Quantitative Methods (MB151): January 2006

Quantitative Methods (MB151): January 2006

· Answer all questions.
· Marks are indicated against each question.
1. A graphical method of representing states of nature and courses of action involved in decision making is
referred to as a
(a) Decision tree (b) Histogram (c) Scatter diagram
(d) Frequency distribution (e) Probability distribution.
(1 mark)
< Answer >
2. In a moderately asymmetrical distribution, the mode and mean are 32.1 and 35.4 respectively. Find out
the value of median.
(a) 32.1 (b) 34.3 (c) 35.2 (d) 36.6 (e) 37.8.
(1 mark)
< Answer >
3. Calculate the harmonic mean from the following data:
3834 382 63 8 0.4 0.03 0.009 0.0005
(a) 0.003726 (b) 0.004929 (c) 0.005283 (d) 0.006721 (e) 0.007214.
(1 mark)
< Answer >
4. A cyclist covers his first 5 km at an average speed of 10 kmph another 3 km at 8 kmph and the last 2 km
at 5 kmph. The average speed of entire journey is
(a) 6.28 kmph (b) 7.15 kmph (c) 7.84 kmph (d) 8.22 kmph (e) 8.94 kmph.
(1 mark)
< Answer >
5.
If
2
xlogx (x - 4x+5) = (x - 1) then x =
(a) 0 (b) 1 (c) 2 (d) 4 (e) 5.
(1 mark)
< Answer >
6. Mr. Roy buys 100 units of the Unit Trust of India at Rs.10.30 per unit. He buys another lot of 200 at
Rs.10.40 per unit. At Rs.10.50 per unit, he takes up another lot of 400 and a further lot of 300 at
Rs.10.80 per unit. He watches as the price goes down and wants to take up as many units at Rs.10.25
per unit as would make the average cost of his holding to Rs.10.50 per unit.
Find the number of units he purchased at the lowest price of Rs.10.25 per unit.
(a) 100 (b) 200 (c) 300 (d) 400 (e) 500.
(2 marks)
< Answer >
7. In how many ways may 6 boys and 6 girls sit around a round table so that any two boys may never sit
together?
((A)(a) 30 (b) 120 (c) 720 (d) 86,400 (e) 5,18,400.
(1 mark)
< Answer >
8. What is the mode for the following data?
Marks 10 15 20 25 30 35 40
Frequency 8 12 36 25 28 18 9
(a) 20 (b) 25 (c) 30 (d) 35 (e) 40.
(1 mark)
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9. Find the value of the median from the data given below:
Weight (kg) 93-98 98-103 103-108 108-113 113-118 118-123 123-128
No. of students 2 5 12 17 14 6 3
(a) 108 kg (b) 111 kg (c) 113 kg (d) 114 kg (e) 118 kg.
(1 mark)
< Answer >
10.
Which of the following is not true with regard to the regression relationship ˆY = a + bX?
(a) The point (X, Y) always lies on the regression line
(b) The expected value of the residuals is zero
(c) The mean of the fitted values of Y is the same as the mean of the observed values of Y
(d) There are always as many points above the fitted line as there are below it
(e) The regression line minimizes the sum of the squared residuals.
(1 mark)
< Answer >
11. Which of the following is/are not true with regard to correlation analysis?
(a) It deals with the relationship between two or more series
(b) It compares the variation in two or more series
(c) It proves the presence of a cause and effect relationship between two or more series
(d) It helps in determining the degree of association between two or more series
(e) Both (a) and (b) above.
(1 mark)
< Answer >
12. Mandira Bedi, human resource manager of a major Television manufacturing company feels that, the
longer a group of employees work together on an assembly line, the higher the daily output rate is. She
has gathered the following data for a group of employees who worked together for 10 days:
Daily output
(units) 4 7 5 6 8 2 3 10 9 12
Days worked
together 1 2 3 4 5 6 7 8 9 10
Find the rank correlation coefficient for the above data.
(a) 0.3051 (b) 0.4162 (c) 0.5273 (d) 0.6384 (e) 0.7495.
(2 marks)
< Answer >
13. Which of the following represents the proportion of variation in the dependent variable that is explained
by the regression line?
(a) Coefficient of determination (b) Coefficient of correlation
(c) Coefficient of variation (d) Standard error of estimate
(e) Standard deviation.
(1 mark)
< Answer >
14.
If logx 5, logy 5 and logz 5 are in A.P then 2logxz z =
(a) logyx (b) logxy (c) logyz (d) logzy (e) logz x.
(1 mark)
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15. In a partially destroyed laboratory record of an analysis of correlation data, the following results only
are legible:
Variance of X = 9
Regression equations: 8X – 10Y + 66 = 0 and 40X – 18Y = 214.
One of these two regression equations is for Y on X (with X as the independent variable) and the other
is for X on Y (with Y as the independent variable).
Find the mean value of Y.
< Answer >
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(a) 13 (b) 15 (c) 16 (d) 17 (e) 18.
(1 mark)
16. Genesis Systems Ltd. produces different types of products in the field of information technology. The
research and development department of the company has been witnessing a growth in the research
activities over the past few years. The data on research expenditure in the recent past are given below:
Year 1999 2000 2001 2002 2003 2004
Research expenditure
(Rs. in lakh)
40 47 54 62 69 77
Find the slope of the linear equation that describes the trend in the expenditure on research by the
company.
[Use coded value of year = 2(X – X ) where X represents the year]
(a) 1.5 (b) 2.6 (c) 3.7 (d) 4.8 (e) 5.9.
(1 mark)
< Answer >
17. According to the ‘method of least squares’ criterion, the regression line should be drawn on the scatter
diagram in such a way that
(a) The sum of the squared values of the vertical distances from each plotted point to the line is
maximum
(b) The sum of the squared values of the vertical distances from each plotted point to the line is
minimum
(c) The sum of the squared values of the horizontal distances from each plotted point to the line is
maximum
(d) The sum of the squared values of the horizontal distances from each plotted point to the line is
minimum
(e) The sum of the squared values of the vertical distances from each plotted point to the line is equal
to zero.
(1 mark)
< Answer >
18. Which of the following is/are false about index numbers?
(a) Index numbers are expressed as a percentage
(b) Index number for the base year is always 100
(c) Index number has a unit of the variable that is being compared
(d) Index number is calculated as a ratio of the current value to a base value
(e) Both (a) and (b) above.
(1 mark)
< Answer >
19. Which of the following statements is false ?
(a) An unweighted aggregates price index does not attach any importance to the consumption or usage
levels of the items covered
(b) Laspeyres price index is an unweighted aggregates price index
(c) Paasches price index is a weighted aggregates price index
(d) A price index reflects the overall change in price for the basket of goods considered
(e) Fisher’s ideal price index is the geometric mean of Laspeyres and Paasches price indices.
(1 mark)
< Answer >
20. For a basket of commodities the price of each commodity in a year X is three times the price in the base
year. What is the Paasches price index for the year X?
(a) 100 (b) 150 (c) 75 (d) 300 (e) 600.
(2 marks)
< Answer >
21. If the average deviation from the arithmetic mean is used as a measure of dispersion then, it will always
indicate that the dispersion of the data set is
(a) A positive value (b) A negative value
(c) Either a positive or negative value (d) Equal to zero
(e) A multiple of the arithmetic mean.
< Answer >
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(1 mark)
22. Consider a batch of N cricket balls. Each cricket ball may be defective or non-defective. The experiment
involves selecting a cricket ball and checking whether it is defective or non-defective. If this experiment
is repeated without replacing the balls then the probability distribution function obtained from such an
experiment can be appropriately described by
(a) The binomial distribution
(b) The hypergeometric distribution
(c) The normal distribution
(d) The lognormal distribution
(e) The t-distribution.
