Quantitative Methods (MB151): July 2005

Quantitative Methods (MB151): July 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 is not true about median?
(a) Median is not strongly affected by the extreme values
(b) It can 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 >
3. For a set of ‘n’ numbers one of the averages is ‘A’. If ‘nA’ is equal to the sum of all the ‘n’ numbers,
then A is a
(a) Quartile (b) Geometric Mean (c) Arithmetic Mean
(d) Median (e) Mode.
(1 mark)
< Answer >
4. If every item in the data set is increased by the same quantity, then the standard deviation of the data
set
(a) Remains the same
(b) Increases by the same quantity by which every data item is increased
(c) Decreases by the same quantity by which every data item is increased
(d) Increases by the square root of the same quantity by which every data item is increased
(e) Decreases by the square root of the same quantity by which every data item is increased.
(1 mark)
< Answer >
5. The logarithm of a number
(a) Is always expressed with respect to base 10
(b) Is always expressed with respect to base 1
(c) Is always expressed with respect to base e
(d) Is equal to the base which must be raised to a given exponent in order to get the number
(e) Is equal to the exponent to which a given base must be raised in order to get the number.
(1 mark)
< Answer >
6. 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 >
7. 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 the
(a) Range of the original data set plus the constant quantity
(b) Range of the original data set minus the constant quantity
(c) Range of the original data set
< Answer >
2
(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)
8. 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.
(1 mark)
< Answer >
9. 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 >
10. The slope of the simple linear regression equation (X is the independent variable and Y is the
dependent variable) represents the
(a) Mean value of Y when X = 0
(b) Change in mean value of Y per unit change in X
(c) True value of Y for a fixed value of X
(d) Variance of the values of X
(e) Variance of the values of Y for a fixed value of X.
(1 mark)
< Answer >
11. Which of the following is not correct about standard error of estimate?
(a) It measures the reliability of the estimating equation
(b) It is the measure of variation of the observed values around the regression line
(c) The smaller the value of the standard error of estimate, the farther are the observations from the
regression line
(d) The larger the value of the standard error of estimate, the farther are the observations from the
regression line
(e) If the value of the standard error of estimate is zero, then there is no variation of the observations
from the regression line.
(1 mark)
< Answer >
12. The value of the coefficient of correlation between two variables can take values in the range of
(a) 0 and 1 (b) –1 and 1 (c) –1 and 0 (d) 0 to ∞ (e) –∞ to ∞.
(1 mark)
< Answer >
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
(b). The coefficient of determination is always negative
(c). The coefficient of determination is always zero
(d). The coefficient of determination always has the same sign as the coefficient of correlation
(e). The magnitude of the coefficient of determination is always greater than the coefficient of
correlation.
(1 mark)
< Answer >
14. The logarithm of 1 with respect to any base is equal to
(a) –10 (b) –1 (c) 1 (d) 0 (e) 10.
(1 mark)
< Answer >
15. 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
< Answer >
3
(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)
16. 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 >
17. 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 >
18. Weighted average of relatives price index for various years can be readily compared if
(a) Base year values are used as weights
(b) Current year values are used as weights
(c) Base year prices are used as weights
(d) Current year prices are used as weights
(e) Sum of base year and current year quantities are used as weights.
(1 mark)
< Answer >
19. The value index number measures the
(a) Change in prices of a basket of commodities from one period to another
(b) Change 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?
(a) It does not consider the base year prices
< Answer >
4
(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. Two events are said to be mutually exclusive if
(a) The sum of their probabilities is less than 1.00
(b) The sum of their probabilities is greater than 1.00
(c) They contain every possible outcome of an experiment
(d) They cannot occur at the same time
(e) The sum of their probabilities is equal to zero.
(1 mark)
< Answer >
24. If the probability of occurrence of one event is not affected by the occurrence of another event and vice
versa then the two events are said to be
(a) Collectively exhaustive (b) Independent
(c) Dependent (d) Mutually exclusive
(e) Non-mutually exclusive.
(1 mark)
< Answer >
25. Bayes’ theorem helps the statistician to calculate
(a) Subjective probability (b) Classical probability
(c) Posterior probability (d) Central tendency (e) Dispersion.
(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.
(1 mark)
< Answer >
27. In a binomial distribution the probability of getting zero or more number of successes is equal to
(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)
< Answer >
28. 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 >
29. 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 >
5
30. Which of the following is the graphical plot of the values of the dependent and independent variables,
in the context of regression analysis?
(a) Scatter diagram (b) Frequency polygon
(c) Histogram (d) p chart (e) Ogive.
(1 mark)
< Answer >
31. When sampling without replacement from a finite population such that the probability of success is not
constant from trial to trial, the data follow a
(a) Binomial distribution (b) Uniform distribution (c) Normal distribution
(d) Continuous distribution (e) Hypergeometric distribution.
(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. Which of the following is true about binomial distribution?
(a) It is a continuous distribution
(b) The probability of the outcome of any trial varies over time
(c) Each trial will have more than two possible outcomes
(d) The outcomes of the trials are statistically independent of each other
(e) It is always a symmetrical distribution.
(1 mark)
< Answer >
34. Which of the following best describes the expected value of a discrete random variable?
(a) It is the geometric average of all possible outcomes of the variable
(b) It is the simple average of all possible outcomes of the variable
(c) It is the weighted average of all possible outcomes of the variable
(d) It is the outcome of the variable, which has the highest probability of occurrence
(e) It is the highest probability of occurrence in probability distribution of the random variable.
(1 mark)
< Answer >
35. 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 >
36. Which of the following is false with regard to standard error of mean?
(a) It is less than the standard deviation of the population
(b) It decreases as the sample size increases
(c) It measures the variability of the mean from sample to sample
(d) It is the standard deviation of the sampling distribution of mean
(e) It is the standard deviation of the sample.
(1 mark)
< Answer >
