. Is the difference significant? Use a 0.05 level of significance.
b) Intelligence tests on two groups of boys and girls gave the following results. Examine if the
difference is significant. Use a 0.05 level of significance.
Mean S.D. Size
Girls 70
10
70
Boys 75
11
100
7.
a) The lifetime of electric bulbs for a random sample of 10 from a larger shipment gave the
following data:
Item
1
2
3
4
5
6
7
8
9 10
Life in ‘1000s of hrs.
4.2 4.6 3.9 4.1 5.2 3.8 3.9 4.3 4.4 5.6
Can we accept the hypothesis that the average life time of bulb is 4000 hrs? Use a 0.05 level of
significance.
b) The average numbers of articles produced by two machines per day are 200 and 250 with
standard deviation 20 and 25 respectively on the basis of records of 25 days production. Can
you regard both the machines as equally efficient at 5% level of significance?
8.
A toll gate is operated on a freeway where cars arrive according to a Poisson distribution with
mean frequency of 2.2 cars per minute. The time of completing payment follows an
exponential distribution with mean of 30 seconds. Find:
a) The idle time of the counter
b) Average number of cars in the system
c) Average number of cars in the queue
d) Average time that a car spends in the system
e) Average time that a car spends in the queue
The probability that a car spends more than 35 seconds in the system.
Note :- Statistical tables are required ( Areas under the Standard Normal Curve from 0 to Z,
Percentile Values tp for Student’s t- distribution with v degrees of freedom, Percentile Values
2p for the Chi-Square distribution with v degrees of freedom, 95th Percentile Values (0.05
levels), F0.95 for the F- Distribution and 99th Percentile Values(0.01 levels), F0.99 for the F
Distribution).
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R07
Code No: X0521
SET - 2
II B. Tech I Semester Supplementary Examinations Dec - 2013
PROBABILITY AND STATISTICS
(Com. to CSE, IT)
Time: 3 hours
Max. Marks: 80
Answer any FIVE Questions
All Questions carry Equal Marks
1.
a) If A,B and C are three events , then prove that
P( A B C) P( A) P(B) P( C) P( A B) P( A C) P(B C)
P( A B C).
b) Determine the probability of the following event:
A non-defective bolt will be found, if out of 600 bolts already examined, 12 were defective.
c) Suppose colored balls are distributed in three indistinguishable boxes as follows:
Box I Box II Box III
Red
2
4
3
3
1
4
White
Blue
5
3
3
10
8
10
Total
A box is selected at random from which a ball is selected at random. What is the probability
that the ball is red?
2.
a) Check whether the following can define probability distributions, and explain your answer :
2
for x 0,1,2,3
f ( x ) 5 x
6

b) The shelf life, in days, for bottles of a certain prescribed medicine is a random variable
having the density function
20,000
, x 0
3
f ( x )  ( x 100)
0 ,
elsewhere
Find the probability that a bottle of this medicine will have a shell life of
i) at least 200 days
ii) anywhere from 80 to 120 days.
3.
a) Find the mean and variance of the Poisson distribution.
b) If the heights of 300 students are normally distributed with mean 68.0 inches and standard
deviation 3.0 inches, how many students have heights
i) Greater than 72 inches,
ii) Less than or equal to 64 inches,
iii) Between 65 and 71 inches inclusive.
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Code No: X0521
4.
5.
R07
SET - 2
Let U be a variable that stands for any of the elements of the population 3, 7, 8 and
U be a
1
2
variable that stands for any of the elements of the population 2, 4. Compute:
a) 
b) 
c) 
d) 
e) 
U
U
U U
U U
U
1
2
1
2
1
2
1
f) 
g) 
h) 
U
UU
U U
2
2
2
1
1
2
2
Verify that i) 
  
ii) 
 