(1 mark)
< Answer >
23. Let A, B and C are three arbitrary events. Which of the following expressions signifies the occurren ce
of both A and B, but not C?
(a) A and B and C (b) A and B and C
(c) A and Band C (d) A or B or C (e) A or B or C.
(1 mark)
< Answer >
24. A and B are two independent events such that P(A) = 0.3, P(B) = k and P(A or B) = 0.8, then the value
of k is given by
(a) 1/3 (b) 2/3 (c) 2/7 (d) 5/7 (e) 2/5.
(1 mark)
< Answer >
25. The events S and T are independent and are equiprobable such that P(S and T) = p, where
p > 0, then P(S) = ?
(a) p (b) p (c) p 2 (d) 2p (e) p/2.
(1 mark)
< Answer >
26. Which of the following is/are the condition(s) for applying the Baye’s theorem for computing posterior
probabilities of certain events?
I. The events must be non-mutually exclusive.
II. The events must be mutually exclusive.
III. The events must not be collectively exhaustive.
IV. The events must be collectively exhaustive.
(a) Only (I) above (b) Only (II) above
(c) Only (III) above (d) Both (I) and (III) above
(e) Both (II) and (IV) above.
(1 mark)
< Answer >
27. If the covariance between two variables is positive then, the coefficient of correlation between them will
be in the range of
(a) –1.00 to 0 (b) –0.50 to 0 (c) –0.25 to 0 (d) 0 to 2.00 (e) 0 to 1.00.
(1 mark)
< Answer >
28. There are 25 students in a class, which consists of 14 boys and 11 girls. 5 students of the class were
absent on a particular day. What is the probability that maximum two girls were absent?
(a) 0.2072 (b) 0.3768 (c) 0.416 (d) 0.584 (e) 0.6232.
(2 marks)
< Answer >
29. Every morning an athlete throws a fair die. If the die shows a number greater than 2 then, he does a
random number of press -ups before having his breakfast. If, however, the die shows a number less than
or equal to 2 then, he has his breakfast without doing any press-up. If this process is observed for 21
days then, which of the following statements is correct?
(a) The number of days on which the athlete does press-ups follows a discrete uniform distribution
with mean 11
< Answer >
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(b) The number of days on which the athlete does press-ups follows a hypergeometric distribution
with mean 10
(c) The number of days on which the athlete does press -ups follows a binomial distribution with mean
14
(d) The number of days on which the athlete does press-ups follows a chi-square distribution with 14
degrees of freedom
(e) The number of days on which the athlete does press-ups follows an unknown probability
distribution.
(1 mark)
30. A multiple regression equation has
(a) Multiple dependent variables
(b) One independent variable
(c) One dependent variable
(d) A standard error of estimate equal to zero
(e) A standard error of estimate equal to 1.
(1 mark)
< Answer >
31. Which of the following is false with regard to a hyper geometric distribution?
(a) The trials are not independent
(b) The probability of success is variable
(c) The outcomes can be labeled as success or failure
(d) The composition of the population remains unchanged
(e) The population is finite.
(1 mark)
< Answer >
32. A random variable X, has the following probability distribution:
X 20 40 60 80 100
Probability 0.10 0.20 0.30 0.25 0.15
Let Y = X – 25
What is the expected value of Y?
(a) 20 (b) 25 (c) 38 (d) 45 (e) 63.
(1 mark)
< Answer >
33. The maximum number of successes that may be observed in a binomial distribution
(a) Is equal to the mean of the distribution
(b) Is equal to the number of trials
(c) Is equal to the standard deviation of the distribution
(d) Is equal to the variance of the distribution
(e) Cannot be known.
(1 mark)
< Answer >
34. The variance of a random variable is expressed
(a) In terms of the unit of the random variable
(b) In terms of t he square of the unit of the random variable
(c) In terms of the square root of the unit of the random variable
(d) Without any unit at all
(e) In terms of the same unit in which its mean is expressed.
(1 mark)
< Answer >
35. Rahul automobiles, wants to study the consumer preference in terms of the features that vehicle owners
of the city want to have in their vehicle. It divided the city in fifty sectors, each representing the
characteristics of the whole city. Now they randomly selected five sectors and interviewed each
household of these sectors for their preferences. They plan to forward the result of this survey to the
vehicle producers to make them aware about the consumer preferences. This method of sampling is
termed as
(a) Cluster sampling
< Answer >
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(b) Stratified sampling
(c) Systematic sampling
(d) Judgmental sampling
(e) Simple random sampling.
(1 mark)
36. Which of the following is true ?
(a) The standard error of mean increases as the sample size increases
(b) The standard error of mean decreases as the sample size increases
(c) The standard error of mean for sample sizes of 2 and above, is more than the standard deviation of
population
(d) The standard error of mean for sample sizes of 2 and above, is equal to the standard deviation of
population
(e) The standard error of mean for sample sizes of 2 and below, is more than the standard deviation of
population.
(1 mark)
< Answer >
37. Which of the following is not a type of random sampling?
(a) Simple random sampling (b) Stratified sampling (c) Systematic sampling
(d) Judgmental sampling (e) Cluster sampling.
(1 mark)
< Answer >
38. The following details are available with regard to a hypothesis test on population mean:
H0: m = 9
H1: m ¹ 9
n = 25
s2 = 256
x = 2.45
Significance level = 0.05
The population is normally distributed. It is later known that the true population mean is 7.
Which of the following can be said with regard to the test?
(a) There is insufficient information for doing the test
(b) The normal distribution should be used
(c) The test does not lead to either type I or type II error
(d) The test leads to a type I error
(e) The test leads to a type II error.
(2 marks)
< Answer >
39. In the graphical method of solving linear programming problems the feasible region is the set of all
points
(a) Which do not satisfy any of the constraints
(b) Which satisfy exactly one of the constraints
(c) Which satisfy all the constraints
(d) At which the objective function has the same value
(e) At which the objective function is equal to zero.
(1 mark)
< Answer >
40. Which of the following conditions indicates the existence of multiple optimal solutions when a linear
programming problem is solved by the graphical method?
(a) One of the constraints is parallel to the horizontal axis
(b) The objective function is parallel to the vertical axis
(c) The objective function is parallel to one of the edges of the feasible region which is in the direction
of optimal movement of the objective function
(d) If two or more constraints are parallel to each other
(e) If there is a redundant constraint present in the problem.
(1 mark)
< Answer >
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41. What is the value of K in the following series?
2 + 8 + 14 + 20 + 26 + ………….+ K = 574
(a) 70 (b) 74 (c) 80 (d) 86 (e) 90.
(2 marks)
< Answer >
42. If a, b and c are in A.P then the expression a3 + 4b3 + c3 =
(a) 6a3 + 18ad + 24ad2 + 12d3 (b) a3 + 3a2d + 4ad2 + 6d3
(c) 6a3 + 18a2d + 24ad2 + 12d3 (d) 3a3 + 12a2d + 18ad2 + 12d 3
(e) 6a3 + 18a2d + 21ad2 + 12d3.
(1 mark)
< Answer >
43. The sum of the series of 243, 324, 432, … upto n terms is
(a) 36 (4n – 3n) (b) 36 – n (4n – 3n) (c) 3n (4n – 3n)
(d) 3n – 6 (3n – 4n) (e) 36 – n (3n – 4n).
(2 marks)
< Answer >
44.
The sum of the series
2 31 5 6 1 ...tothe25th termis
2 2
+ + + +
(a) 100 (b) 200 (c) 400 (d) 500 (e) 600.
(1 mark)
< Answer >
45. The pth term of an A.P is q and the qth term is p. The rth term would be
(a) p + r (b) q + r (c) p + q + r (d) p – q – r (e) p + q – r.
(2 marks)
< Answer >
46. The first and the last term of an A.P are –4 and 146 respectively, also the sum of the series is 7171.
Find the number of terms in the series of an A.P.