37. Which type of sampling is appropriate when the population consists of well-defined groups such that
the elements within each group are heterogeneous and each group is a representative of the population
as a whole?
(a) Simple random sampling (b) Cluster sampling (c) Stratified sampling
(d) Systematic sampling (e) Judgmental sampling.
< Answer >
6
(1 mark)
38. 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 >
39. 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 >
40. 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 >
41. If the sum of four numbers in arithmetic progression is 32 whereas the sum of their squares is 276. The
lowest of the four numbers would be
(a) 2.5 (b) 3.0 (c) 3.5 (d) 4.0 (e) 5.0.
(2 marks)
< Answer >
The ratio of the sum of first three terms of a geometric progression and the sum of the first six terms of
the same geometric progression is 125:152. The common ratio of the series is
(a) 1
3
(b) 1
5
(c) 3
5
(d) 5
3
(e) 23
.
(1 mark)
< Answer >
43. The product of three numbers in geometric progression is 125 and the sum of their products taken in
pairs is 1
87
2
. The highest of the three numbers is
(a) 5 (b) 7 (c) 10 (d) 12 (e) 15.
(2 marks)
< Answer >
44. The pth term of an harmonic progression is qr and the qth term is rp. The rth term of the harmonic
progression is
(a)
p
q (b) pq (c)
q
p (d)
rp
q (e)
rq
p .
(2 marks)
< Answer >
45. Three numbers a, b, and c are in arithmetic progression. P is the geometric mean between a and b and,
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 >
46. The sum of first 8 and 19 terms in an arithmetic progression are 64 and 361 respectively, the sum of
first 15 terms of the series is
< Answer >
7
first 15 terms of the series is
(a) 110 (b) 144 (c) 196 (d) 225 (e) 289.
(2 marks)
47. The 1st term of an harmonic progression is 5/7 and 4th term is 10/29. The 20th term of the harmonic
progression is
(a) 10
81
(b) 10
109
(c) 21
61
(d) 25
71
(e) 10
89
.
(1 mark)
< Answer >
48.
If
n n
C10 = C15 , the value of
27
Cn is
(a) 165 (b) 190 (c) 220 (d) 351 (e) 2925.
(1 mark)
< Answer >
49. How many different words ending and beginning with a consonant can be made by rearranging letters
of EQUATION? The words may not have any meaning.
(a) 720 (b) 1440 (c) 2160 (d) 4320 (e) 6480.
(1 mark)
< Answer >
50. There are 5 professors and 10 students out of whom a committee of 2 professors and 3 students is to be
formed. The number of ways in which this committee can be formed is
(a) 252 (b) 450 (c) 780 (d) 920 (e)
1200.
(1 mark)
< Answer >
51. A guard of 12 persons is to be formed from a group of 16 soldiers in all possible ways. How many
times three particular soldiers are together on guard?
(a) 286 (b) 343 (c) 468 (d) 576 (e) 715.
(1 mark)
< Answer >
52. Three men have 4 coats, 5 waistcoats, and 6 caps. In how many ways can they wear them?
(a) 2880 (b) 7200 (c) 124400 (d) 172800 (e) 224000.
(1 mark)
< Answer >
53.
If (lbog- ca) = (lcog- ab) = (lao-gb c);the value of aabbcc would be
(a) a (b) b (c) c (d) 1 (e) 0.
(1 mark)
< Answer >
54. If log3 (2x – 1) = 4, the value of x would be
(a) 6.5 (b) 41 (c) 32.5 (d) 2.5 (e) 22.5.
(1 mark)
< Answer >
55. The following distribution shows the distribution of wages of 500 workers in a factory:
Weekly wages (in Rs.) Number of workers
Below 300 60
300 - 350 110
350 - 400 160
400 - 450 90
450 - 500 46
500 - 550 24
550 - 600 10
< Answer >
8
550 - 600 10
What is the median wage earned by the workers in the factory? (Select the nearest figure)
(a) Rs.325.70 (b) Rs.350.25 (c) Rs.374.84 (d) Rs.395.30 (e) Rs.412.50.
(2 marks)
56. If 5 is subtracted from every item in a data set then the coefficient of variation of the resulting data set
is 10%. If 5 is added to every item of the same data set then the coefficient of variation of the resulting
data set is 6%. The coefficient of variation of the original data set is
(a) 6.0% (b) 7.5% (c) 10.0% (d) 12.5% (e) 15.6%.
(2 marks)
< Answer >
57. A group of salesmen from the same industry consists of some sales men who have 5 years of
experience and others who have 10 years of experience. Sixty percent of the salesmen in the group
have 5 years of experience and their average salary is Rs.7,000 per month. The average salary for the
entire group is Rs.9,000.
What is the average salary of the salesmen who have 10 years of experience?
(a) Rs.4,200 (b) Rs.4,800 (c) Rs.8,000 (d) Rs.10,000 (e) Rs.12,000.
(1 mark)
< Answer >
58. The geometric mean of a set of five numbers is 3. If each of the five numbers is multiplied by 2, then
the geometric mean of the resulting values will be
(a) 2 (b) 3 (c) 2/3 (d) 3/2 (e) 6.
(1 mark)
< Answer >
59. The following distribution shows the ages of 100 persons in a group:
Age group (in years) Number of persons
20 - 25 3
25 - 30 16
30 - 35 22
35 - 40 18
40 - 45 14
45 - 50 10
50 - 55 7
55 - 60 6
60 - 65 4
What is the average age of the persons in the group?
(a) 25.5 years (b) 28.4 years (c) 32.6 years (d) 39.3 years (e) 45.2 years.
(1 mark)
< Answer >
60. A group consists of 150 children. The group is divided into three subgroups viz., A, B and C, in the
ratio of 2:5:3 respectively. The average age of the children in the subgroup A is 8 years. The average
age of the children in the subgroup B is 10 years. The average age of the children in the subgroup C is
12 years. What is the average age of all the 150 children in the group?
(a) 8.5 years (b) 9.3 years (c) 10.2 years (d) 10.9 years (e) 11.5
years.
(1 mark)
< Answer >
61. Two fair dice are thrown. What is the probability that one of them gives an even number less than 5,
and the other one gives an odd number less than 4?
(a) 1
9
(b) 2
9
(c) 1
3
(d) 4
9
(e)5
9
.
(1 mark)
< Answer >
9
62. Three balls are drawn successively from a box containing 6 red balls, 4 white balls and 5 blue balls.
What is the probability that they are drawn in the order red, white and blue if each of the ball drawn is
not replaced?
(a) 2/91 (b) 3/91 (c) 4/91 (d) 4/225 (e) 8/225.
(1 mark)
< Answer >
63. Plant A of a company employs 5 production and 3 maintenance engineers while plant B employs 4
production and 5 maintenance engineers. A plant is chosen randomly and two engineers are selected
randomly from that plant. What is the probability that one is a production engineer and the other is a
maintenance engineer?
(a) 0.35 (b) 0.40 (c) 0.45 (d) 0.50 (e) 0.55.
(2 marks)
< Answer >
64. Five students viz, P, Q, R, S and T are independently trying to solve a problem in mathematics. The
probabilities that they will be able to solve the problem are 2
5
, 2
3
, 1
6
, 1
2
and 3
8
respectively.
What is the probability that the problem will be solved?
(a) 5.2% (b) 0.83% (c) 25.5% (d) 56.7% (e) 94.8%.
(1 mark)
< Answer >
65. The wind speed at a wind energy station is approximately normally distributed with a mean of 25 miles
per hour and a standard deviation of 7 miles per hour. When wind speeds exceed 45 miles per hour, the
station produces too much electricity for the current transmission lines and must be shut down. What
percentage of the time will the station be shut down?
(a) 0.21% (b) 2.1% (c) 9.79% (d) 28.6% (e) 50%.
(1 mark)
< Answer >
66. A biased coin has the probability of giving a head when tossed, equal to 0.60. What is the probability
of getting exactly three heads in 4 tosses?
(a) 0.2546 (b) 0.3456 (c) 0.4654 (d) 0.5346 (e) 0.6354.
(1 mark)
< Answer >
67. If a fair die is thrown twenty times then, what is the probability of obtaining a minimum of 3 in a
maximum of 17 throws?
