Quantitative Methods (MB151) : October 2005

Question Paper

Quantitative Methods (MB151) : October 2005

· Answer all questions.
· Marks are indicated against each question.
1. Which of the following is used to represent data in the form of pictures, where the size of the picture
is proportional to the magnitude of the data?
(a) Bar chart (b) Line chart (c) Pictogram
(d) Pie chart (e) Scatter diagram.
(1 mark)
< Answer >
2. Which of the following statements is true?
(a) The tallest rectangle in a histogram represents the modal class of the distribution
(b) In a symmetrical distribution the mean, median and mode are unequal
(c) The medians of two sets of data can be combined mathematically
(d) The mode can not be determined graphically
(e) The mode is always uniquely defined.
(1 mark)
< Answer >
3. Which of the following is not true about median?
(a) Median is not strongly affected by the extreme values
(b) It can be also be calculated from grouped data
(c) It can also be calculated for qualitative data
(d) It can not be computed for grouped data having open-ended classes
(e) It involves a time consuming process for raw data as the data has to be arranged.
(1 mark)
< Answer >
4. The standard deviation of a data set
(a) Is expressed in the same unit as the observations in the data set
(b) Is expressed in the square of the unit of the observations in the data set
(c) Is expressed in the square root of the unit of the observations in the data set
(d) Is expressed in a different unit from the unit in which the observations in the data set are
expressed
(e) Is always expressed as a percentage of the mean of the data set.
(1 mark)
< Answer >
5. Which of the following is true?
(a) In a logarithmic function the logarithmic operation is applied on the dependent variable
(b) In a logarithmic function the base of the logarithm is always 10
(c) In a logarithmic function the independent variable cannot be negative
(d) In a logarithmic function the base of the logarithm can be any real number
(e) In a logarithmic function the dependent variable cannot be negative.
(1 mark)
< Answer >
6. If a constant quantity is subtracted from every observation in a data set then the range of the
resulting set of values will be equal to
(a) Range of the original data set plus the constant quantity
(b) Range of the original data set minus the constant quantity
< Answer >
(c) Range of the original data set
(d) Range of the original data set multiplied by the constant quantity
(e) Range of the original data set divided by the constant quantity.
(1 mark)
7. Which of the following measures is not affected by the presence of extreme values in a data set?
(a) Range (b) Arithmetic mean
(c) Standard deviation (d) Variance (e) Median.
(1 mark)
< Answer >
8. Which of the following is not true about mean absolute deviation?
(a) It is obtained by calculating the absolute deviations of each observation from mean
(b) It is a more comprehensive measure compared to range
(c) It is conducive to further algebraic treatment
(d) It cannot be computed for distributions with open end classes
(e) It can also be computed for grouped or continuous frequency distribution.
(1 mark)
< Answer >
9. Which of the following distributions should be used in hypothesis testing when the population
standard deviation is not known and the sample size is less than 30?
(a) Chi-square distribution (b) Poisson distribution
(c) Normal distribution (d) Binomial distribution
(e) t -distribution.
(1 mark)
< Answer >
10. Which of the following is true when the slope of a regression line is negative?
(a) The correlation coefficient between the dependent and independent variables is 1
(b) The correlation coefficient between the dependent and independent variables, lies between
0 and 1
(c) There is a negative correlation between the dependent and independent variables
(d) The regression line is parallel to the horizontal axis
(e) The regression line passes through the intersection of the horizontal and vertical axes.
(1 mark)
< Answer >
11. If bYX is the slope coefficient of regression line of Y on X, and bXY is the slope coefficient of
regression line of X on Y then which of the following is true?
(a) bYX is positive implies that bXY is positive
(b) bYX is positive implies that bXY is negative
(c) bYX and bXY are reciprocals
(d) The product of bYX and bXY is zero
(e) The sum of bYX and bXY is one.
(1 mark)
< Answer >
12. The coefficient of variation cannot be meaningfully used to compare the variability of two or more
sets of data, when
< Answer >
(a) The standard deviation is zero for one or more sets of data
(b) The standard deviation is 1 for one or more sets of data
(c) The mean is zero for one or more sets of data
(d) The mean is 1 for one or more sets of data
(e) The mean and standard deviation are equal for one or more sets of data.
(1 mark)
13. Which of the following is true with regard to a given coefficient of correlation and its
corresponding coefficient of determination?
(a) The coefficient of determination is always greater than or equal to zero, and less than or
equal to 1
(b) The coefficient of determination is always less than zero
(c) The coefficient of determination is always equal to 1
(d) The coefficient of determination always has the same sign as the coefficient of correlation
(e) The magnitude of coefficient of determination is always higher than the magnitude of
coefficient of correlation.
(1 mark)
< Answer >
14. To which of the following distributions is Bienayme and Chebyshev’s rule applicable?
(a) Normal distribution
(b) Positively skewed distributions
(c) All symmetrical distributions
(d) Negatively skewed distributions
(e) Any type of distribution.
(1 mark)
< Answer >
15. The Y intercept in a simple linear regression equation (X is the independent variable and Y is the
dependent variable) is equal to the
(a) True value of Y when X = 0
(b) Change in estimated value of Y per unit change in X
(c) Estimated value of Y when X = 0
(d) Standard deviation of the values of X
(e) Mean of the values of X.
(1 mark)
< Answer >
16. If the points in a scatter diagram show a very little spread and a tendency to rise from the left to the
right along the horizontal axis, then we can conclude that
(a) There is perfectly positive correlation
(b) There is perfectly negative correlation
(c) There is no correlation
(d) There is high degree of positive correlation
< Answer >
(e) There is low degree of negative correlation.
(1 mark)
17. In which of the following conditions will the problem of multicollinearity arise in a multiple
regression relationship given as : Yˆ = a + b1X1 + b2X2?
(a) There is a high degree of correlation between the variables Y and X1
(b) There is a high degree of correlation between the variables Y and X2
(c) There is a high degree of correlation between the variables X1 and X2
(d) Both the regression coefficients b1 and b2 are positive
(e) The intercept a is positive.
(1 mark)
< Answer >
18. The weighted average of price relatives using base values as weights is same as the
(a) Unweighted aggregates price index (b) Unweighted aggregates quantity index
(c) Laspeyres price index (d) Laspeyres quantity index
(e) Paasche’s price index.
(1 mark)
< Answer >
19. 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 >
20. Which of the following weighted price index numbers uses only the quantity measures for the
current period as weights?
(a) Laspeyre’s 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 >
21. If all the terms of an arithmetic progression are multiplied by a constant quantity the resulting terms
will always form
(a) A geometric progression
(b) A harmonic progression
(c) An arithmetic progression
(d) Either a geometric progression or a harmonic progression
(e) Either a geometric progression or an arithmetic progression.
(1 mark)
< Answer >
22. Which of the following is true with regard to Fisher’s ideal price index? < Answer >
(a) It does not consider the base year prices
(b) It does not consider the base year quantities
(c) It does not consider the current year prices
(d) It does not consider the current quantities
(e) It is the geometric mean of the Laspeyres’ and Paasche’s price indices.
(1 mark)
23. Events A and B are dependent. The joint probability of the events A and B is
(a) Equal to the product of the marginal probabilities of the events A and B
(b) Not equal to the product of the marginal probabilities of the events A and B
(c) Equal to the sum of the marginal probabilities of the events A and B
(d) Equal to the difference between the marginal probabilities of the events A and B
(e) Is always equal to 1.