(a) 51 (b) 63 (c) 77 (d) 101 (e) 117.
(1 mark)
< Answer >
47.
The sum of the series 8, 4 2, 4, … ¥ is
(a) 16 (b) 14 (c) 18 (d) 8(2 + 2) (e) 8 2 .
(1 mark)
< Answer >
48.
If ´ 4n 2n
P3 = 28 P2 , then n =
(a) 5 (b) 4 (c) 3 (d) 2 (e) 1.
(2 marks)
< Answer >
49. The number of ways all the letters of the word COMBINATION can be arranged is
(a)
11
11 P (b)
11
P11
2! (c)
11
11
3
P
(2!) (d)
11
C11 (e)
11
11
3
C
(2!) .
(1 mark)
< Answer >
50. Three persons enter into a railway carriage and there are 8 seats available. In how many ways can they
seat themselves?
(a) 24 (b) 40 (c) 56 (d) 48 (e) 336.
(1 mark)
< Answer >
51. There are 5 routes for journey from station A to station B. In how many different ways can a man go
from A to B and return, if for returning any of the routes is taken?
(a) 5 (b) 10 (c) 15 (d) 20 (e) 25.
< Answer >
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(1 mark)
52. How many telephone connections can be allotted with 6 digits from the natural numbers 1 to 9
inclusive?
(a) 5,11,441 (b) 5,31,441 (c) 729 (d) 1,00,77,696 (e) 216.
(1 mark)
< Answer >
53.
If
1
2
6
= a log
the value of a is equal to
(a) 2 (b) 4 (c) 8 (d) 16 (e) 64.
(1 mark)
< Answer >
54. The logarithm of 5832 to base 3 2 is
(a) 2 (b) 4 (c) 6 (d) 8 (e) 12.
(1 mark)
< Answer >
55. Which of the following statements is most appropriate if certain events are mutually exclusive and
collectively exhaustive?
(a) Each of the events has a zero probability
(b) Some of the events will definitely have zero probability
(c) The sum of the probabilities of the events will be equal to 1
(d) The sum of the probabilities of the events will be less than 1
(e) The sum of the probabilities of the events will be more than 1.
(1 mark)
< Answer >
56. A continuous random variable is one which
(a) Can assume only real values
(b) Can assume only even values
(c) Can assume only odd values
(d) Can assume any value within a given range
(e) Can assume only a limited number of values.
(1 mark)
< Answer >
57. From the following data calculate arithmetic mean:
Marks 0 – 10 10 – 20 20 – 30 30 – 40 40 – 50 50 – 60
No. of students 5 10 25 30 20 10
(a) 30 (b) 31 (c) 33 (d) 35 (e) 38.
(1 mark)
< Answer >
58. Three numbers a, b and c are in arithmetic progression (A.P). p is the geometric mean between a and b.
q is the geometric mean between b and c. What is the arithmetic mean between p2 and q2?
(a) a2 (b) b2 (c) c2 (d) p (e) q.
(2 marks)
< Answer >
59. The average deviation of all items in a data set from zero is equ al to
(a) Arithmetic mean (b) Geometric mean
(c) Quartile deviation (d) Standard deviation (e) Variance.
(1 mark)
< Answer >
60. Which of the following measures is based only on two observations in a data set?
(a) Arithmetic mean (b) Harmonic mean
(c) Range (d) Mean absolute deviation
(e) Standard deviation.
(1 mark)
< Answer >
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61. If a fair die is thrown ten times then, what is the probability of getting an even number greater than 2,
six times?
(a) 0.0569 (b) 0.33 (c) 0.67 (d) 0.9431 (e) 0.60.
(1 mark)
< Answer >
62. A random variable Z has the following probability distribution:
Z 40 60 80 100 120
Probability 0.15 0.25 0.30 0.20 0.10
If twenty values of Z are observed then how many observations are expected to be less than 80?
(a) 2 (b) 4 (c) 6 (d) 8 (e) 10.
(2 marks)
< Answer >
63. Northern Suppliers Ltd. supplies processed agricultural materials to agro-based industries. At present,
the company has submitted quotations in response to tenders invited by 6 parties for a specific material.
The probability that it will be able to secure supply contract with exactly one party is 18.66% and the
probability that it will be able to secure supply contracts with exactly two parties is 31.10%. The
invitations for tenders are independent of each other and the probability of securing supply contract with
the party inviting the tender is assumed to remain fixed over time.
What is the probability that the company will secure supply contracts with at least 3 parties?
(a) 4.67% (b) 18.66% (c) 31.10% (d) 45.57% (e) 54.43%.
(3 marks)
< Answer >
64. Which of the following refers to the random (chance) occurrences that can affect the outcome of an
individual's decision?
(a) Payoffs (b) States of nature (c) Decision alternatives
(d) Probabilities (e) Expected values.
(1 mark)
< Answer >
65. Assume z is a standard normal random variable.
P(-2.0 < z < -1.0) =
(a) 0.8185 (b) 0.1469 (c) 1.0000 (d) 0.1359 (e) 0.4772.
(1 mark)
< Answer >
66. If three fair coins are tossed simultaneously ten times then, what is the likelihood that heads will be
observed on the three coins simultaneously, on three throws?
(a) 0.09204 (b) 0.90796 (c) 0.125 (d) 0.875 (e) 0.00195.
(1 mark)
< Answer >
67. The expected value of a binom ial probability distribution is
(a) Always equal to the variance (b) Always more than the variance
(c) Always less than the variance (d) Always equal to zero
(e) Always equal to 1.
(1 mark)
< Answer >
68. The academic council of Ohio University has decided to bring some changes to its existing grading
system. The university wants to determine what proportion of the students is in favour of new grading
system. The total number of students enrolled in the university is 60,000 and the university feels that at
least 36,000 of them support the changes. What should be the sample size that will enable the university
to be 89.9% certain of estimating the true proportion in the favour of a new system within ± 0.02?
(a) 1075 (b) 1411 (c) 1613 (d) 1680 (e) 1711.
(2 marks)
< Answer >
69. Each day, the Indian Customs Service has historically intercepted about Rs.28 million in contraband
goods being smuggled into the country, with a standard deviation of Rs.16 million per day. On 64
randomly chosen days in 1992, the Indian Customs Service intercepted an average of Rs.30.3 million in
< Answer >
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contraband goods. The Customs Commissioner wants to test at a significance level of 5% whether the
smuggling has increased above its historic level.
Which of the following is an appropriate alternative hypothesis for the above test?
(a) The average daily smuggling intercepted is same as the historic level
(b) The average daily smuggling intercepted is not same as the historic level
(c) The average daily smuggling intercepted is more than the historic level
(d) The average daily smuggling intercepted is less than the historic level
(e) The sample mean is incorrect.
(1 mark)
70. The Andhra Theatres Association knows that a certain hit movie ran an average of 84 days in each city
and the corresponding standard deviation was 10 days. The manager of Guntur district was interested in
comparing the movie’s popularity in his region with that in all of Andhra’s other theatres. He randomly
chose 75 theatres in his region and found that they ran the movie to an average of 80.5 days.
The manager wants to test whether there was a significant difference in the length of the picture’s run
between theatres in the Guntur district and all of Andhra’s other theatres. He is ready to use 1 percent
significance level.
What is/are the critical value(s) for the above test?
(a) 2.33 (b) - 2.33 (c) ± 1.64 (d) ± 1.96 (e) ± 2.575.
(1 mark)
< Answer >
71. The following details are available for a hypothesis test on population proportion:
H0: p = 0.60
H1: p < 0.60
Sample size = 360
Sample proportion = 0.57
Significance level = 0.05
It is later known that the true population proportion is 0.60.
Which of the following can be said with regard to the test?