U U
U
U
U U  U
U
1
2
1
2
1
2
1
2
a) The mean weight obtained from a random sample of size 100 is 64 gms. The S.D. of the
weight distribution of the population is 3 gms. Set up 99% confidence limits of the mean
weight of the population.
b) The numbers 24.4, 18.9, 12.8, 20.5, 19.1, 15.2, 21.7, 14.6 from a random sample of values
normally distributed random variable. Find a 95% confidence interval for the mean  of X.
6.
a) A random sample of 900 wooden sticks has a mean 3.4 cms. Can it be reasonably regarded
as a sample from a large population of mean 3.2 cms and standard deviation 2.3 cms? Use a
0.05 level of significance.
b) Two random samples of sizes 1000 and 2000 of farms gave an average yield of 2000 kg and
2050 kg respectively. The variance of wheat farms in the country may be taken as 100 kg.
Examine whether the two samples differ significantly in yield. Use a 0.05 level of
significance.
7.
a) The 9 items of a sample have the following values 45, 47, 50, 52, 48, 47, 49, 53, 51.Does the
mean of these values differ significantly from the assumed mean 47.5? Use a 0.05 level of
significance.
b) Two salesmen represent affirm in a certain company. One of them claims that he makes
larger sales than the other. A sample survey was made and the following results were
obtained:
No. of sales
1st Salesman(18) 2nd Salesman(20)
Average sales
$175
$210
S.D.
$25
$20
Find if the average sales differ significantly. Use a 0.05 level of significance.
8.
a) Derive the average number of customers in the system, In (M
/ M / 1) : ( / FCFS ) model.
b) A ticket issuing office is being manned by a single server. Customers arrive to purchase
tickets according to a Poisson process 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 the value
of P , L and W where L andW denote the expected line length and waiting time in the
n
s
s
s
s
system respectively.
Note :- Statistical tables are required ( Areas under the Standard Normal Curve from 0 to Z,
Percentile Values tp for Student’s t- distribution with v degrees of freedom, Percentile Values
2p for the Chi-Square distribution with v degrees of freedom, 95 th Percentile Values (0.05
levels), F0.95 for the F- Distribution and 99th Percentile Values(0.01 levels), F0.99 for the F
Distribution).
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R07
Code No: X0521
SET - 3
II B. Tech I Semester Supplementary Examinations Dec - 2013
PROBABILITY AND STATISTICS
(Com. to CSE, IT)
Time: 3 hours
Max. Marks: 80
Answer any FIVE Questions
All Questions carry Equal Marks
1.
a) If A and Ac are complementary events, then prove that P ( A ) P(
c
A ) 1.
b) Determine the probability of the following event:
The sum 7 or 8 or 12 appears in a single toss of a pair of fair dice.
c) Three machines A, B and C produce respectively 50%, 30% and 20% of the total number of
items of a factory. The percentages of defective output of these machines are 3%, 4% and
5%. If an item is selected at random, find the probability that the item is defective.
2.
a) Check whether the following can define probability distribution, and explain your answer :
1
f ( x ) 4 for x 3,4,5,6

b) If X is a continuous random variable with probability density function given by
kx
f ( x)

if 0 x 2
2k
if 2 x 4
kx 6 k if 4 x 6.
i) Find k and
ii) The distribution function F ( x).
3.
a) If 3% of the electric bulbs manufactured by a company are defective, find the probability
that in a sample of 100 bulbs i) 0 ii) 1
iii) 2
iv) 3 v) 4
vi) 5 bulbs will be defective.
b) Out of 2000 families with 4 children each, how many would you except to have
i) 2 boys ii) 1 or 2 girls iii) no girls. Assume equal probabilities for boys and girls.
4.
A population consists of the five numbers 2, 3, 6, 8, 11. Consider all possible samples of size
two that can be drawn with replacement from this population. Find
a) The population mean,
b) The population standard deviation,
c) The mean of the sampling distribution of means,
d) The standard deviation of the sampling distribution of means.
5.
a) Determine a 95% confidence interval for the mean of a normal distribution with variance
2
 156.25 , using a sample of n50 values with mean x72.4.
b) Determine 99% confidence interval for the mean of contents of soft drink bottles if contents
of 7 such soft drink bottles are 10.2 ,10.4,9.8,10.0,9.8,10.2,9.6 ml.
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R07
Code No: X0521
SET - 3
6.
a) A sample of 1000 students from a university was taken and their average weight was found
to be 112 pounds with a standard deviation of 20 pounds. Could the mean weight of students in
the population be 120 pounds? Use a 0.05 level of significance.
b) A sample of heights of 6400 soldiers has a mean of 67.85 inches and a S.D. of 2.56 inches
while another sample of heights of 1600 sailors has a mean of 68.55 inches with S.D. of
2.52 inches. Do the data indicate that the sailors are on the average taller than soldiers? Use a
0.05 level of significance.
7.
a) The following values given the lengths of 12 samples of Egyptian cotton taken from a
shipment : 48,46,49,46,52,45,43,47,47,46,45,50. Test if the mean length of the shipment can be
taken as 46. Use a 0.05 level of significance.
b) A study is conducted to compare the length of time between men and women to assemble a
certain product. Past experience indicates that the distribution of times for both men and
women is approximately normal but the variance of the times for women is less than that for
men. A random sample of times for 11 men and 14 women produced the following data:
Men
n 11
Women
n 14
1
2
s  6.1
s 5.3
1
2
2
Test the hypothesis that
significance.
8.
2