(1 mark)
< Answer >
24. If two events A and B are independent then, the conditional probability of event A given that event
B has occurred, is equal to
(a) Joint probability of events A and B
(b) Conditional probability of event B given event A
(c) Marginal probability of event B
(d) Marginal probability of event A
(e) Zero.
(1 mark)
< Answer >
25. Which of the following are the conditions for applying the Bayes’ theorem for computing posterior
probabilities of certain events?
(a) The events must be non-mutually exclusive
(b) The events must be mutually exclusive
(c) The events must not be collectively exhaustive
(d) The events must be collectively exhaustive
(e) Both (b) and (d) above.
(1 mark)
< Answer >
26. Which of the following is true with regard to the classical approach to probability?
(a) It assumes that the outcomes are not equally likely
(b) The probability of an event is determined after performing the experiment large number of
times
(c) The probability of an event is determined before performing the experiment
(d) It assumes that all possible outcomes of the experiment are not known
(e) The classical approach cannot be used to find out the probability of mutually exclusive
events.
< Answer >
(1 mark)
27. If the coefficient of correlation between two variables lies between –1 and 0, then the covariance
between them is
(a) Positive (b) Negative (c)
Zero
(d) Equal in magnitude to the variances of both the variables
(e) Equal in magnitude to the standard deviations of both the variables.
(1 mark)
< Answer >
28. The probability of the occurrence of an event is expressed as a number which lies between
(a) 0 and 1 (b) 1 and 2 (c) –1 and 0 (d) –2 and –1 (e) 1 and
infinity.
(1 mark)
< Answer >
29. If A, B and C are three events such that P(A).P(B).P(C) > 0 and P(B) > P(A), which of the
following is always true?
(a) P(A)> P(B) (b) P(A)> P(B)
(c) P(A)> P(B) (d) P(B)> P(A) (e) P(B|C) >P(A|C) .
(1 mark)
< Answer >
30. Which of the following measures represents the scatter of the values in a data set?
(a) Arithmetic mean (b) Geometric mean (c)
Mode
(d) Median (e) Standard deviation.
(1 mark)
< Answer >
31. Which of the following is false with regard to a hypergeometric 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 continuous random variable is one which
(a) Can assume only integer 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 >
33. In a binomial distribution the probability of getting zero or more number of successes is equal to < Answer >
(a) 0
(b) The probability of getting zero success
(c) The probability of getting successes in all the trials
(d) 1 minus the probability of getting successes in all the trials
(e) 1.
(1 mark)
34. The sampling distribution of the mean is a distribution of
(a) Means of individual populations
(b) Observations within a population
(c) Observations within a sample
(d) Means of all possible samples of a specific size taken from a population
(e) Means of samples of a specific size taken from different populations.
(1 mark)
< Answer >
35. As the sample size increases
(a) The variation of the samp le mean from the population mean becomes larger
(b) The variation of the sample mean from the population mean becomes smaller
(c) The variance of the sample becomes less than the variance of the population
(d) The standard deviation of the sample becomes more than the standard deviation of the
population
(e) The standard deviation of the sample comes close to zero.
(1 mark)
< Answer >
36. The standard error of estimate
(a) Is a measure of central tendency for the distribution of the values estimated by the
regression equation
(b) Measures the variability of the observed values of the dependent variable around their mean
(c) Measures the variability of the observed values of the dependent variable around the
regression line
(d) Measures the variability of the values of the independent variable around their mean
(e) Measures the variability of the values of the independent variable around the regression
line.
(1 mark)
< Answer >
37. In which of the following methods of sampling the population is divided into groups so that the
elements in each group are homogeneous and the groups vary from each other?
(a) Judgmental sampling (b) Stratified sampling
(c) Systematic sampling (d) Cluster sampling (e) Random
sampling.
(1 mark)
< Answer >
38. For a given hypothesis test, if we do not reject the null hypothesis, and the null hypothesis is true, < Answer >
then which of the following can be stated?
(a) Neither type I error nor type II error has been committed
(b) Type I error has been committed
(c) Type II error has been committed
(d) Both type I and type II error have been committed
(e) The test statistic is incorrect.
(1 mark)
39. Which of the following is true with regard to the simplex method of solving a linear programming
problem (LPP) on profit maximization?
(a) At the optimal solution the slack variables are always equal to zero
(b) The constraints which contain ‘£’ are converted into equations by adding slack variables
(c) There can be only one feasible solution to a LPP
(d) The slack variables make positive contributions to profit
(e) The slack variables can only assume the values 0 or 1.
(1 mark)
< Answer >
40. In the graphical method of solving linear programming problems if there is a unique optimal
solution, then the optimal solution
(a) Is always found at the center of the feasible region
(b) Is always at the origin
(c) Lies outside the feasible region
(d) Is located at one of the corner points of the feasible region
(e) Always lies on one of the two axes.
(1 mark)
< Answer >
41. If the ratio of the sum of m terms and n terms of an A.P. is m2 : n2, then the ratio of its mth and nth
terms will be
(a) (2m + 1) : (2n + 1)
(b) (2m - 1) : (2n + 1)
(c) (2m - 1) : (2n - 1)
(d) (2m + 1) : (2n - 1)
(e) (m + 1) : (n + 1).
(2 marks)
< Answer >
42. S is the sum and P is the product of the first n terms in a Geometrical Progression (G.P). R is the
sum of the reciprocals of the first n terms of the G.P. Sn=
(a) P2 (b) Rn (c) Pn (d) P. Rn (e) P2 Rn.
(2 marks)
< Answer >
43. The ratio of harmonic mean of two numbers to the geometric mean of the same numbers is 4:5. The
actual numbers are in the ratio of
< Answer >
(a) 1:2 (b) 1:3 (c) 1:4 (d) 2:3 (e) 2:5.
(2 marks)
44. If mth term of an H.P is n and nth term of the HP is m. The (m+n)th term of the H.P would be
(a) m
n
(b) n
m
(c) m
(m+n)
(d) mn
(m+n)
(e) (m - n)
.
(m + n)
(1 mark)
< Answer >
45. If the sums of first 8 and first 19 terms of an A.P. are 64 and 361 respectively, find the sum of its n
terms.
(a) n2 +1 (b) n2 -1 (c) (n2 -1)/2 (d) n2 (e) (n2
+1)/2.
(1 mark)
< Answer >
46. A class consists of a number of boys whose ages are in A.P. with a common difference of four
months. If the youngest boy is just eight years old and if the sum of the ages is 168 years, find the
number of boys.
(a) 12 (b) 14 (c) 16 (d) 18 (e) 20.
(1 mark)
< Answer >
47. m-1
n
n+1
m
x =m
n ,
1-n
m
m+3
n
y = n
m and
5
n
1-2n
m
z = m
n
The value of xyz is
(a)
m
n
m
n (b)
n
m
m
n (c)
m
n
n
m (d)
n
m
n
m (e) n m ´m n .
(1 mark)
< Answer >
48.
If
n n - 3
6 3 C : C =33:4 . The value of n would be
(a) 8 (b) 9 (c) 10 (d) 11 (e) 12.
(2 marks)
< Answer >
49. In a class of 8 students there are three girls. In how many different ways can they be arranged in a
row so that no two of the three girls are seated next to each other?
(a) 24 (b) 48 (c) 72 (d) 120 (e) 144.
(1 mark)
< Answer >
50. The number of words that can be formed by rearranging the letters of the word
CORRESPONDENCE is
(a) 14! (b)
3
14!