(a) 0.01759 (b) 0.01428 (c) 0.98572 (d) 0.98241 (e) 0.9997.
(2 marks)
< Answer >
68. The following details are available with regard to a hypothesis test on population mean:
H0: μ=9
H1: μ ≠ 9
n = 25
s2 = 256
x = 2.25
Significance level = 0.05
The population is normally distributed. It is later known that the true population mean is 9.
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 >
69. The following details are available with regard to a test of hypothesis for the population mean: < Answer >
10
H0: 60
x 72
60
μ =
=
σ =
Test statistic = 1.20
What is the sample size?
(a) 6 (b) 10 (c) 12 (d) 36 (e) 72.
(1 mark)
70. The following details are available with regard to a hypothesis test on population mean:
H0: μ=10
H1: μ>10
n = 64
σ2 = 256
x =13.50
Significance level = 0.05
It is later known that the true population mean is 12. Which of the following can be said with regard to
the test?
(a) There is insufficient information for doing the test
(b) The t 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 >
71. The following details are available with regard to a hypothesis test on a population mean:
0
1
2
H : 20
H : 20
81
n 36
μ =
μ ≠
σ =
=
The null hypothesis is rejected if x ≤ 17.06or x ≥ 22.94 . What is the probability of committing a
type I error?
(a) 0.5 (b) 0.05 (c) 0.025 (d) 0.005 (e)
1.00.
(2 marks)
< Answer >
72. The following information are available with regard to a sampling distribution of mean:
Probability that the sample mean is more than 51 = 0.1587
Population mean = 50
Population variance = 36
It is assumed that the Central Limit Theorem will be applicable.
With what sample size is the sampling distribution of mean associated?
(a) 30 (b) 36 (c) 18 (d) 54 (e) 60.
(2 marks)
< Answer >
73. The following details are available with regard to a regression analysis (Y is the dependent variable)
with 5 observations:
Y 18 24 20 25 32
ˆY
20 22 18 26 30
What is the standard error of estimate?
(a) 2.38 (b) 7.67 (c) 17 (d) 39.5 (e) 289.
(1 mark)
< Answer >
11
74. The following details are available with regard to a simple regression relationship between variables X
and Y in which Y is the dependent variable:
( )2 Σ Yˆ −Y = 8750
( )2 Σ Y−Yˆ = 2250
What is the coefficient of determination?
(a) 0.2571 (b) 0.2045 (c) 0.7955 (d) 3.89 (e) 1.00.
(1 mark)
< Answer >
75. The following details are available with regard to the variables X and Y:
Σ(X−X)(Y−Y) = 1200
( )2 Σ X−X = 2500
( )2 Σ Y−Y = 900
If a regression relationship is derived using X as the independent variable then what will be the slope of
the regression line?
(a) 0.48 (b) 2.083 (c) 2.78 (d) 1.33 (e) 1.67.
(1 mark)
< Answer >
76. The average price of a group of five commodities in the current year is Rs.75 and the unweighted
aggregates price index for the group for the current year is 125. What is the average price of the
commodities in the base year?
(a) Rs.30 (b) Rs.40 (c) Rs.50 (d) Rs.60 (e) Rs.80.
(1 mark)
< Answer >
77. The following data are available with regard to a basket of goods:
Good P0Q0 P1Q0
A 160 320
B 500 600
C 560 700
D 380 380
What is Laspeyres price index for the basket of goods?
(a) 100 (b) 125 (c) 80 (d) 120 (e) 150.
(1 mark)
< Answer >
78. The following details are available with regard to a group of goods:
Good P1Q0 P0Q0 1
0
Q
Q
A 640 320 3/4
B 1200 1000 1/2
C 1400 1120 5/7
D 760 760 13/19
What is Paasches price index for the group of goods?
(a) 65 (b) 153.85 (c) 126.21 (d) 81.25 (e) 123.08.
(1 mark)
< Answer >
12
79. The following data pertains to the consumption of materials by a bakery.
Inputs Units Prices (in Rs.) Quantities Used 1993 1996 1993 1996
Flour Kilogram 15 30 500 700
Eggs Dozen 8 14 100 70
Milk Litres 6 17 200 120
Sugar Kilogram 10 16 50 70
The unweighted average of relatives price index for the year 1996 considering the year 1993 as the
base year, is
(a) 123.41 (b) 199.22 (c) 201.44 (d) 204.58 (e) 307.31.
(1 mark)
< Answer >
80. The following information are available with regard to a basket of goods:
P1Q1 0.8125 P Q 0 0
Σ
Σ =
P1Q0 1.9417
P0Q1
Σ
=
Σ
What is Fisher’s ideal price index for the basket of goods?
(a) 81.25 (b) 125.6 (c) 194.17 (d) 238.98 (e) 157.76.
(1 mark)
< Answer >
81. Calculate the coefficient of correlation between variables X and Y, from the following data:
X 9 8 7 6 5 4 3 2 1
Y 15 16 14 13 11 12 10 8 9
(a) 0.91 (b) 0.93 (c) 0.95 (d) 0.97 (e) 0.99.
(2 marks)
< Answer >
82. The following table shows the ages and blood pressure of 8 persons.
Age (in years) 52 63 45 36 72 65 47 25
Blood pressure 62 53 51 25 79 43 60 33
What would be the expected blood pressure of a person who is 49 years old? Find with the help of an
appropriate regression analysis.
(a) 47.289 (b) 48.301 (c) 49.502 (d) 50.612 (e) 51.625.
(2 marks)
< Answer >
83 In a normal distribution, 30.85% of the items are under 45 and 8.08% are over 64. Find the standard
deviation of the distribution.
(a) 8 (b) 9 (c) 10 (d) 11 (e) 12.
(1 mark)
< Answer >
84. The estimated regression relationship between variables X (X is the independent variable) and Y
is: Yˆ =6+3X
Let M = Y + 2
What will be the estimated regression relationship between M and X (X is the independent variable)?
(a) ˆM = 6 + 2X (b) ˆM = 8 + 3X (c) ˆM = 6 + 3X (d) ˆM = 8 + X (e) ˆM = 6 + X.
(1 mark)
< Answer >
85. The coefficient of correlation between variables X and Y is 0.70. The regression sum of squares is
161.21 and the number of observations is 10. What is the standard error of estimate for the simple
regression relationship between X (independent variable) and Y (dependent variable)?
< Answer >
13
(a) 4.096 (b) 16.779 (c) 20.974 (d) 4.58 (e) 4.32.
(1 mark)
14
Suggested Answers
Quantitative Methods (MB151): July 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 : (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 >
3. Answer : (c)
Reason : A = n
Sumof all items
∴ Sum of all items = nA.
< TOP >
4. Answer : (a)
Reason : Standard deviation is independent of change of origin i.e., it remains unchanged
even if all the items in the data set are increased or decreased by the same quantity.
< TOP >
5. Answer : (e)
Reason : The logarithm of a number is equal to the exponent to which a given base must be
raised in order to get the number. The logarithm of a number is not expressed with
respect to base zero or one. The logarithm of a number may be expressed with
respect to base 10 or other positive real number not equal to 1.
< TOP >
6. 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 the observations.
< TOP >
7. 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 >
8. 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 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.
< TOP >
9. 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 >
10. Answer : (b)
Reason : a. The slope of the simple regression equation does not represent the mean value
of Y when X = 0.
b. The slope of the simple regression equation represents the change in average
value of Y per unit change in X.
c. The slope of the simple regression equation does not represent the true value of
< TOP >
15
Y for a fixed value of X.
d. The slope of the simple regression equation does not represent the variance of
the values of X.
e. The slope of the simple regression equation does not represent variance of the
values of Y for a fixed value of X.
11. Answer : (c)
Reason : The smaller the value of the standard error of estimate, the closer are the
observations to the regression line. All other statement are true for the standard error