2
2
1 2 against the alternative that 1 2 . Use 0.05 level of
a) Explain briefly the main characteristics of Queuing system?
b) Assume that both arrival rate and service rate following Poisson distribution. The arrival rate
and service rate are 25 and 35 customers/hour respectively, at a single window in RTC
reservation counter. Find  , L , L , W and W
s
q
s
q
Note :- Statistical tables are required ( Areas under the Standard Normal Curve from 0 to Z,
Percentile Values tp for Student’s t- distribution with v degrees of freedom, Percentile Values
2
p for the Chi-Square distribution with v degrees of freedom, 95th Percentile Values (0.05
levels), F0.95 for the F- Distribution and 99th Percentile Values(0.01 levels), F 0.99 for the F
Distribution).
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Code No: X0521
R07
SET - 4
II B. Tech I Semester Supplementary Examinations Dec - 2013
PROBABILITY AND STATISTICS
(Com. to CSE, IT)
Time: 3 hours
Max. Marks: 80
Answer any FIVE Questions
All Questions carry Equal Marks
P Ac B c) 1 P ( A B ) P (A ) P( B ).
(
b) Determine the probability of the following event:
A king, ace, jack of clubs or queen of diamonds appear in drawing in a single card from a well
shuffled ordinary deck of cards.
c) Box I contains 10 white and 3 black balls while Box II contains 3 white and 5 black balls.
Two balls are drawn at random from Box I and placed in Box II. Then 1 ball is drawn at
random from Box II. What is the probability that it is a white ball?
1.
a) Prove that
2.
a) Check whether the following can define probability distribution, and explain your answer :
x 1
f ( x )  25
for x 1,2,3,4,5.

(2 x 3) if 0 x 2
b) If f ( x
is the probability density function of the continuous
k
0 therwise
)

0
random variable X ,find k and find the distribution function
F (x).
3.
a) If the probability of a bad reaction from a certain injection is
0.001. Determine the
probability that out of 2000 individuals more than two will get a bad reaction.
b) Find the mean and standard deviation of a normal distribution in which 7% of the items are
under 35 and 89% are under 63.
4.
A population consists of the four numbers 3, 7, 11, 15. Consider all possible samples of size
two that can be drawn without replacement from this population. Find
a) The population mean,
b) The population standard deviation,
c) The mean of the sampling distribution of means,
d) The standard deviation of the sampling distribution of means.
5.
a) Determine a 95% confidence interval for the mean of a normal distribution with variance
2
 400 , using a sample of n100 values with mean x250.
b) The numbers 124.9, 127.9, 130.9, 128.9, 122.9 from a random sample of values normally
distributed random variable. Find a 99% confidence interval for the mean  of X.
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Code No: X0521
R07
SET - 4
6.
a) The heights of college students in a city are normally distributed with a standard deviation of
6 cms. A sample of 1000 students has mean height 158 cms. Test the hypothesis that the
mean height of college students in the city is 160 cms. Use a 0.05 level of significance.
b) The numbers of accidents per day were studied for 144 days in town A and for 100 days in
town B and the following information is obtained.
Mean number of accidents S.D.
Town A 4.5
1.2
Town B 5.4
1.5
Is the difference between the mean accidents of the two towns statistically significant? Use a
0.05 level of significance.
7.
a) A filling machine is expected to fill 5 kg of powder into bags. A sample of 10 bags gave the
following weights: 4.7, 4.9, 5.0, 5.1, 5.4, 5.2, 4.6, 5.1, 4.6 and 4.7. Test whether the machine is
working properly. Use a 0.05 level of significance.
b) An instructor has two classes, A and B in the particular subject. Class A has 16 students
while Class B has 25 students. On the same examination, although there was no significant
difference in mean grades, class A had a standard deviation of 9 while class B had a
standard deviation of 12. Can we conclude at the 0.05 level of significance that the
variability of class B is greater than that of A?
8.
a) Explain (M / M / 1 ) : ( / FCFS ) Queuing model.
b) At a public telephone booth the arrivals are on the average 15 per hour. A call on the average
takes 3 minutes. If there is just one phone,
i) What is the expected number of callers in the booth at any time?
ii) For what proportion of time is the booth expected to be idle.
Note :- Statistical tables are required ( Areas under the Standard Normal Curve from 0 to Z,
Percentile Values tp for Student’s t- distribution with v degrees of freedom, Percentile Values
2p for the Chi-Square distribution with v degrees of freedom, 95 th Percentile Values (0.05
levels), F0.95 for the F- Distribution and 99th Percentile Values(0.01 levels), F0.99 for the F
Distribution).
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