(a) There is insufficient information for doing the test
(b) The chi-square distribution should be used
(c) The test does not lead to either type I or type II error
(d) The test leads to a type I error
(e) The test leads to a type II error.
(1 mark)
< Answer >
72. Magnus Products Ltd. manufactures and sells domestic products. The marketing manager of the
company is concerned about the sales behavior of a kitchen-knife set manufactured and sold by the
company. In his opinion the number of advertisements and price per set are significant determinants of
sales. He has collected the following data from the records of the company for the past three years:
Sales
(’000 sets sold)
Number of
Advertisements
Price per set
(Rs.)
30 2 120
58 5 110
67 9 135
79 12 125
14 8 140
If Y denotes sales (in thousands), X1 denotes the number of advertisements and X2 denotes the price per
set (in Rupees), then obtain a multiple regression relationship of the form Y = a + b1 X1 + b2 X2.
What is the value of coefficient a in the above regression relationship?
(a) 145.246 (b) 152.876 (c) 185.147 (d) 215.358 (e) 251.469.
(2 marks)
< Answer >
73. You are working as a purchase manager of a company. The following information has been supplied to
you by two manufacturers of electric bulbs on the basis of sample data:
Company A Company B
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Mean life time (in hours) 1,300 1,288
Standard deviation of life tim e (in hours) 82 93
Sample Size 100 100
Your quality manager hypothesizes that there is no significant difference between the mean lives of the
electric bulbs of the two companies. You wish to test this hypothesis at 5% level of significance.
What is the value of the appropriate test statistic for this test?
(a) 0.658 (b) 0.758 (c) 0.858 (d) 0.968 (e) 1.078.
(1 mark)
74. Calculate the coefficient of correlation from the following data:
Marks in Accountancy 48 35 17 23 47
Marks in Statistics 45 20 40 25 45
(a) 0.429 (b) 0.438 (c) 0.447 (d) 0.456 (e) 0.465.
(1 mark)
< Answer >
75. The covariance of a random variable with another random variable is
(a) Negative if the random variables are directly correlated
(b) Positive if the random variables are inversely correlated
(c) Zero if the random variables are independent
(d) Equal to the product of the standard deviations of the random variables
(e) Equal to the product of the expected values of the random variables.
(1 mark)
< Answer >
76. The following data are collected with regard to some commodities:
Commodity Quantity (Units) Price (Rs./Unit)
1994 2004 1994 2004
A 15 20 17 21
B 20 20 190 220
C 25 30 30 31
What is the unweighted aggregates price index for the year 2004 considering the year 1994 as the base
year?
(a) 83.57 (b) 114.77 (c) 116.67 (d) 119.67 (e) 87.13.
(1 mark)
< Answer >
77. Which of the following price indices will tend to underestimate the rise in prices?
(a) Laspeyres Index (b) Paasches Index
(c) Fisher’s Index (d) Value Index
(e) Quantity Index.
(1 mark)
< Answer >
78. Which of the following is a quantity index?
(a) Index number of industrial production
(b) Index number of wholesale prices
(c) Index number of industrial security prices
(d) Consumer price index for industrial workers
(e) Index number of distributor prices.
(1 mark)
< Answer >
79. The value index number measures the
(a) Changes in prices of a basket of commodities from one period to another
(b) Changes in quantities consumed of a basket of commodities over a period of time
(c) Change in the total monetary value of a basket of commodities over a period of time
(d) Change in the retail prices of various commodities
(e) Change in the general price level in a country.
(1 mark)
< Answer >
12
80. Which of the following weighted price index numbers uses only the quantity measures for the current
period as weights?
(a) Laspeyres price index (b) Paasches price index
(c) Fisher’s ideal price index (d) Fixed weight aggregates price index
(e) Weighted aggregates index.
(1 mark)
< Answer >
81. Which of the following parameters must be specified in order to describe a binomial distribution?
(a) Number of trials only
(b) Number of successes desired
(c) Minimum number of successes required
(d) Probability of success in any trial only
(e) Both number of trials and probability of success in any trial.
(1 mark)
< Answer >
82. From the following data obtain the regression equation of Y on X (independent variable)
X 6 2 10 4 8
Y 9 11 5 8 7
(a) ˆY = 11.9 – 0.65X (b) ˆY = 0.65 + 11.9X
(c) ˆY = -11.9 + 0.65X (d) ˆY = 11.9 + 0.65X
(e) ˆY = 0.65 –11.9 X .
(1 mark)
< Answer >
83 A normal probability distribution
(a) Is a discrete probability distribution (b) Is a continuous probability distribution
(c) Must always have a variance of 1 (d) Must always have a mean of zero
(e) Must always have a mean of 1.
(1 mark)
< Answer >
84. The minor of an element of the matrix is defined as in one of the following ways:
(a) Deleting the row in which the element is present
(b) Deleting the rows and columns in which the element is not present
(c) Deleting the diagonal elements of the element present
(d) Deleting the row and column in which the element is present
(e) Deleting the column in which the element is present.
(1 mark)
< Answer >
85. The speed of a train while crossing signals is approximately normally distributed with mean of 50 miles
per hour and a standard deviation of 5 miles per hour. Then what will be the percent of speed of train
that is below 42 miles per hour?
(a) 4.78 % (b) 3.48 % (c) 5.91 % (d) 5.48 % (e) 4.84 %.
(1 mark)
< Answer >
13
Suggested Answers
Quantitative Methods (MB151): January 2006
1. Answer : (a)
Reason : a. A decision tree is a graphical method which represents states of nature and courses of
action.
b. A histogram is a graphical representation of a frequency distribution.
c. A scatter diagram shows the distribution of data points in regression analysis.
d. A frequency distribution is a distribution of data along with their frequencies.
e. A probability distribution is a distribution of values of a random variable along with
their respective probabilities.
< TOP >
2. Answer : (b)
Reason :
Mode = 3 Median – 2 Mean
or 32.1 = 3 Median – 2 ´ 35.4
or 3 Median = 32.1 + 70.8
or Median =
102.9
3 = 34.3.
< TOP >
3. Answer : (a)
Reason : Calculation of harmonic mean
X 1/X X 1/X
3834 0.0003 0.4 2.5
382 0.0027 0.03 33.3333
63 0.0159 0.009 111.1111
8 0.1250 0.0005 2000
1
2147.0883
x
Sæ ö = ç ÷
è ø
H.M. =
N 8
0.003726.
1 2147.0883
x
= =
Sæ ö ç ÷
è ø
< TOP >
4. Answer : (c)
Reason : Weighted harmonic mean is appropriate here
The various speeds are 10 kmph, 8 kmph and 5 kmph
Hence H.M. =
10 10 10
1 1 1 1 3 2 51
5 3 2
10 8 5 2 8 5 40
= =
æ ´ ö +æ ´ ö +æ ´ ö + + ç ÷ ç ÷ ç ÷
è ø è ø è ø
=
10 40
7.84kmph
51
´
=
< TOP >
5. Answer : (c)
Reason : We have,
2
2 log
2
log ( 4 5)
( 4 5) 1,
, 5 6 0
,( 2)( 3) 0
2 3
- + ( 1)
Þ - + = - =
- + =
- - =
Þ =
= -
éëQ a ùû
x
m
x x
x x x a m
orx x
or x x
x or
x x
< TOP >
6. Answer : (b) < TOP >
14
Reason : Let x be the number of unit purchased at Rs.10.25. The total number of units purchased at an
average price per unit of Rs.10.50
= 100 + 200 + 400 + 300 + x
= 1000 + x
\ Value of units = Rs.10.50 ´ (1000 + x) = 10,500 + 10.50x
But the value of unit held by him
= (Rs.100 ´ 10.30 + Rs.200 ´ 10.40 + Rs.400 ´ 10.50 + Rs.300 ´ 10.80 + Rs.x ´ 10.25)
= Rs.10550 + Rs.10.25x
From the given data, we have
10500 + 10.50x = 10550 + 10.25x
Þ 10.50x – 10.25x = 10550 – 10500
Þ 0.25x = 50
Þ x = 200
\ He purchased 200 units at Rs.10.25.