of estimate.
Therefore the correct answer is (c).
< TOP >
12. Answer : (b)
Reason : The coefficient of correlation can take values in the range of –1 to 1. Therefore the
correct answer is (b).
< TOP >
13. Answer : (a)
Reason : The coefficient of determination is the square of the coefficient of regression.
Therefore it takes only positive values.
Therefore the correct answer is (a)
< TOP >
14. Answer : (d)
Reason : (d) The logarithm of 1 with respect to any base is equal to 0 , because any quantity
raised to the exponent of 0 is equal to 1.
(a), (b), (c) and (e) are incorrect conclusions because raising any quantity to these
exponents will not give 1.
< TOP >
15. 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 >
16. 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 >
17. Answer : (c)
Reason : Any multiple regression equation consists of only one dependent variable and more
than one independent variables.
< TOP >
18. Answer : (a) < TOP >
16
Reason : The weighted average of relatives price index with the base year values used as
weights, can be readily compared because the weights in the base year and current
year remain the same (and for every year the same weights are used).
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 : (d)
Reason : If two events cannot occur at the same time then they are called mutually exclusive
events.
< TOP >
24. Answer : (b)
Reason : If A and B are two events such that the occurrence of event A does not influence the
occurrence of event B and, the occurrence of event B does not influence the
occurrence of event A then A and B are said to be independent events.
< TOP >
25. Answer : (c)
Reason : Baye’s theorem helps the statistician to calculate posterior probability.
< 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.
(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.
< TOP >
27. Answer : (e)
Reason : In a binomial distribution the probability of getting zero or more number of
< TOP >
17
= P(x = 0) + P(x = 1) + … P(x = n) = 1.00.
28. 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 >
29. 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 >
30. Answer : (a)
Reason : a. The graphical plot of the values of the dependent and independent variables, in
the context of regression analysis, is called scatter diagram.
b. A frequency polygon is a graphical representation of a frequency distribution
which uses straight lines to join the top mid points of the rectangles in a
histogram.
c. A histogram is a graphical representation of a frequency distribution.
d. A p chart is a quality control chart.
e. An ogive is a graphical plot of a cumulative frequency distribution
< TOP >
31. Answer : (e)
Reason : a. The data may follow a binomial distribution if the probability of success is
constant from trial to trial.
b. The data may follow a uniform distribution if the probabilities of the outcomes
of the trial are equal.
c. The data may follow a normal distribution if the probability of success is
constant from trial to trial.
d. The data may follow a continuous distribution if the outcome is any value
within a given range of values.
e. When sampling is done without replacement from an finite population such
that the probability of success is not constant from trial to trial, the data follow
a hypergeometric distribution.
< 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 : (d)
Reason : The outcome of each trial in a binomial distribution is independent of each other.
All other statement are false for a binomial distribution.
Therefore the correct answer is (d).
< TOP >
34. Answer : (c)
Reason : a. The expected value of a discrete random variable is not a geometric average of
the outcomes of the variable.
b. The expected value of a discrete random variable is not a simple average of the
outcomes of the variable.
c. The expected value of a discrete random variable is a weighted average of the
outcomes of the variable.
d. The expected value of a discrete random variable is not the outcome, which
has the highest frequency.
e. The expected value of a discrete random variable is not the highest probability
< TOP >
18
of occurrence in the distribution of the random variable
35. 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 >
36. Answer : (e)
Reason : a. The standard error of mean is less than the population standard deviation(σ)
because it is equal to σ/√n.
b. From above we can see that it will decrease as the sample size increases.
c. The standard error is a measure of the variability of the mean across various
samples of the same size taken from the population.
d. It is the standard deviation of the distribution of means of all possible samples
of a specific size that can be taken from the population.
e. From above we can see that the standard error of mean is not the standard
deviation of the sample.
< TOP >
37. Answer : (b)
Reason : a. Simple random sampling may not be appropriate when the population is
known to consist of well defined groups such that the elements within each
group are heterogeneous and each group is a representative of the population
as a whole, because even if the sample is random it may not reflect the nature
of the population.
b. When the population is known to consist of well-defined groups such that the
elements within each group are heterogeneous and each group is a
representative of the population as a whole, the cluster sampling is appropriate.
c. When the population is known to consist of well-defined groups such that the
elements within each group are homogeneous and the groups vary from each
other significantly, the stratified sampling is appropriate.
d. When the population is known to consist of well defined groups such that the
elements within each group are heterogeneous and each group is a
representative of the population as a whole, because even if the sample is
random it may not reflect the true nature of the population.
e. When the population is known to consist of well defined groups such that the
elements within each group are heterogeneous and each group is a
representative of the population as a whole, judgmental sampling may not be
appropriate because the representativeness of the sample depends upon the
knowledge and judgment of the decision maker
< TOP >
38. 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 >
39. 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 >
40. Answer : (c)
Reason : The feasible region is the set off all points which satisfy all the constraints in the
LPP.
< TOP >
19
41. Answer : (e)
Reason : Lets the numbers in the A.P be (m-3d), (m-d), (m+d), and (m+3d)
So, (m -3d) + (m - d) + (m + d) + (m + 3d) = 32
Or, 4m = 32
Or, m = 8
2 2 2 2
2 2 2 2 2 2 2
2
2 2
2
2
2
(m-2d) +(m-d) + (m+d) +(m+2d) =276
Or,m -6md+9d +m -2md+d +m +2md+d +m
+6md+9d =276
Or, 4m + 20 d = 276
Puttingm=8we get
4×64+20d = 276
Or, 20d =276 -256=20
Or, d = 1
Or, d = ±1
So the lowest number would be8-3×1= 5
< TOP >
42. Answer : (c)
Reason :
3
3
3
3
3
3 3
3
6 6 3 3
8
1 125
,
( 1) 152
, (125 125) 152
, 125 27
27
,
125
3
,
5
( 1)
( 1) ( 1) ( 1) 125
( 1) ( 1) ( 1)( 1) 152
( 1)
=
+
+ =
=
=
=
−
− − −
= = = =
− − + −
−
Or
r
Or r
Or r
Or r
Or r
a r
S r ar r
S a r a r r r
r
< TOP >
43. Answer : (c)
Reason :
< TOP >
20
3
2
2
m
Let the numbers be , m, mr.
r
The product of the numbers is m 125; or, m = 5.
The sum of the products in pair is
m m 175
.m + m.mr+ mr. =
r r 2
1 175
25 r 1
r 2
Or,
Or,
Or,
2r + 2r + 2 = 7r
2r - 5r + 2 = 0
=
+ + =  
 