(2!) 3! (c)
4
14!
(2!) 3! (d)
2
14!
(2!) 3! (e)
14!
.
(2!)(3!)
(1 mark)
< Answer >
51. n1 n 1
Cr Cr 1 + +
+ - =
< Answer >
(a)
n 1
Pr +
(b)
n 1
Pr 1 +
- (c) ( )
n 2
Pr
r 1 !
+
- (d)
n 2
Pr
r!
+
(e) ( )
n 2
Pr
r 1 !
+
+ .
(2 marks)
52. A round table conference is to be held between 10 Indo-Pak delegates. In how many ways they can
be arranged to sit so that both the high commissioners are never seated together?
(a)725760 (b) 362880 (c) 282240 (d) 189260 (e)
144520.
(1 mark)
< Answer >
53. logab + logac = 0
This implies that
(a) b = c (b) b = -c (c) b + c
= 1
(d) b – c = 1 (e) b and c are reciprocals.
(1 mark)
< Answer >
54. 1 1 1 1
log 1 log 1 log 1 ...... log 1
2 3 4
- + - + - + + - = æ ö æ ö æ ö æ ö
çè ÷ø çè ÷ø çè ÷ø çè ÷ø n n n n n
(a) - 1/2 (b) 1/2 (c) 1 (d) - 1 (e) 2.
(1 mark)
< Answer >
55. The observations from a population are presented in the following frequency distribution:
Class interval Numbers of observations
10 – 20 5
20 – 30 15
30 – 40 17
40 – 50 25
50 – 60 18
60 – 70 10
70 – 80 6
80 – 90 4
Total number of observations
in the population
100
What is the standard deviation of the population?
(a) 14.25 (b) 17.23 (c) 20.18 (d) 22.16 (e)
24.36.
(2 marks)
< Answer >
56. The variance of a set of data is 3. If every item is doubled then, what will be the variance of the
resulting data set?
(a)
3
4 (b)
3
2 (c) 3 (d) 6 (e) 12.
(1 mark)
< Answer >
57. The data below represents the amount of grams of carbohydrates in a serving of breakfast cereal:
11, 15, 23, 29, 19, 22, 21, 20, 15, 25, 17.
The mean carbohydrates in this sample is ________ grams.
< Answer >
(a) 15.25 (b) 19.73 (c) 21.42 (d) 21.70 (e)
20.56.
(1 mark)
58. The market value of an investment has undergone the following growth rates in different years:
Year 1 2 3 4 5 6 7
Growth rate
(%)
5 8 –
10
–
4
6 10 7
Which of the following statements is/are correct?
I. The appropriate average that should be applied for calculating the average growth rate over
the period of seven years is arithmetic mean.
II. The appropriate average that should be applied for calculating the average growth rate over
the period of seven years is the harmonic mean.
III. The appropriate average that should be applied for calculating the average growth rate over
the period of seven years is the geometric mean.
IV. The average growth rate over the seven year period is 3.14%.
V. The average growth rate over the seven year period is 18.2%.
VI. The average growth rate over the seven year period is 2.91%.
(a) Only (I) (b) Only (II) (c) Both (I) and
(IV)
(d) Both (II) and (V) (e) Both (III) and (VI).
(2 marks)
< Answer >
59. What is the harmonic mean of the following quantities?
20, 15, 30, 22, 12, 25, 18, 16
(a) 0.055 (b) 0.437 (c) 18.313 (d) 19.02 (e)
19.75.
(1 mark)
< Answer >
60. The average salary of the male employees in a firm was Rs.5,200 and that of females was Rs.4,200.
If the mean salary of the employees was Rs.5,000, what was the percentage of male and female
employees?
(a) 90:10 (b) 80:20 (c) 70:30 (d) 60:40 (e)
50:50.
(1 mark)
< Answer >
61. If the letters of the word VICTORY are arranged in a row at random, the probability that no two
vowels may come together is
(a) 2/7 (b) 3/7 (c) 4/7 (d) 5/7 (e) 6/7.
(1 mark)
< Answer >
62. If the letters of the word MISSISSIPI are arranged in a row at random, the probability that all S’s
come together is
(a) 1/7 (b) 1/10 (c) 1/3 (d) 1/30 (e) 2/15.
< Answer >
(1 mark)
63. Two dice are thrown simultaneously. The probability that the sum of the numbers on them is less
than or equal to 5 is
(a) 1/9 (b) 1/6 (c) 2/9 (d) 5/18 (e) 1/3.
(1 mark)
< Answer >
64. Experiment one results in the mutually exclusive and collectively exhaustive events A, B and C.
Experiment two results in the mutually exclusive and collectively exhaustive events D and E. The
joint probabilities of the events that may result from these two experiments are given below:
A B C
D 0.12 0.075 0.12
E 0.28 0.225 0.18
Which of the following statements are correct?
I. P(A) = 0.30 II. P(B) = 0.40
III. P(C) = 0.30 IV. P(D) = 0.685
V. P(E) = 0.685 VI. P(D|C) = 0.25
VII.P(E|B) = 0.75 VIII. P(D|A) = 0.30
IX. D is independent of the events A, B and C respectively.
X. E is dependent on the events A, B and C respectively.
(a) (I), (IV) and (VI) (b) (IV), (VI), (VII) and (X)
(c) (II), (III), (IV), (VI) and (IX) (d) (III), (V), (VII), (VIII) and (X)
(e) (I), (II), (IV), (VI) and (IX).
(2 marks)
< Answer >
65. For the standard normal probability distribution, the area to the left of the mean is
(a) greater than 0.5 (b) less than 0.5 (c) equal to 1
(d) equal to 0.5 (e) greater than 1.
(1 mark)
< Answer >
66. The probability of failure in a Bernoulli experiment is 0.60. The experiment is repeated 25 times.
The mean of the binomial distribution of the number of successes is
(a) 10 (b) 20 (c) 30 (d) 40 (e) 50.
(1 mark)
< Answer >
67. A random variable, Y, has the following probability distribution:.
Y 20 40 60 80 100
Probability 0.15 0.20 0.30 0.25 0.10
15 values of Y have been observed. What is the probability that less than 12 observations are such
that 40 £ Y £ 80?
(a) 0.8441 (b) 0.1559 (c) 0.0668 (d) 0.4613 (e)
0.5387.
(2 marks)
< Answer >
68. 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 = 15
Significance level = 0.05
The population is normally distributed. It is later known that the true population mean is 11.
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 doesnot 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 >
69. In which of the following conditions a Type I error is said to have occurred, in testing a hypothesis?
(a) The sample statistic is incorrectly calculated
(b) The sample variance is incorrectly calculated
(c) Null hypothesis is rejected though it is true
(d) Null hypothesis is accepted though it is false
(e) The sample size is small.
(1 mark)
< Answer >
70. The following details are available for a hypothesis test on population proportion:
H0: p = 0.40
H1: p ¹ 0.40
Sample size = 360
Sample proportion = 0.36
Significance level = 0.05
It is later known that the true population proportion is 0.40.
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 doesnot lead to either type I or type II error
< Answer >
(d) The test leads to a type I error
(e) The test leads to a type II error.
(2 marks)
71. The following details are available with regard to a hypothesis test on means of two populations:
n1 = 64
2
s1 = 256
n2 = 36
22
s =144
The samples collected from the two populations are independent.
What is the standard error of difference between means?
(a) 2.828 (b) 4 (c) 8 (d) 16 (e)
1.414.