7. Answer : (d)
Reason : Let us fix a girl to one particular seat, other 5 girls can sit in remaining five seats in 5!
Ways. The six boys can be seated in six empty seats between two consecutive girls’ seats
in 6! Ways. So, total number of ways the boys and girls may be arranged to seat is
5!6!=(1´ 2 ´ 3 ´4 ´5)(1 ´2 ´ 3 ´ 4 ´ 5´ 6) =86400 .
< TOP >
8. Answer : (a)
Reason : Since it is difficult to say by inspection on to which is the model value, we prepare grouping
and analysts tables.
M 10 15 20 25 30 35 40
F 8 12 36 35 28 18 9
Since the marks 20 occurs maximum no. of times i.e., 36, hence the value of mode is = 20.
< TOP >
9. Answer : (c)
Reason :
No.of students
(f)
Cumulative
frequency (cf)
2 2
5 7
12 19
17 36
14 50
6 56
3 59
Lm = 108, N = 59
F = 12, fm = 17, W = 5
Md =
m
m
(N 1) /2 (F 1)
W L
f
é + - + ù
ê ú +
ë û
Md =
m
(59 1) /2 (12 1)
W L
17
é + - + ù ê ú + ë û
=
30 13
5 108
17
é - ù ê ú + ë û = 5 + 108 = 113.
< TOP >
10. Answer : (d)
Reason : The statement that there are always as many points above the fitted line as there are below it
is wrong.
< TOP >
15
11. Answer : (c)
Reason : Correlation analysis does not prove the presence of cause and effect relationship between
two or more series. Alternatives (a), (b), (d) and (e) are true.
< TOP >
12. Answer : (c)
Reason : Since the ranks are not given to the actual data, we assign ranks to the data. We assign rank
‘1’ to the highest value in the series. We follow the same rule for the other series also.
Days Worked
Together (X)
Rank
(R1)
Daily Output
(Y)
Rank
(R2)
D D2
1 10 4 8 2 4
2 9 7 5 4 16
3 8 5 7 1 1
4 7 6 6 1 1
5 6 8 4 2 4
6 5 2 10 -5 25
7 4 3 9 -5 25
8 3 10 2 1 1
9 2 9 3 -1 1
10 1 12 1 0 0
D = (R1 – R2) and D2 = (R1 – R2)2
We get, S D2 = 78
Spearman’s rank Correlation Coefficient,
R =
2
2 2
6 D 6 78
1 1 0.5273.
N(N 1) 10(10 1)
å ´
- =- =
- -
< TOP >
13. Answer : (a)
Reason : Coefficient of determination represents the proportion of variation in the dependent variable
that is explained by the regression line.
< TOP >
14. Answer : (b)
Reason : Given that log 5, log 5 and log 5 are in A.P. x y z
5 5 5
5 5 5
5 5 5 5 5
5 5
5 5
2log 5 log 5 log 5
2 1 1
,
log log log
2 log log log
,
log log .log log .log
log log
, 2
log log
,2log log
\ = +
= +
+
= =
=
=
y x z
xz x
or
y x z
z x xz
or
y x z x z
z y
or
xz x
or z y
< TOP >
15. Answer : (d)
Reason : The mean value of the X series and Y series is the point of intersection of the two regression
lines of Y on X and X on Y.
Therefore the points of intersection could be found by solving the two equations:
8X – 10 Y = - 66 and 40 X – 18 Y = 214.
We get the values of X = 13 and Y = 17.
< TOP >
16. Answer : (c)
Reason : Let the following notations be used:
X : Year
x : Coded value of year = 2 (X – X )
Y : Research expenditure (in Rs.)
Year X x =2(X – X) Y xY x2
< TOP >
16
1997 –5 40 –200 25
1998 –3 47 –141 9
1999 –1 54 –54 1
2000 1 62 62 1
2001 3 69 207 9
2002 5 77 385 25
SX = 11997 SY= 349 SxY = 259 Sx2 = 70
11997
X
6
=
= 1999.5
ˆY
= a + bx
b =
2
xY
x
å
å =
259
70 = 3.7.
17. Answer : (b)
Reason : (a) This is a wrong answer. The sum of the squared values of the vertical distances from
each plotted point to the line should be minimum (not maximum).
(b) This is the right answer. According to the ‘method of least squares’ criterion, the
regression line should be drawn on the scatter diagram in such a way that the sum of
the squared values of the vertical distances from each plotted point to the line are
minimum.
(c) This is the wrong answer. The sum of the squared values of the vertical (not the
horizontal) distances from each plotted point to the line should be minimum (not
maximum).
(d) This is the wrong answer. The sum of the squared values of the vertical (not the
horizontal) distances from each plotted point to the line should be minimum.
(e) This is the wrong answer. The sum of the squared values of the vertical distances from
each plotted point to the line should be minimum (not necessarily zero).
< TOP >
18. Answer : (c)
Reason : (I) This is true for index numbers. Index numbers are expressed as a percentage.
(II) This is true for index numbers. Index number for the base year is always 100.
(III) This is not true for index numbers. Index number is a ratio of the price, quantity or
value of the commodities under study and therefore does not have any unit.
(IV) This is true for index numbers. Index number is calculated as a ratio of the current
value to a base value.
(a) so both a and b are correct answers.
< TOP >
19. Answer : (b)
Reason : Laspeyres price index is a weighted aggregates price index. Alternatives (a), (c), (d) and (e)
are true.
< TOP >
20. Answer : (d)
Reason : Given P1 = 3P0, For each commodity
\ Paasches price index =
1 1
0 1
P Q
100
P Q
S
´
S =
0 1
0 1
3PQ
100
P Q
S
´
S
=
0 1
0 1
3 P Q
100
P Q
S
´
S = 300.
< TOP >
21. Answer : (d)
Reason : (a) The average deviation from the arithmetic mean will never be greater than zero.
(b) The average deviation from the arithmetic mean will never be less than zero.
(c) The average deviation from the arithmetic mean can never be less than or greater than
zero.
(d) The sum of the deviations from the arithmetic mean is equal to zero. Hence the average
deviation from the arithmetic mean will always indicate that the dispersion of the data
< TOP >
17
set is equal to zero.
(e) There is no reason why the average deviation from the arithmetic mean will be a
multiple of the arithmetic mean.
22. Answer : (b)
Reason : (a) This is a wrong answer. If the balls are replaced after inspection then it will follow the
Bernoulli process and can be described by the binomial distribution.
(b) This is the right answer. If the Bernoulli experiment is repeated without replacement, it
follows hypergeometric distribution. Therefore the random variables in the given case
are best described by hypergeometric distribution.
(c) This is the wrong answer. Any variable studied for a large population is assumed to
follow a normal distribution. If we draw samples of size more than 30 from any
population the sample means are approximately normally distributed. But in our case
the distribution of the random variable can be best described by hypergeometric
distribution.
(d) This is a wrong answer. If ln (X) is a normally distributed random variable, the X is
said to be a lognormal variable. The random variables in given case are not described
by lognormal distribution.
(e) This is a wrong answer. t-distribution is used for testing of hypothesis when the sample
size is 30 or less than 30 and the population standard deviation is not known. The
random variables in the given case are not described well with the help of tdistribution.
< TOP >
23. Answer : (b)
Reason : Here, (a) AandBandC signifies the occurrence of A but not that of B and C. But
A a n d B a n d C represents the occurrence of both A and B, not that of C. Hence, the option
(b) is the answer. While (c) A a n d B a n d C shows the occurrence of all the three events.