 
2
Or, 2r(r - 2) -1(r - 2) = 0
Or, (r - 2)(2r - 1) = 0
1
So, r = 2 or,
2
So the highest number is 5 2 = 10
2r - 4r - r + 2 = 0
×
44. Answer : (b)
Reason : The pth term of the A.P. is
1
qr i.e. A + (p – 1)d =
1
qr
The qth term of the A.P. is
1
pr i.e. A + (q – 1)d =
1
pr
By subtracting the two equations we get
(p – q)d =
1
qr –
1
pr =
(p - q)
rpq
Or, d =
1
rpq
The rth term would exceed the pth term by (r – p)d. i.e.
(r - p)
rpq
So the value of rth term is
1 (r - p) 1 1 1 1
+ = =
rq rpq rq pq rq pq
+ +
So the rth term of the H.P. is pq.
< TOP >
45. Answer : (b)
Reason : a, b and c are in A.P.
∴ b – a = c – b ⇒ 2b = a + c
p is the G.M. between a and b ⇒ p = ab ⇒ p2 = ab
q is the G.M. between b and c ⇒ q = bc ⇒ q2 = bc
∴ p2 + q2 = ab + bc
or p2 + q2 = b(a + c)
or p2 + q2 = b(2b)
or p2 + q2 = 2b2
< TOP >
21
or b2 =
p2 q2
2
+
∴ b2 is the A.M. between p2 and q2.
46. Answer : (d)
Reason :
( )
( )
( )
8
19
S = 2a+7d 4=64
Or, 2 a +7 d = 16 ..................(i)
19
S = 2a+18d =361
2
Or, 2 a +18d = 38 .................(ii)
By (ii) - (i) weget
11d = 38-16 = 22
Or, d = 2
So, the common difference is 2.
Putting the value of d in equation (i)we get
2a +7× 2 4=64
O
( )
r, a =1
So, thesum of first15 terms is
15 30×15
2 +14× 2 = = 225
2 2
< TOP >
47. Answer : (b)
Reason : The first term of the A.P is 7/5 and 4th term of the A.P. is 29/10
If the common difference is d then
29 7 (29 - 14) 3
3d = = =
10 5 10 2
1
So the value of d =
2
−
So the 20th term in the A.P. is
7 1
+ (20 1)
5 2
7 19 (14 95) 109
+
5 2 10 10
× −
+
= =
So the corresponding term in H.P would be
10
109
< TOP >
48. Answer : (d)
Reason :
n n
r (n-r)
n n
10 15
27 27
n 25
We know c = c
So, when c = c That indicates if r = 10 ; (n-r) = 15
i.e. (n - 10) = 15
n = 10+ 15 = 25
27 26
So, c = c = 351
1 2
∴
×
=
×
< TOP >
22
49. Answer : (d)
Reason : The word EQUATION has total 8 letters: of which 5 vowels and 3 consonants. Two
consonants for beginning and ending are to be selected from 3 consonants of the
given word in
3
2 P ways. Remaining six positions are to be filled with six letters in
6! ways. So, total number of word is
3
P2 .6!= 2 × 3 × 720 = 4320 .
< TOP >
50. Answer : (e)
Reason : 2 professors can be selected from 5 professors in
5
2
5 4
C = = 10
1 2
×
× ways. On the
other hand, 3 students can be selected from 10 students in
10
3
10 9 8
C = = 120
1 2 3
× ×
× ×
ways. So, the 120 ×10 =1200 ways this committee can be formed in the given
condition.
< TOP >
51. Answer : (e)
Reason : The size of the guard group is 12. When three particular persons are together, the
remaining 9 soldiers are to be selected from (16 – 3) or 13 soldiers in
13
9
10 11 12 13
C
1 2 3 4
× × ×
=
× × × or 715 times.
< TOP >
52. Answer : (d)
Reason :
4 5 6
3 3 3
24 60 120
172800
P P P
= × ×
=
× ×
< TOP >
53. Answer : (d)
Reason :
a b c
a b c
log a log b log c
Let k
(b-c) (c-a) (a-b)
log a = k(b-c); log b = k(c-a); log c = k(a-b)
log a b c alog a + blog b + clog c = ak(b-c) + bk(c-a) + ck(a-b) = 0
So, a b c 1
= = =
∴
∴ =
=
< TOP >
54. Answer : (b)
Reason : log3 (2x – 1) = 4 implies 2x-1 = 34
2x = 81 + 1 = 82
x = 41.
< TOP >
55. Answer : (c)
Reason : Median =
m
m
L (N 1) / 2 (F 1) W
f
 + − + 
+ 
 