(1 mark)
< Answer >
72. The following information are available with regard to a sampling distribution of mean:
Population mean = 60
Sample size = 64
Probability that the sample mean is less than 55 = 0.0228
It is assumed that the Central Limit Theorem will be applicable.
What is the population variance?
(a) 6.25 (b) 64 (c) 20 (d) 400 (e) 3600.
(2 marks)
< Answer >
73. The following details are available with regard to a hypothesis test on difference between means of
two populations:
n1 = 16 n2 = 9
The samples are independently collected and it is assumed that the two populations have the same
variance. The pooled estimate of the underlying common population variance is 217.04. What is the
estimated standard error of difference between means?
(a) 6.138 (b) 37.675 (c) 12.276 (d) 3.069 (e)
8.682.
(2 marks)
< Answer >
74. Rola Cola is studying the effect of its latest advertising campaign. People chosen at random were
asked how many cans of Rola Cola they had bought in the past week and how many Rola Cola
advertisements they had seen on television in the past week.
No. of advertisements seen 3 7 4 2 0 4 1
No. of cans of Rola Cola
purchased
10 17 9 4 7 6 3
What is the coeffic ient of correlation for the above sample of the number of advertisement seen and
the number of cans purchased in the past week?
(a) 0.6548 (b) 0.7458 (c) 0.8001 (d) 0.9134 (e)
0.9868.
< Answer >
(1 mark)
75. A fitted regression equation is given by ˆY = 20 + 0.
75X. What is the value of the residual at the point X = 100, Y = 90?
(a) 5 (b) 10 (c) 0 (d) 15 (e) -
5.
(1 mark)
< Answer >
76. The following details are available with regard to a basket of commodities in a year:
Laspeyre’s price index = 140.5
Fisher’s ideal price index = 143.03
What is the Paasche’s price index for the basket of commodities in that year?
(a) 98.23 (b) 101.8 (c) 103.6 (d) 135.6 (e)
145.6.
(1 mark)
< Answer >
77. Consider the following information:
2000 (Base year) 2003
Commodity
Price (Rs./Kg.) Quantity (Kg.) Price (Rs./Kg.) Quantity (Kg.)
Wheat 7.50 5000 10.25 4800
Rice 10.00 6000 14.00 5500
Pulses 25.00 3000 28.00 3500
Sugar 10.60 3000 14.50 2400
Salt 2.60 500 4.50 600
What is Fisher’s ideal price index for the year 2003 considering the year 2000 as the base year?
(a) 112.82 (b) 113.55 (c) 127.35 (d) 128.12 (e)
133.58.
(1 mark)
< Answer >
78. Consider the following information:
Commodity
100
P
P
0
1 ´ Pn Qn
Rice 118.42 10,000
Wheat 130.77 4,050
Salt 160 1,300
< Answer >
Sugar 145 9,600
Pulses 137.78 13250
The weighted average of relatives price index is
(a) 118.42 (b) 130.77 (c) 134.54 (d) 137.78 (e)
139.20.
(1 mark)
79. For a basket of commodities the price of each commodity in a year, X, is double the price in the
base year. What is the Laspeyres price index for the year X?
(a) 100 (b) 150 (c) 200 (d) 250 (e) 50.
(1 mark)
< Answer >
80. The price of an item in the year 1991 (base year) was Rs.16 per unit and its price in the year 2001
was Rs.20 per unit. What is the price relative for the item for the year 2001?
(a) 80 (b) 4 (c) 36 (d) 125 (e) 320.
(1 mark)
< Answer >
81. The covariance between two variables X and Y is 256. If the variance of variable X is 625 and the
coefficient of correlation between X and Y is 0.64, the variance of the variable Y is approximately
(a) 4 (b) 8 (c) 16 (d) 64 (e) 256.
(1 mark)
< Answer >
82. Given the following data, the regression line is :
X Y
1 5
6 15
(a) Y = (1/2)X + 10 (b) Y = 0.35X + 7 (c) Y = 2X + 3
(d) Y = 3X + 2 (e) Y = 2X +2.
(1 mark)
< Answer >
83 The percentage of area under the normal distribution curve that falls within ± 2s limits from the
mean is approximately
(a) 9.55% (b) 95.5% (c) 68.33% (d) 50% (e)
2.0%.
(1 mark)
< Answer >
84. The following details are available with regard to a regression relationship:
Sum of the observations of the independent variable = 120
Sum of the observations of the dependent variable = 425
Number of data points = 10
Slope of the regression line = 2.5
What is the estimated value of the dependent variable if the independent variable is equal to 10?
(a) 2.5 (b) 12.5 (c) 25 (d) 37.5 (e) 42.5.
< Answer >
(2 marks)
85.
For a simple linear regression equation ( )2 å Y-Yˆ =2025 and the standard error of estimate is 15.
What is the number of observations? (Y is the dependent variable)
(a) 8 (b) 9 (c) 10 (d) 11 (e) 15.
(2 marks)
< Answer >
Suggested Answers
Quantitative Methods (MB151): October 2005
1. Answer : (c)
Reason : Pictograms represents the data in the form of pictures. The data is presented using
appropriate pictures and the size indicates the magnitude of the data.
Therefore the correct answer is (c)
< TOP >
2. Answer : (a)
Reason : (a) The tallest rectangle in a histogram represents the modal class.
(b) In a symmetrical distribution mean, median and mode are equal.
(c) The median can not be mathematically manipulated; hence medians of two sets of
data can not be combined.
(d) The median can be determined graphically.
(e) The mode is not uniquely determined when more than one observations have the
highest frequency.
< TOP >
3. Answer : (d)
Reason : The median can be computed for the data having open-ended classes unless the data falls
in the open-ended class. All other statements are true for median.
Therefore the correct answer is (d).
< TOP >
4. Answer : (a)
Reason : (a) The standard deviation of a data set is expressed in the same unit as the observations
in the data set.
(b), (c), (d) and (e) are incorrect with regard to the standard deviation.
< TOP >
5. Answer : (c)
Reason : a. In a logarithmic function the logarithmic operation is applied on the independent
variable.
b. In a logarithmic function the base of the logarithm may be any number other than 10.
c. In a logarithmic function the independent variable cannot be negative.
d. In a logarithmic function the base of the logarithm cannot be zero or negative
numbers.
< TOP >
6. Answer : (c)
Reason : If a constant is subtracted from every observation in a data set then the range of the
resulting set of values will be equal to the range of the original data set.
< TOP >
7. Answer : (e)
Reason : a. Range is based on the two extreme values in the data set. Hence it is affected by the
extreme values in the data set.
b. Arithmetic mean is based on all the values. Hence this is also affected by the
presence of extreme of values.
c. Standard deviation is based on all the values. Hence this is also affected by the
presence of extreme of values.
d. Variance is based on all the values. Hence this is also affected by the presence of
< TOP >
extreme of values.
e. Median is middle value of a data array. So it is not affected by the presence of the
extreme values.
8. Answer : (c)
Reason : Mean deviation is not conducive to further algebraic treatment. All other statements are
true.
Therefore the correct answer is (c)
< TOP >
9. Answer : (e)
Reason : a. The chi-square distribution is applied for comparing more than two samp le
proportions.
b. The F-distribution is applied for comparing more than two sample means.
c. The normal distribution is applied if the sample is large (or the population standard
deviation is known if the sample is small).
d. In the given situation the binomial distribution is not appropriate.
e. The t-distribution is appropriate for hypothesis testing when the population standard
deviation is not known and the sample size is less than 30.