The option (d) represents that A o r B o r C that means the occurrence of only one of the
three events. Similarly option (e) says A or B occurs but not C.
< TOP >
24. Answer : (d)
Reason : Here, P(A) = 0.3 and P(B) = k while, P(A or B)
= P(A) + P(B) – P(A).P(B) (as the events are independent) = 0.8
Therefore, 0.3 + k – 0.3´k = 0.8 or, 0.7´k = 0.5 or, k = 5/7.
< TOP >
25. Answer : (a)
Reason : We have P(S) = P(T) = x (say) and P(S and T) = p while the events are independent.
Therefore, x2 = p and hence, we get x = p . Hence, the option (a) is correct.
< TOP >
26. Answer : (e)
Reason : The events for which the Bayes’ theorem may be applied for computing posterior
probabilities must be mutually exclusive and collectively exhaustive.
< TOP >
27. Answer : (e)
Reason : The coefficient of correlation is defined as
( )
X y
cov x,y
r =
s ×s
Now, cov(x, y) > 0 and the standard deviation is the positive square root of variance. \ r >
0 and hence 0 < r £ 1 (because r cannot be more than 1).
< TOP >
28. Answer : (d)
Reason : The number of absentees who are girls follows a hypergeometric distribution. The following
details are available:
N = No. of elements in the population = 25
r = No. of elements in the population labelled success = No. of girls = 11
n = No. of trials = No. of absent students = 5
x = Desired number of successes
< TOP >
18
In a hypergeometric distribution P(x) =
r (N-r)
x (n-x)
N
n
C C
C
11 (25-11) 11 (25-11)
1 (5-1) 2 (5-2)
25 25
5 5
14 14
4 3
25 25
5 5
C C C C
P(x 2) =P(x 1) + P(x = 2) = +
C C
11× C 55× C
= +
C C
= 0.2072 + 0.3768
= 0.584
£ =
29. Answer : (c)
Reason : The number of days on which the athlete does press -ups follows a binomial distribution with
mean = n.p
n = Number of trials = 21
p = P(X > 2) = P(X = 3 or 4 or 5 or 6) =
4 2
6 3
=
Hence mean =
2
np = × 21 = 14
3
< TOP >
30. Answer : (c)
Reason : Any multiple regression equation consists of only one dependent variable and more than one
independent variables.
< TOP >
31. Answer : (d)
Reason : In a hypergeometric distribution the composition of the population changes from trial to
trial.
< TOP >
32. Answer : (c)
Reason : E(Y) = E(X– 25) = E(X) – 25
E(X) = (20 ´ 0.10) + (40 ´ 0.20) + (60 ´ 0.30) + (80 ´ 0.25) + (100 ´ 0.15)
= 63
\ E(Y) = 63 – 25 = 38.
< TOP >
33. Answer : (b)
Reason : The maximum number of successes that may be observed in a binomial distribution is equal
to the number of trials
< TOP >
34. Answer : (b)
Reason : The variance of a random variable is expressed in terms of the square of the unit of the
random variable.
< TOP >
35. Answer : (a)
Reason : (a) This is the right answer. In cluster sampling, we divide the population into groups, or
clusters , and then select a random sample of these clusters. We assume that these
individual clusters are representative of the population as a whole. Then we choose a
certain number of clusters randomly. Every household of these clusters is interviewed.
(b) This is a wrong answer. In stratified sampling we divide the population into relatively
homogeneous groups, called strata. Then we use one of two approaches. Either we
select at random from each stratum a specified number of elements corresponding to
the proportion of that stratum in the population as a whole or we draw an equal number
of elements from each stratum and give weight to the results according to the stratum’s
proportion of total population.
(c) This is a wrong answer. In systematic sampling, elements are selected from the
population at a uniform interval that is measured in time, order, or space. In this
method of sampling each element has an equal chance of being selected but each
sample does not have an equal chance of being selected.
(d) This is a wrong answer. In the judgmental sampling the sample is selected according to
the judgment of the investigators or experts.
< TOP >
19
(e) This is a wrong answer. In simple random sampling we do not divide the population in
groups. We select a certain number of elements randomly. Here each possible sample
has an equal chance of being selected and each item in the entire population also has an
equal chance of being selected.
36. Answer : (b)
Reason : Standard error of mean is defined as x n
s = s
. This implies that, as the sample size
increases, the standard error of mean decreases.
< TOP >
37. Answer : (d)
Reason : Since all the options (a), (b), (c) and (e) are of random sampling so, where option (d) is a non
random sampling technique.
Therefore the correct answer is (d).
< TOP >
38. Answer : (e)
Reason : H0 : m = 9
H1 : m ¹ 9
The sample is small and the population variance is not known. The sample variance is
specified.
The population is normally distributed. Hence we should use the t distribution with 25 – 1 =
24 d.o.f.
x s =
s
n =
256
25 = 3.20
z = x
x -m
s =
2.45 9
3.20
-
= –2.047
At a = 0.05, the critical values are ±2.064.
The test statistic is more than the left tail critical value. So it falls in the acceptance region.
\ We accept H0. But the true mean is 7. So H0 is false. Hence the test leads to a type II
error.
< TOP >
39. Answer : (c)
Reason : The feasible region is the set off all points which satisfy all the constraints in the LPP.
< TOP >
40. Answer : (c)
Reason : (a) and (b) are not the indicators of multioptimality.
(c) In the graphical method of solving a LPP the situation of multioptimality arises when
the objective function is parallel to one of the edges of the feasible region which is in
the direction of the optimal movement of the objective function.
(d) and (e) are not correct because they only assume non-negative values.
< TOP >
41. Answer : (c)
Reason : Let K be the nth term.
The given series is in A.P
First term, a = 2
Common difference, d = 6
\ K = a + (n – 1)d = 2 + 6(n – 1)
By the question:
Sn =
n ( )
2a n 1 d
2
éë + - ùû
n ( )
574 2 2 n 1 6
2
Þ = éë ´ + - ùû
or
( ) n
574 4 6 n 1
2
= éë + - ùû
< TOP >
20
or 574 =
n [ ] n [ ]
4 6n 6 6n 2
2 2
+ - = -
or 1148 = 6n2 – 2n
or 6n2 – 2n – 1148 = 0
or 3n2 – n – 574 = 0
\ n =
b b2 4ac
2a
- ± -
=
(1 ) ( 1)2 4 3 ( 574)
2 3
- - ± - - ´ ´ -
´
=
1 1 6888
6
± +
=
1 83
14
6
±
=
or
82
6
-
Since the number of terms, n cannot be negative, n = 14
\ K = 2+ 6 (14 – 1) = 80.
42. Answer : (c)
Reason : Let d be the common difference of this A.P. so that b = a + d and c = a + 2d
Now, a3 + 4(b)3 + c3
= a3 + 4(a + d)3 + (a + 2d)3
= a3 + 4(a 3 + 3a2d + 3ad2 + d3) + (a3 + 6a2d + 12ad2 + 8d3)
= a3 + 4a3 + 12a2d + 12ad2 + 4d3 + a3 + 6a2d + 12ad2 + 8d3
= 6a3 + 18a2d + 24ad2 + 12d3.
< TOP >
43. Answer : (b)
Reason :
324 4
243 3
=
;
432 4
324 3
=
\ The series is a G.P. with a = 243, r =
4
1
3
>
\
( )
n
n
4
243 1
a r 1 3 Sn
r 1 4 1
3
ìæ ö ü íç ÷ - ý - îè ø þ = =
- -
=
n
n
4
243 1
3
1
3
æ ö
ç - ÷
è ø
=
n n
n
243 3 4 3
3
æ - ö ´ç ÷
è ø
=
6 ( n n )
n
3 4 3
3
-
= 36 – n (4n – 3n)
< TOP >
44. Answer : (d)
Reason : Here the first term a is 2, the common difference is
1
1
2. The last term is not known while the
number of terms or n is 25.