The position of the median is the
N 1th
2
+
item i.e.,
500 1
2
+
= 250.5th item.
Class
Frequency
(f)
Cumulative Frequency
(F)
Less than 300 60 60
300-350 110 170
350-400 160 330
400-450 90 420
< TOP >
23
450-500 46 466
500-550 24 490
550-600 10 500
∴ Median class is 350-400.
Lm = 350 N = 500 F = 170 fm = 160 W = 50
∴ Median = 350 +
501/ 2 (170 1) 50
160
 − +   ×  
= Rs.374.84.
56. Answer : (b)
Reason : Let the average and standard deviation of the original data let be x and s.
Average of all items ‘x – 5’ = n
Σ(x − 5)
= n
5
n
x Σ
−
Σ
= n
x − n5
= x – 5.
Standard deviation of all items ‘x – 5’ = s
(This is because the value of standard deviation remains the same if each
observation in a series is increased or decreased by the same quantity).
Given:
100
x 5
s ×
− = 10
or 100s = 10 x – 50
or 10 x – 100s = 50 …… ……… (A)
Average of all items ‘x + 5’ = n
Σ(x + 5)
= n
5
n
x Σ
+
Σ
= n
x + n5
= x +5
Standard deviation of all items ‘x + 5’ = s
(This is because the value of standard deviation remains the same if each
observation in a series is increased or decreased by the same quantity).
Given :
100
x 5
s ×
+ = 6
or 100s = 6 x + 30
or 6 x – 100s = –30 ……………….. (B)
Subtracting equation (B) from equation (A) we get:
(10 x – 100s) – (6 x – 100s) = 50 – (–30)
or 4 x = 80
or x = 4
80
= 20
∴ x = 20.
Putting the value of x in equation (A) we get :
10 (20) – 100s = 50
or 200 – 100s = 50
or 100s = 150
or s = 100
150
= 1.50
∴s = 1.50.
Coefficient of variation of the original data set = x
s
× 100
=
100
20
1.50 ×
= 7.5%.
< TOP >
24
57. Answer : (e)
Reason : Let the following notations be used:
N1 : Number of sales men with 5 years experience
N2 : Number of sales men with 10 years experience
∴X2 :Average salary of sales men with 10 years experience.
By the question –
1 2 2
1 2
N 7000 N X
9000
N N
× +
=
+
or
1 2
2
1 2 1 2
N N
(7000) X 9000
N N N N
+ =
+ +
or (0.6 × 7000) + (1– 0.6) X2 = 9000
or 0.4 X2 = 9000 – 4200 = 4800
or 2
X 4800 Rs.12000.
0.4
= =
< TOP >
58. Answer : (e)
Reason : Geometric mean = (Product of the numbers)1/n
Where n = Number of numbers
∴ (Geometric mean)n = Product of the numbers (before multiplication by 2)
= 35
Each number is multiplied by 2.
∴ Product of the resulting values = 25 × 35
∴ Geometric mean of the resulting values = (25 × 35)1/5 = 2 × 3 = 6.
< TOP >
59. Answer : (d)
Reason :
Class
Mid-value
(m)
Frequency
(f)
f × m
20-25 22.5 3 67.5
25-30 27.5 16 440
30-35 32.5 22 715
35-40 37.5 18 675
40-45 42.5 14 595
45-50 47.5 10 475
50-55 52.5 7 367.5
55-60 57.5 6 345
60-65 62.5 4 250
Σf = 100 Σf × m = 3930
∴ Mean =
f m 3930
f 100
Σ ×
=
Σ = 39.3 years
< TOP >
60. Answer : (c)
Reason : Subgroup A : NA =
2 150
2 5 3
×
+ + = 30
A x = 8 years
< TOP >
25
Subgroup B : NB =
5 150
2 5 3
×
+ + = 75
B x = 10 years
Subgroup C : NC =
3 150
2 5 3
×
+ + = 45
C x = 12 years
∴ Average age of the entire group =
( ) ( ) ( ) A A B B C C
A B C
N .x N .x N .x
N N N
+ +
+ +
=
(30 8) (75 10) (45 12)
30 75 45
× + × + ×
+ + =
1530
150 = 10.20 years.
61. Answer : (b)
Reason : Probability that the first dice gives an even number less than 5 and the second dice
gives an odd number less then 4 =
2 2
6 6
×
=
1
9
Probability that the second dice gives an even number less than 5 and the first dice
gives an odd number less than 4 =
2 2
6 6
×
=
1
9
Since these events are mutually exclusive, the probability that one of the dice gives
an even number less than 5 and the other one gives an odd number less than 4 =
1
9 +
1
9 =
2
9
< TOP >
62. Answer : (c)
Reason : If each ball is not replaced, then R, W, and B are dependent events and
P{RWB} = P{R} P{W I R} P{B I WR} = (6/15)(4/14)(5/13) = 4/91.
< TOP >
63. Answer : (e)
Reason : Let us denote the events as follows:
A1: Plant I is being selected while A2: Plant 2 is being selected
B: Selection of two persons where one is a production engineer and another is a
maintenance engineer.
Now, the required probability of selecting one production engineer and one
maintenance engineer is possible if either of the following two mutually events
happens:
(i) A1∩B happens, (ii) A2∩B happens
Now, from the given situation, P(A1) = P(A2) = 0.5
P(B/A1) = Probability of selecting one production engineer and one maintenance
engineer in a selection of two engineers from the first plant.
Therefore, P(B/A1) =
5 3
1 1
8
2
5 3 2 15
8 7 28
× × ×
= =
×
C C
C .
Similarly, P(B/A2) =
4 5
1 1
9
2
5 4 2 5
8 9 9
× × ×
= =
×
C C
C .
Therefore, the required probability will be P(B) = P(A1) × P(B/A1) + P(A2) ×
P(B/A2) =
1 15 1 5
2 28 2 9
× + ×
=
275
504 = 0.5456 ≅ 0.55 (approx.).
< TOP >
64. Answer : (e)
Reason : Let the probability that the students P, Q, R, S and T will be able to solve the
problem be denoted by P(P), P(Q), P(R), P(S), and P(T) respectively.
< TOP >
26
Given:
P(P) =
2
5 P(Q) =
2
3 P(R) =
1
6 P(S) =
1
2 P(T) =
3
8
Probability that none of the students will be able to solve the problem
=
1 2 1 2 1 1 1 1 1 3 3 1 5 1 5 5
5 3 6 2 8 5 3 6 2 8 96
 −  −  −  −  −  = × × × × =      
     