< TOP >
10. Answer : (c)
Reason : a. When the slope of a regression line is negative the correlation coefficient need
not be 1.
b. When the slope of a regression line is negative, there is a negative correlation
between the variables; hence the correlation coefficient lies between –1 and 0.
c. When the slope of a regression line is negative, there is a negative correlation
between the variables.
d. When the slope of a regression line is zero, the regression line will be parallel to the
horizontal axis.
e. When the y-intercept of a regression line is zero, the regression line passes through
the intersection of horizontal and vertical axes.
< TOP >
11. Answer : (a)
Reason : If byx is the slope of regression line of Y on X and bxy is the slope of regression line of X
on Y then, byx is positive implies that bxy is positive.
< TOP >
12. Answer : (c)
Reason : Coefficient of variation = (standard deviation / mean) × 100
(a) A standard deviation equal to zero implies that there is no deviation in the data set.
The same will be reflected by the c.v. provided mean is not equal to zero.
(b) Even when the standard deviation is 1 the c.v. can be meaningfully used for
comparison of variability provided mean is not equal to zero.
(c) Hence it cannot be meaningfully used for comparison of variability when mean of
one or more data sets is zero.
(d) When the mean is equal to 1, the c.v. can be meaningfully used for comparison of
variability.
(e) When the mean and standard deviation are equal for one or more sets of data, the c.v.
can be meaningfully used for comparison of variability.
< TOP >
13. Answer : (a)
Reason : a. The coefficient of determination is the square of coefficient of correlation (r). Hence
it will always be ³ 0 and £ 1.
b. From above we can see that the coefficient of determination cannot be less than zero.
c. The coefficient of determination is will be equal to 1, only if the coefficient of
correlation is equal to –1 or 1; in other cases it will be > 0 and < 1.
d. The coefficient of determination is always positive; The coefficient of correlation
may be negative.
e. Since -1 £ r £ 1, the coefficient of determination which is the square of coefficient
of correlation (r), will always have magnitude less than or equal to r
< TOP >
14. Answer : (e)
Reason : Bienayme -Chebyshev theorem is applicable to any type of distribution regardless of its
shape.
< TOP >
15. Answer : (c)
Reason : a. The Y intercept of the regression line does not represent the true value of Y
when X= 0.
b. The Y intercept of the regression line does not represent the change in average value
of Y per unit change in X.
c. The Y intercept of the regression line represents the mean value of Y when X = 0.
d. The Y intercept of the regression line does not represent the standard deviation of the
values of X.
e. The Y intercept of the regression line does not represent mean of the values of X
< TOP >
16. Answer : (d)
Reason : If the trend of the points in a scatter diagram show a very little spread and show a rising
tendency from bottom left hand corner to top right hand corner then we can conclude that
there is high degree of positive correlation.
Therefore the correct answer is (d).
< TOP >
17. Answer : (c)
Reason : In a multiple regression relationship, the problem of multicollearity arises when there is a
high degree of correlation between the independent variables. In this case the problem of
multicollinearity will arise if there is a high degree of correlation between the variations of
X1 and X2.
< TOP >
18. Answer : (c)
Reason : (a) This is the wrong answer. The weighted average of price relatives using base values
as weights is not same as the unweighted aggregates price index.
(b) This is the wrong answer. The weighted average of price relatives using base values
as weights is not same as the unweighted aggregates quantity index.
(c) This is the right answer. The weighted average of price relatives using base values as
weights is same as the Laspeyres price index.
(d) This is the wrong answer. The weighted average of price relatives using base values
as weights is not same as the Laspeyres quantity index.
(e) This is the wrong answer. The weighted average of price relatives using base values
< TOP >
as weights is not same as the Paasche’s price index.
19. 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).
< TOP >
20. Answer : (b)
Reason : Paasche price index uses the quantity measures for the current period as weights.
Therefore the correct answer is (b)
< TOP >
21. Answer : (c)
Reason : If all the terms of an arithmetic progression are multiplied by a constant the resulting terms
will always form an arithmetic progression with the first term multiplied by the constant as
well as the common difference multiplied by the constant. The resulting series will neither
be in a geometric series or a harmonic series because nature of the resulting terms will not
satisfy their requirements.
< TOP >
22. Answer : (e)
Reason : a. Fisher’s ideal price index considers base year prices.
b. Fisher’s ideal price index considers base year quantities.
c. Fisher’s ideal price index considers current year prices.
d. Fisher’s ideal price index considers current year prices.
e. Fisher’s ideal price index is the geometric mean of the Laspeyres and Paasche’s price
indices
< TOP >
23. Answer : (b)
Reason : (a) & (b) For two dependent events A and B, the joint probability of the events A and B is
not equal to the product of their marginal probabilities.
(c) For two dependent events A and B, the joint probability of the events A and B is not
equal to the sum of their marginal probabilities.
(d) For two dependent events A and B, the joint probability of the events A and B is not
equal to the difference between their marginal probabilities.
(e) For two dependent events A and B, the joint probability of the events A and B is not
always equal to 1.
< TOP >
24. Answer : (d)
Reason : If two events A and B are independent then, the conditional probability of event A given
event B is equal to marginal probability of event A because the occurrence of event B does
not influence the occurrence of event A.
< TOP >
25. Answer : (e)
Reason : The events for which the Baye’s theorem may be applied for computing posterior
probabilities must be mutually exclusive and collectively exhaustive.
< TOP >
26. Answer : (c)
Reason : (a) The classical approach to probability assumes that the outcomes are equally likely.
(b) In the relative frequency approach to probability the probability of an event is
determined after performing the experiment large number times.
(c) In the classical approach to probability the probability of an event is determined
before performing the experiment.
< TOP >
(d) The classical approach to probability assumes that all possible outcomes of the
experiment are known.
(e) The classical approach can be used to find out the probability of mutually exclusive
events.
27. Answer : (b)
Reason : In the coefficient of correlation between two variables lies between –1 and 0 then the
covariance between them is negative.
< TOP >
28. Answer : (a)
Reason : The probability of the occurrence of an event is expressed as a number which lies between
0 and 1.
< TOP >
29. Answer : (a)
Reason : If P(B) > P(A) then – P(A) > – P(B) i.e. 1 – P(A) > 1– P(B) i.e. P(A) > P(B) Hence,
the answer.
< TOP >
30. Answer : (e)
Reason : The standard deviation represents the scatter of the values in a data set.
Arithmetic mean, geometric mean, harmonic mean and median are measures of central
tendency.
< TOP >
31. Answer : (d)
Reason : In a hypergeometric distribution the composition of the population changes from trial to
trial.
< TOP >
32. 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.
< TOP >
33. Answer : (e)
Reason : In a binomial distribution the probability of getting zero or more number of successes
= P(x = 0) + P(x = 1) + … P(x = n) = 1.00.
< TOP >
34. Answer : (d)
Reason : a. The sampling distribution of mean is not a distribution of means of individual
populations.
b. The sampling distribution of mean is not a distribution of observations within a
population
c. The sampling distribution of mean is not a distribution of observations within a
sample.
d. The sampling distribution of mean is a distribution of means of all possible samples
of a specific size taken from a population.
e. The sampling distribution of mean is not a distribution of means of samples of a
specific size taken from different population
< TOP >
35. Answer : (b)
Reason : a. As the sample size increases the variation of the sample mean from the population
mean becomes smaller.