\ Sn =
n [ ]
2a (n 1)d
2
+ -
or S25 =
25 3 25 [ ]
2 2 (25 1) 4 36 500.
2 2 2
é ´ + - ù = + = êë úû
< TOP >
45. Answer : (e)
Reason : Let a be the first term and d the common difference of the given series. We are given
q = a + (p – 1)d ------------- (1)
< TOP >
21
p = a + (q – 1)d ------------- (2)
Subtracting (2) from (1), we get
q – p = pd – qd
or q – p = d(p – q)
d(p – q) =– (p – q)
d = –1
Substituting the value of d in (1), we get
a + (p – 1) (–1) = q
q = a – p + 1
a = p + q – 1
The rth term = a + (r – 1) d
= (p + q – 1) + (r – 1) ( –1)
= p + q – 1 – r + 1 = p + q – r.
46. Answer : (d)
Reason : We have a = –4, L = 146 and Sn = 7171
Let n be the number of terms in the A.P.
\ Sn =
n
(a L) 7171
2
+ =
Þ
n ( )
4 146
2
- +
= 7171 or
n
142 7171
2
´ =
or n = 101.
< TOP >
47. Answer : (d)
Reason : The series in a G.P. with a = 8 and r =
1
2
\
a 8 8 2
S
1 r 1 2 1 1
2
a
= é ù = = êë - úû - -
=
82 2 1
2 1 2 1
´ +
- + =
8(2 2 )
2 1
+
- = 8(2 + 2)
< TOP >
48. Answer : (d)
Reason : Given that
4 2
3 28. 2 nP = nP
or,
(4 )! 28.(2 )!
(4 3)! (2 2)!
n n
n n
=
- -
or, (4n)(4n - 1)(4n - 2) = 28. (2n) (2n - 1)
or, 8n (4n –1)(2n – 1) = 28 (2n) (2n – 1)
or, (4n – 1) = 7
or, n = 2.
< TOP >
49. Answer : (c)
Reason : In the word COMBINATION the total number of letters are 11
The number of O’s are = 2
The nu mber of I’s are = 2
The number of N’s are = 2
\Total number of permutations are =
11 11
11 11
(2!) (2)! (2)! (2!)3
=
´ ´
P P
= 4989600
< TOP >
50. Answer : (e)
Reason : Since there are 8 vacant seats, the first man can choose any one of these 8 seats. There are
< TOP >
22
thus 8 ways of filling the first seat, when that one is occupied 7 seats are left, therefore, the
second man can occupy any one of the 7 seats. The third man can now seat himself in one of
the remaining 6 seats.
\ Number of ways in which 3 persons can occupy 8 seats is 8 ´ 7 ´ 6 = 336.
51. Answer : (e)
Reason : The man can go from A to B in 5 different ways, for he may take any one of the 5 routes.
When he has done so in any of the 5 ways he may return in 5 different ways. i.e., there are 5
different ways for returning
\ The total number of different ways are 5 ´ 5 = 25.
< TOP >
52. Answer : (b)
Reason : As per the rule of counting, the total number of telephone connection can be 9 ´ 9 ´ 9 ´ 9 ´
9 ´ 9 = 96 = 5,31,441.
< TOP >
53. Answer : (c)
Reason : a
1
log 2
6
=
Þ
1
a 6 = 2
Þ a = ( )6
2 = 23 = 8.
< TOP >
54. Answer : (c)
Reason : Let us take log3 2 5832 = x
\(3 2)x = 5832 = 8 ´ 729
= 23 ´ 36 = ( 2)6 ´ 36
=
(3 2)6 , Hence x = 6.
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55. Answer : (c)
Reason : If certain events are mutually exclusive and collectively exhaustive then only one of them
can happen at a time and the entire sample space is divided among the two events. Hence the
sum of their probabilities will be equal to 1.00.
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56. Answer : (d)
Reason : A continuous random variable is one which can assume any value within a given range. The
values that may be assumed by such a variable are not restricted to integers, even values, odd
values or any other limited number of values.
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57. Answer : (c)
Reason : Calculation of Arithmetic mean
Marks Mid-point
m
No. of student
f fm
0 – 10 5 5 25
10 – 20 15 10 150
20 – 30 25 25 625
30 – 40 35 30 1050
40 – 50 45 20 900
50 – 60 55 10 550
N = 100 Sfm = 3,300
x =
fx 3,300
33
N 100
S
= =
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58. Answer : (b)
Reason : a, b and c are in A.P.
\ b – a = c – b Þ 2b = a + c
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23
p is t he G.M. between a and b Þ p = ab Þ p2 = ab
q is the G.M. between b and c Þ q = bc Þ q2 = bc
\ p2 + q2 = ab + bc
or p 2 + q2 = b(a + c)
or p 2 + q2 = b(2b)
or p 2 + q2 = 2b2
or b2 =
p2 q2
2
+
\ b2 is the A.M. between p2 and q2.
59. Answer : (a)
Reason : (a) The average deviation of all the items from zero is equal to the arithmetic mean.
(b), (c), (d) and (e). The geometric mean, quartile deviation, standard deviation, and variance
cannot be calculated on the basis of average deviation from zero.
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60. Answer : (c)
Reason : Range (= Highest value – Lowest value) is based only on two observations in a data set.
Arithmetic mean, harmonic mean, mean absolute deviation and standard deviation are based
on all t he observations.
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61. Answer : (a)
Reason :
10
6
Even numbers > 2 are 4 and 6
P(Even numbers > 2) = P(4 or 6)
2 1
= =
6 3
Let X be the number of throws which give 4 or 6.
1
X follows a binomial distribution with n = 10 and p =
3
P(X = 6) = C
\
\
\ 6 10 - 6
6 4
10
6
p (1 - p)
1 2
= C
3 3
= 0.0569
æ ö æ ö
ç ÷ ç ÷
è ø è ø
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62. Answer : (d)
Reason : Our desired event is Z < 80. P(Z < 80) = 0.15 + 0.25 = 0.40
Let X be the number of times Z assumes values less than 80. Then X is binomially
distributed with
number of trials = 20 and probability of success = 0.40
The expected number of observations less than 80 = E(Z) = n.p = 20 x 0.40 = 8
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63. Answer : (d)
Reason : This is a case of Bernoulli process.
For a Bernoulli process the probability of ‘r’ successes in ‘n’ trials =
r n r
n C .p . q
r
-
Given:
P(r = 1) = 18.66% = 0.1866
P(r = 2) = 31.10% = 0.3110 n = 6
Let the probability of securing the supply contract with the party inviting the tender be ‘p’.
\ Probability of not securing the supply contract, q = 1–p
P(r = 1) =
1 6 1
6C p (1 p)
1
- -
= 0.1866 (given)
or 6p (1 – p)5 = 0.1866 (A)
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24
And, P(r = 2) =
2 6 2
6C p (1 p)
2
- -
= 0.3110 (given)
or 15p2 (1 – p)4 = 0.3110 (B)
Dividing equation (B) by equation (A) we get:
5
2 4
6p (1 p)
15p (1 p)
-
-
= 0.1866
0.3110
or 1 p
p
2
5
-
´
= 0.1866
0.3110
or 1 p
p
- =
÷ø
ö
çè
æ
0.1866
0.3110
5
2
= 0.66667
or p = 0.66667 (1 – p) or 1.66667p = 0.66667
or p = 1.66667
0.66667
= 0.4000012 @ 0.40
\ Probability of securing supply contract = 0.40
\ Probability of not securing supply contract, q = 1 – 0.40 = 0.60
Probability of securing supply contracts with a maximum of 2 parties i.e., P (r £ 2)
= P (r = 0) + P (r = 1) + P (r = 2)
P (r = 2) = 0.3110 (given)
P (r = 1) = 0.1866 (given) P(r = 0) =
0 6 0
6C p (1 p)
0
- -
= 0.0467
\p (r £ 2) = 0.0467 + 0.1866 + 0.3110 = 0.5443
P (r ³ 3) = P (r = 3) + P (r = 4) + P(r = 5) + P (r = 6) = 1 – P (r £ 2)
= 1 – 0.5443 = 0.4557 i.e., 45.57%.