∴ Probability that the problem will be solved = 1 –
5
96 =
91
96 = 0.948 i.e. 94.8%.
65. Answer : (a)
Reason : Percentage of the time will the station be shut down = P(x > 45)
=
P z 45 25
7
 −   > 
 = P(z > 2.86) = 1 - (0.5 + 0.4979) = 0.0021 i.e. 0.21%
< TOP >
66. Answer : (b)
Reason : Given that p= 0.60 , q= 0.40, n = 4,
P(r) =
n r nr
Crp q −
P (3) =
4 3 4 3
C3 (0.60) (0.40) × × −
= 0.3456
< TOP >
67. Answer : (d)
Reason : Let X be the number obtained on any throw. The desired event is X = 3 or 4 or 5 or
6 i.e, X ≥ 3 .
This can be considered as success.
P(Success) = P(X 3) 4 2
6 3
∴ ≥ = =
Let Y denote the number of throws in which success ( X ≥ 3 ) happens. Y follows a
binomial
distribution with p =
2
3 and number of trials = 20
20 18 2 20 19 20 20
18 19 20
18 2 19
P(Y 17) = 1 - [P(Y = 18) + P(Y = 19) + P(Y = 20)]
= 1 - [ C p (1 - p) + C p (1 - p) + C p ]
= 1 - [190 2 1 + 20 2 1 + 2
3 3 3 3 3
≤
×    ×            
        