< TOP >
b. From above we can see that (b) is correct.
c. It can not be said with certainty that as the sample size increases the variance of the
sample becomes less than the variance of the population.
d. It can not be said with certainty that as the sample size increases the standard
deviation of the sample becomes less than the standard deviation of the population.
e. It can not be said with certainty that as the sample size increases the standard
deviation of the sample comes close to zero.
36. Answer : (c)
Reason : The standard error of estimate is a measure of the variability of the observed values of the
dependent variable around the regression line.
< TOP >
37. Answer : (b)
Reason : In the stratified method of sampling the population is divided into groups such that
elements in each group are homogeneous and the groups vary from each other.
< TOP >
38. Answer : (a)
Reason : Since the null hypothesis is true and it has been accepted there is no instance of either type
I or type II error.
< TOP >
39. Answer : (b)
Reason : (a) At the optimal solution the slack variables need not be equal to zero.
(b) The constraints which contain the £ sign are converted into equations by adding slack
variables.
(c) There can be more than one feasible solution to a LPP.
(d) The slack variables make zero contribution towards profit.
(e) The slack variables can only assume any non-negative value.
< TOP >
40. Answer : (d)
Reason : In the graphical method of solving linear programming problems if there is a unique
optimal solution, then the optimal solution is located at one of the corner points of the
feasible region.
< TOP >
41. Answer : (c)
Reason : Given that
2
2
m
n
S m
S n
=
2
2
( /2)[2 ( 1) ]
( /2)[2 ( 1) ]
2 ( 1)
2 ( 1)
m a m d m
n a n d n
a m d m
a n d n
+ -
\ =
+ -
+ -
\ =
+ -
Simplifying the above relation we get d = 2a
Now
( 1) ( 1)2 2 1
( 1) ( 1)2 2 1
m
n
T a m d a m a m
T a n d a n a n
+ - + - -
= = =
+ - + - -
< TOP >
42. Answer : (e) < TOP >
Reason : S =
a (1 rn )
1 r
-
-
P = a ´ ar ´ ar2 ……….´arn – 1 = an r1 + 2 +…………n – 1
=
n( n 1)
an r 2
-
[1+2+……n-1] =
( )
n 1 ( ) n 1[ ] n ( n 1)
2 1 n 2 1 2 n 2
2 2 2
- - -
éë ´ + - ùû = + - =
\ P2 = a2nrn(n-1)
R = r 2 n 1
1 1 1 1
.......
a a ar ar - + + +
=
1 n
1
1 r
a 1 1
r
éæ ö ù êç ÷ - ú êëè ø úû ´
-
= ( )
n
n
r 1 r
a 1 r r
-
´
-
\
( )
( )
( )
( )
n n
2 n 1
n
S a 1 r a 1 r r a r
R 1 r r 1 r
-
- -
= ´ =
- -
\
( ) n
2n n n 1 2
n
S a r P
R
= - =
\ Sn = P2 Rn
43. Answer : (c)
Reason :
< TOP >
2
2
Let the numbers be a and b.
2ab
HM = ; GM = ab
(a+b)
2ab
HM (a+b) 2 ab 4
=
GM ab ( ) 5
Or, 2a + 2b = 5 ab
a b
Or, 2 2 5
b a
a 2
Let = z So, the equation would be 2z + 5
b z
Or, 2z + 2 = 5z
Or, 2z -5z+ 2 = 0
O
= =
+
+ =
=
a b
2
2
r, 2z -4z -1z+ 2 = 0
Or, 2z(z - 2) -1(z - 2) = 0
Or, (z - 2)(2z - 1) = 0
1
So, z = or 2
2
a 1
So, z 4 or
b 4
= =
44. Answer : (d)
Reason : The mth term of the A.P. is 1/n and the nth term of the A.P is 1/m.
Let the first term of the AP be A and common difference be D
So, A + (m – 1)D = 1/n
And, A + (n – 1)D = 1/m
By subtracting these two equations we get
(m – n)D = 1/n – 1/m = (m – n)/mn
Or, D = 1/mn
So, m+n th term would exceed mth term by nD
So the (m+n)th term is nD + 1/n = 1/n + 1/m = (m+n)/mn
So the corresponding term in H.P. would be mn/(m+n)
< TOP >
45. Answer : (d)
Reason : Given that S8 = 64.
\2a +7d = 16
S19 = 361 \2a + 18d = 38. Solving a =1 and d= 2.
Sn= (n/2)[2´1 + (n-1) 2] = n2 .
< TOP >
46. Answer : (c)
Reason : The give A.P has its first term as, a = 8 years and common difference, d = 1/3 years
Now given that the sum Sn = (n/2)[2´8 + (n –1) (1/3)] = 168
Simplifying the above equation we get n2 + 47n – 1008 = 0
Or, (n – 16) (n + 63) = 0
\n = 16. (we ignore the negative value of n).
< TOP >
47. Answer : (b)
Reason :
m-1 1-n 5
n m n
n+1 m+3 1-2n
m n m
m-1 m+3 5
- +
n n n
n+1 1-n 1-2n
- +
m m m
1
n n
1 m
m
m n m
xyz= × ×
n m n
m
=
n
m m
= =
n n
æ ö
ç ÷
è ø
æ ö
ç ÷
è ø
< TOP >
48. Answer : (d)
Reason :
n (n - 3)
6 3
n
6
(n - 3)
3
n (n - 3)
6 3
c : c = 3 3 : 4
c 33
Or,
c 4
Or, 4 c = 33 c
n! (n-3)! (n-3)!
Or, 4 33 33
6!(n-6)! 3!(n-3-3)! 3!(n-6)!
(n-2)(n-1)n
Or, 4 33
4 5 6
Or, (n-2)(n-1)n=30 33 = 3 10 3 11 = 9 10 11 = (11-2)(1
=
´ ´
´ = ´ = ´
´ =
´ ´
´ ´ ´ ´ ´ ´ 1-1)(11)
So the value of n = 11
< TOP >
49. Answer : (d)
Reason : Five boys can be arranged to seat in 5! ways. Now there are six places remaining in the
row for the girls. So, the girls’ position can be arranged in
6
3P = 120 ways.
Pictorial Presentation:
´ B ´ B ´ B ´ B ´ B ´
B stands for boys position in the row, ´ stands for could-be girls’ position.
< TOP >
50. Answer : (c)
Reason : In the word ‘CORRESPONDENCE’ the total number of letters are 14, C’s are 2, O’s are
2, R’s are 2 E’s are 3 and N’s are 2.
\Total no. of ways =
4
(14)!
(2!) .(3)!
< TOP >
51. Answer : (d)
Reason :
n1 n 1
Cr Cr 1 + +
+ -
=
( )
( )
( )
( ) ( )
( )
( )
( )
( ) ( )
n 1 ! n 1 ! n 1! n 1 !
r! n-r+1! r-1 ! n+1-r+1 ! r! n-r+1! r-1 ! n-r+2 !
+ + + +
+ = +
=
( )
( ) ( )
n 1 ! 1 1
r-1 ! n-r+1 ! r n r 2
+ é ù ê + ú ë - + û
=
( )
( ) ( ) ( )
n 1 ! n r 2 r
r-1 ! n-r+1! r n r 2
+ é - + + ù
´ê ú êë - + úû
=
( )
( ) ( )
( )
( )
n 1! n 2
r-1 ! n-r+1 ! r n r 2
+ +
´
- +
=
( ) ( )
( ) ( ) ( )
n 1! n+2
r 1 ! r n r 1 ! n-r+2
+ ´
éë - ´ ùû éë - + ´ ùû
=
( )
( )
n 2 !
r! n+2-r !