64. Answer : (b)
Reason : a. The payoffs represent the outcomes of certain decisions.
b. The states of nature refer to the chance occurrences that can affect the outcome of an
individual’s decision
c. The decision alternative courses of action that are available to the decision maker.
d. The probabilities refer to the likelihood of the chance occurrences or the states of
nature.
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65. Answer : (d)
Reason : P(-2.0 < z < -1.0) = 0.4772 – 0.3413 = 0.1359.
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66. Answer : (a)
Reason :
Let success be defined as obtaining head in all the three coins
simultaneously.
1
On any coin P(Head) =
2
1 1 1 1
P(Success) = × × =
2 2 2 8
Let X denote the number of successes.
X follows a binomial distribut
\
\
3 10 - 3
10
3
1
ion with n = 10 and p =
8
1 1
P(X 3) = C 1 - = 0.09204
8 8
= æ ö æ ö ç ÷ç ÷
è øè ø
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67. Answer : (b)
Reason : The expected value of a binomial probability distribution is always more than the variance.
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68. Answer : (c)
Reason : The z value in the normal table for 89.9 percent confidence level is 1.64. We want our
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25
estimate to be within 0.02, so,
Z sp= 0.02
And z = 1.64
Then 1.64 sp = 0.02
Substituting the value of sp in the equation,
1.64
pq
0.02
n
=
or,
pq
n = 0.0122
As the university feels that the proportion of students favoring the new system is
36000
60000
= 0.60. We can take this value as the estimate of the population parameter.
Therefore,
0.6 0.4
0.0122
n
´
=
or, n =
2
0.6 0.4
(0.0122)
´
= 1612.4697
Therefore a sample size of 1613 would be an appropriate to estimate the proportion of the
students favoring the new system in an interval of ± 0.02 at a confidence level of 89.9%.
69. Answer : (c)
Reason : We set the null hypothesis as: The average daily smuggling intercepted is same as the
historic level
We set the alternative hypothesis as: The average daily smuggling intercepted is more than
the historic level.
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70. Answer : (e)
Reason : H0 : m = 84 H1 : m ¹ 84
This is a two tailed test. The sample size more than 30, and the population standard deviation
is known. Hence we use normal distribution for the given test.
The area under the rejection region is 0.005 in each tail. The appropriate critical values for
this test are ± 2.575.
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71. Answer : (c)
Reason : H0: p = 0.60
H1: p < 0.60
This is a large sample test of proportion. So we can use the normal approximation to the
binomial.
p s =
( ) 0 0 p 1 p
n
-
=
0.60 0.40
360
´
= 0.02582
\ z =
p 0.60 0.57 0.60
0.02582 0.02582
- -
=
= –1.162
At a = 0.05, the critical value in the left tail is –1.645.
The test statistic is more than the left tail critical value. So it falls in the acceptance region.
\ We accept H0. The true proportion is 0.60. So H0 is true. Hence the test does not lead to
type I or type II error.
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72. Answer : (d)
Reason : Let the following notations be used:
Y : Sales (’000 sets)
X1 : Number of advertisements
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26
X2 : Price per set.
The multiple regression relationship is–
Yˆ
= a + b1X1 + b2X2
å Y = na + b 1 å X1 + b2 åX2
å X1Y = a å X1+ b1
2
å X1 + b2 å X1X2
åX2Y = a å X2+ b1 å X1X2 + b2
22
å X
We obtain the following values:
å Y = 248 å X1 = 36 å X2 = 630
å X1Y = 2013
2
å X1 = 318
22
å X = 79950
å X 2Y = 30860 å X1X 2 = 4625 n = 5
\ We get the following equation :
248 = 5a + 36b1 + 630b2 –– (A)
2013 = 36a + 318b1 + 4625b2 –– (B)
30860 = 630a + 4625b1 + 79950b2 –– (C)
Solving the above three equations we get , a = 215.358
b1 = 6.413
b2 = –1.682.
73. Answer : (d)
Reason : Given that 1 x = 1300, s1 = 82, n1 = 100.
2 x = 1288, s2 = 93, n2 = 100
The estimated standard error for the difference between means is,
1 2
2 2 2 2
1 2
1 2
(82) (93)
ˆ 12.399
100 100 - s= + = + = x x
s s
n n
The test statistic for this test is,
( ) ( )
1 2
1 2 1 2 (1300 1288) 0
0.968.
12.399 -
- - m -m - -
= = =
sx x
x x
z
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74. Answer : (a)
Reason : Lets marks in Accountancy be denoted by X and marks in Statistics by Y.
X 48 35 17 23 47 åX = 170
Y 45 20 40 25 45 åY = 175
x 14 1 -17 -11 13 åx = 0
x2 196 1 289 121 169 åx2 = 776
y 10 -15 5 -10 10 åy = 0
y2 100 225 25 100 100 åy2 =550
xy 140 -15 -85 110 130 åxy =280
Where, x = X - X and y = Y - Y
X = åX / n = 170 / 5 = 34, Y = åY/ n = 175/5 = 35.
Now, r =
2 2
xy 280
0.429.
x y 776 550
å
= =
å å ´
< TOP >
75. Answer : (c)
Reason : The covariance of a random variable with another random variable is zero if the random
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27
variables are independent.
76. Answer : (b)
Reason : Unweighted aggregates price index =
1
0
P
100
P
S
´
S
=
(21 220 31)
100
(17 190 30)
+ +
´
+ + = 114.77.
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77. Answer : (b)
Reason : Because people tend to spend less on goods when their prices are rising, Paasches index
which bases on current weights, produces an index w hich has a downward bias..
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78. Answer : (a)
Reason : Index number of industrial production is a quantity index. The other indices are price
indices.
< TOP >
79. Answer : (c)
Reason : The value index measures the change in the total monetary value over a period of time.
Therefore the correct answer is (c).
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80. Answer : (b)
Reason : Paasches price index uses the quantity measures for the current period as weights.
Therefore the correct answer is (b).
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81. Answer : (e)
Reason : In order to describe a binomial distribution the number of trials and probability of success
in any trial must be specified.
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82. Answer : (a)
Reason : We obtain the regression equation using the following table:
X Y XY X2
6 9 54 36
2 11 22 4
10 5 50 100
4 8 32 16
8 7 56 64
åX = 30 åY = 40 åXY = 214 åX2 = 220
The regression equation of Y on X is ˆY = a + bX
To determine the values of a and b the following two normal equations are to be solved:
å Y = na + b å X
å XY = a å X + b å X2
Substituting the values, we get :
40 = 5 a + 30 b
214 = 30 a + 220 b
The values of a = 11.9 and b = -0.65
Y = 11.9 – 0.65 X.
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83. Answer : (b)
Reason : A normal probability distribution is a continuous probability distribution
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84. Answer : (d)
Reason : The minor of an element of the matrix is defined as deleting the row and column in which
the element is present then the matrix is called the minor of an element.
< TOP >
85. Answer : (d)
Reason : P (x<42) =
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28
42 50
P z P(z 1.6) 1 P(z 1.6)
5
1 P(z 1.6)
1 (0.5 0.4452)
1 0.9452
0.0548 i.e.5.48%.
æ - ö ç < ÷= < - = - > -
è ø
= - <
= - +
= -
=
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