+
=
n 2
Cr +
Now ,
( )
( )
n 2
r
n 2 !
P
n 2 r !
+ +
=
+ -
\
n 2
n 2 r
r
P
C
r!
+
+ =
\
n 2
n1 n 1 r
r r 1
P
C C
r!
+
+ +
+ - =
< TOP >
52. Answer : (c)
Reason : Let us fix a delegate to one particular seat. Now, other 9 delegates can sit in remaining
nine seats in 9! Ways. As two high commissioners are always seated next to each other we
consider the group of these two delegates as one unit so the remaining 8 (One is fixed to a
particular seat.) can occupy other 8 seats in 8! Ways. But these two delegates can
interchange seats among themselves in 2! Ways. So, total number of ways 10 delegates
can be arranged to sit around a round table so that two of them are always next to each
other in 8!2!= 40320 ´ 2 =80640 ways.
Now, the number of ways these particular two delegates never get to sit next to each other
in (9!- 8!2!)=8!(9-2) = 40320 ´ 7 = 282240 ways.
< TOP >
53. Answer : (e)
Reason : (ii) b = c cannot be implied from the given condition.
(iii) b = -c cannot be implied from the given condition.
(iv) b + c = 1 cannot be implied from the given condition.
(iv) b – c = 1 cannot be implied from the given condition.
(v) Let logab = k and logac = m.
Hence b = ak and c = am
logab + logac = 0 implies that k + m = 0.
\k = -m
Þ ak = a-m Þ ak = 1/ am Þ ak. am = 1 Þ b.c = 1 Þ b = 1/c and vice versa.
Hence they are reciprocals.
< TOP >
54. Answer : (d)
Reason : The given expression can be rewritten as,
1
1 2 3 1
log log log ....... log
2 3 4
1 2 3 1
log .......
2 3 4
1
log log
log 1
-
-
+ + + +
-
= ´ ´ ´ ´
= =
=- = -
æ ö
ç ÷
è ø
æ ö
ç ÷
è ø
n n n n
n
n n
n
n
n
n
n
n
n
n
< TOP >
55. Answer : (b)
Reason : Standard deviation of population, s =
f (x )2
f
S -m
S
Class Mid-point Frequency (f) fx x - x ( )2 x - x ( )2 f x- x
10-20 15 5 75 –31 961 4805
20-30 25 15 375 –21 441 6615
30-40 35 17 595 –11 121 2057
40-50 45 25 1125 –1 1 25
50-60 55 18 990 9 81 1458
60-70 65 10 650 19 361 3610
70-80 75 6 450 29 841 5046
80-90 85 4 340 39 1521 6084
100 4600 29700
Mean,
fx 4600
x 46
f 100
S =
=
=
S
< TOP >
Standard Deviation =
29700
100 = 17.23.
56. Answer : (e)
Reason : For a given data set, if each observation is multiplied by a constant then, the standard
deviation of the resulting data set is equal to the product of the standard deviation of the
original data set and the constant. In this case standard deviation of the original data set
= 3
\ Standard deviation of the resulting data set = 2 3
\Variance of the resulting data set = ( )2
2 3
= 4 ´ 3 = 12.
< TOP >
57. Answer : (b)
Reason : Mean = åX / N = 11+ 15+ 23+ 29+ 19+ 22+ 21+ 20+ 15+ 25+ 17/ 11 = 217/ 11 =
19.73.
< TOP >
58. Answer : (e)
Reason : The appropriate average for quantities that change over time is the geometric mean.
\ The average growth rate over the seven year period
= Geometric mean of the quantity ratios – 1
= [1.05´1.08 ´0.90´ 0.96´1.06´ 1.10´1.07]1 / 7 – 1
= 0.0291
i.e. 2.91%.
< TOP >
59. Answer : (c)
Reason : Harmonic mean of a set of ‘n’ quantities = i
n
1
x
å
=
8
1 1 1 1 1 1 1 1
20 15 30 22 12 25 18 16
+ + + + + + +
= 18.313.
< TOP >
60. Answer : (b)
Reason : Let n1 and n2 are the number of male and female employees while 1 x and 2 x are the
average salaries of the male and female employees. Let x be the average salary of the
composite group. It is known that
1 1 2 2
1 2
nx n x
x
n n
+
=
+
Hence, 5000 (n1 + n2) = 5200 ´ n1 + 4200 ´ n2 or, 200 n1 = 800 n2 or, n1
= 4n2
Therefore, the ratio of male and female employees is 80:20.
< TOP >
61. Answer : (d)
Reason : The total number of ways the letters of the word VICTORY can be arranged is n(S) = 7!
The number of ways two vowels always come together is = 2!.6!
Therefore total number of ways when the vowels do not come together is n (E) = 7! – 2!.6!
= 5.6!
Therefore the probability of favorable event =
( ) 5
( ) 7
=
n E
n S
< TOP >
62. Answer : (d)
Reason : The total number of ways the letters of the word MISSISSIPI can be arranged n(S) =
10!
4!4!
The number of ways the letters can be arranged when all S’s come together is n(E) =
7!
4!
The required probability =
( ) 1
( ) 30
=
n E
n S
< TOP >
63. Answer : (d)
Reason : Total number of possible event = 36.
The set of favorable event is E = {x, y}, such that (x + y) £ 5,
E = {(1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (4, 1)}
Thus total number of favorable event = 10.
Therefore required probability = P(E) =
( ) 10 5
( ) 36 18
= =
n E
n S
< TOP >
64. Answer : (d)
Reason : P(A) = P(A and D) + P(A and E) = 0.12 + 0.28 = 0.40
Similarly,
P(B) = 0.075 + 0.225 = 0.30
P(C) = 0.12 + 0.18 = 0.30
P(D) = P(A and D) + P(B and D) + P(C and D) = 0.12 + 0.075 +0.12 = 0.315
Similarly,
P(E) = 0.28 + 0.225 + 0.18 = 0.685
P(D|C) =
P(C and D)
P(C) =
0.12
0.30 = 0.40
P(E|B) =
P(B and E)
P(B) =
0.225
0.30 = 0.75
< TOP >
P(D|A) =
P(A and D)
P(A) =
0.12
0.40 = 0.30
P(D|B) =
P(B and D)
P(B) =
0.075
0.30 = 0.25
P(E|A) =
P(A and E)
P(A) =
0.28
0.40 = 0.70
P(E|C) =
P(C and E)
P(C) =
0.18
0.30 = 0.60.
Fro m above we can see that:
P(D|A)¹ P(D) P(D|B)¹ P(D) P(D|C)¹ P(D)
\ D is dependent on the events A, B and C respectively.
P(E|A)¹ P(E) P(E|B) ¹ P(E) P(E|C)¹ P(E)
\ E is dependent on the events A, B and C respectively.
65. Answer : (d)
Reason : For the standard normal probability distribution, the area to the left of the mean is equal to
0.5.
< TOP >
66. Answer : (a)
Reason : Mean of a binomial distribution = n . p
\The mean of the binomial distribution of the number of successes is
= 25(1–0.60) =10
< TOP >
67. Answer : (e)
Reason : The event 40 £ Y £ 80 occurs when
Y = 40 or 60 or 80
\ P(40 £ Y £ 80 ) = 0.20 + 0.30 + 0.25 = 0.75
Let Z donote the number of observations of Y which satisfy the condition 40 £ Y £ 80.
\ Z follows a binomial distribution with number of trials = 15 and probability of success
= 0.75.
\ P (Z <12) = P (Z £11) = 1–P(Z ³ 12)
= 1– [P(Z=12)+ P(Z = 13) + P(Z=14) + P(Z=15)]
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
15 12 15 12 15 13 15 13
12 13
15 14 15 14 15 15 15 15
14 15
1 [ C 0.75 1 0.75 C 0.75 1 0.75
C 0.75 1 0.75 C 0.75 1 0.75 ]
- -
- -
= - - + - +
- + -
= 1–
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
12 3 13 2
14 15 0
455 0.75 0.25 105 0.75 0.25
15 0.75 0.25 1 0.75 0.25
é + +ù
ê ú
ê + ú ë û
= 1– [0.2252 + 0. 1559 + 0.0668 + 0.0134]
< TOP >
= 1 – 0.4613
= 0.5387.
68. 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 =
15 9
3.20
-
= 1.875
At a = 0.05, the critical values are ±2.064.
The test statistic is less than the right tail critical value. So it falls in the acceptance region.
\ We accept H0. But the true mean is 11. So H0 is false. Hence the test leads to a type II
error.
< TOP >
69. Answer : (c)
Reason : c. A type I error occurs if the null hypothesis is rejected though it is true.
d. A type II error occurs if the null hypothesis is accepted though it is false.
a, b & e are incorrect interpretations of type I error.
< TOP >
70. Answer : (c)
Reason : H0: p = 0.40
H1: p ¹ 0.40
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
-
=
0.40 0.60
360
´
= 0.02582.
\ z =
p 0.40 0.36 0.40
0.02582 0.02582
- = -
= –1.55
At a = 0.05, the critical values are ±1.96.
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 0.40. So H0 is true. Hence the test does not lead to
either type I or type II error.
< TOP >
71. Answer : (a)
Reason :
< TOP >
1 2
1 2
2 2
2 1 2
x x
1 2
x x
256 144 8
n n 64 36
8 2.828
-
-
s = s + s = + =
s = =
72. Answer : (d)
Reason : By Central Limit Theorem for large samples the sample mean is approximately normally
distributed with mean = population mean and standard deviation
=
n
s
E (x) = m = 60
x s = n
s
Now, P ( x<55) = 0.0228
\ P (55 < x < 60) = 0.50 – 0.0228 = 0.4772 = P(k < z < 0)
From the tables we find that k = –2.0
\ z = –2 = x
x- m
s or –2 = x
55- 60
s or x s =
5
-2 = –2.50
\ n
s
= –2.50
or 64
s
= –2.50
or s2 = (–2.50)2 ´ 64 = 400.
< TOP >
73. Answer : (a)
Reason :
Estimated standard error of difference between means =
p
1 2
s 1 1 217.04 1 1 6.138
n n 16 9
+ = æ + ö = çè ÷ø
< TOP >
74. Answer : (c)
Reason : Let the number of advertisement shown be X and the number of cans of cola purchased be
Y.
Then åX = 21, åY= 56, n = 7
X = 3, Y = 8,
å(X - X)(Y -Y) = 52,
å(X - X)2 = 32 and
å(Y - Y)2 = 132
< TOP >
The coefficient of correlation r =
2 2
(X X)(Y Y)
(X X) (Y Y)
å - -
å - å -
=
52
32´132 = 0.8001
75. Answer : (e)
Reason : ˆY = 20 + 0.75X.
At the point X=100, Y=90, ˆY = 20 + 0.75 ´ 100 = 75 + 20 = 95 but Y = 90 \Residual
Y -Yˆ = 90 – 95 = –5
< TOP >
76. Answer : (e)
Reason : Fisher’s ideal price index (F) = Lasperes price index(L)´ Paasches price index(P)
Or 143.03 = 140.5´P
\140.5P = 143.032
P
143.032 145.6
140.5
= =
< TOP >
77. Answer : (d)
Reason : Fisher’s ideal price index = 0 1
1 1
0 0
1 0
P Q
P Q
P Q
P Q
å
´ å
å
å
´ 100
SP1Q0 = (10.25 ´ 5000) + (14.00 ´ 6000) + (28.00 ´ 3000) + (14.50 ´ 3000) + (4.50
´ 500) = 265000
SP1Q1 = (10.25 ´ 4800) + (14.00 ´ 5500) + (28.00 ´ 3500) + (14.50 ´ 2400) +
(4.50 ´ 600) = 261700
SP0Q0 = 205600 and SP0Q1 = 205500
\ Fisher’s ideal price index 205500
261700
205600
265000 ´
´ 100 = 128.12
< TOP >
78. Answer : (c)
Reason : Weighted average of relatives price index
n n
n n
0
1
P Q
100 P Q
P
P
S
ú úû
ù
ê êë
é
÷ ÷ø
ö
ç çè
æ
S ´
Commodity
100
P
P
0
1 ´ Pn Qn
n n
0
1 100 P Q
P
P
÷ ÷ø
ö
ç çè
æ
´
< TOP >
Rice 118.42 10,000 11,84,200
Wheat 130.77 4,050 529618.50
Salt 160 1,300 208,000
Sugar 145 9,600 13,92,000
Pulses 137.78 13250 18,25,585
S = 38,200 S 51,39,403.50
( ) ú
û
ù
êë
é
S ´ n n
0
1 100 P Q
P
P
= 51,39,403.50
S PnQn = 38,200.
\ Weighted average of relatives price index for year is
= 38,200
51,39,403.50
= 134.54.
79. Answer : (c)
Reason : Given : P1 = 2P0, for each commodity
\ Laspeyres price index =
1 0
0 0
PQ
100
P Q
S
´
S =
0 0
0 0
2PQ
100
P Q
S
´
S
=
0 0
0 0
2 P Q
100
P Q
S
´
S = 200.
< TOP >
80. Answer : (d)
Reason : The price relative for item A for 2001 is
20 100
16
æ ö´ çè ÷ø = 125. Hence from above
discussion, we can infer that option (d) is correct.
< TOP >
81. Answer : (e)
Reason :
X,Y
X Y
CoV(X,Y)
r =
s s
or
y
x
Cov(X,Y) 256
16
r. 0.64 625
s= = =
s
\ 2
Y s = 162 = 256
< TOP >
82. Answer : (c)
Reason : Y = 2X + 3 Because there are only two data points, a straight line fits the data perfectly.
For the two points, the slope is m =
(15 5)
(6 1)
-
- = 2.
\
Y 5
2
X 1
-
=
- or Y – 5 = 2X –2 or Y = 3 + 2X
< TOP >
83. Answer : (b) < TOP >
Reason : The percentage of area under the normal distribution curve that falls ± 2s limits from the
mean is Approximately is 95.5%.
84. Answer : (d)
Reason : Regression equation : Yˆ = a + bX
a = Y -bX
Given : b = 2.5 åX =120 åY = 425 n =10
\ a =
425 120
2.5 12.5
10 10
æ ö - æ ö = çè ÷ø çè ÷ø
\ Regression equation is ˆY=12.5 + 2.5X
If X = 10, then ( ) ˆY=12.5+ 2.5´10 = 37.5
< TOP >
85. Answer : (d)
Reason :
(Y Yˆ )2
15
n 2
-
=
-
å
\
(Y Yˆ )2
225
n 2
-
=
-
å
or
2025 n 2
225
= -
or n = 9 + 2 = 11
< TOP >
< TOP OF THE DOCUMENT >