[ ]
20
]
= 1 - 0.01428 + 0.00301 + 0.0003
= 0.98241

 

< TOP >
68. Answer : (d)
Reason : H0 : μ = 9
H1 : μ ≠ 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 σ =
s
n =
256
25 = 3.20
z = x
x − μ
σ =
2.25 9
3.20
−
= –2.109
At α = 0.05, the critical values are ±2.064.
The test statistic is less than the left tail critical value. So it falls in the rejection
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27
region.
∴ We reject H0. But the true mean is 9. So H0 is true. Hence the test leads to a type I
error.
69. Answer : (d)
Reason : H0 : μ = 60
x = 72
Test statistic = x
x − μ
σ
Or 1.20 =
72 60
60
n
−
 
 
 
Or
60
n =
72 60
1.20
−
= 10
Or n =
2 60
10
 
 
  = 36.
< TOP >
70. Answer : (c)
Reason : H0 : μ = 10
H1 : μ > 10
The sample is large. So we shall use the normal distribution.
x σ = n
σ
=
256
64 = 2
z = x
x − μ
σ =
13.50 10
2
−
= 1.75
At α = 0.05, the right tail critical value is 1.645.
The test statistic is more than the right tail critical value. So it falls in the rejection
region.
∴ We reject H0. But the true mean is 12. So H0 is false and it is rejected.
∴ This does not lead to either type I error or type II error.
< TOP >
71. Answer : (b)
Reason : Probability of committing a type I error = Probability of rejecting the null
hypothesis when it is true
By the central limit theorem x is normally distributed with mean = μ and variance =
2
n
σ
Hence Probability of rejecting the null hypothesis when it is true =
2 2
P(x 17.06 or x 22.94) P(x 17.06) P(x 22.94)
= P z 17.06 20 P z 22.94 20
n n
P z 2.94
81
36
≤ ≥ = ≤ + ≥
   
 −   −   ≤  +  ≥ 
 σ   σ 
   
   
 
 −  =  ≤ 
 
 
 
P z 2.94
81
36
P(z 1.96) P(z 1.96)
= 0.025 + 0.025 = 0.05
 
 
+  ≥ 
 
 
 
= ≤− + ≥
< TOP >
28
72. Answer : (b)
Reason : By Central Limit Theorem for large samples the sample mean is approximately
normally distributed with mean = population mean and standard deviation
=
n
σ
P (x >51) = 0.1587 = P(z > k)
∴ P (0 < z < k) = 0.50 – 0.1587 = 0.3413
From standard normal table we can see that k = 1.00
∴ z = 1 = x
x − μ
σ or 1 = x
51− 50
σ or x σ =
1
1 =
1
∴ n
σ
= 1 or n =
2
1
 σ 
 
  = σ2 = 36.
< TOP >
73. Answer : (a)
Reason : ˆy represents the estimated value of the dependent variable(y)
Se =
(Y Yˆ )2
n 2
Σ −
−
Y 18 24 20 25 32
ˆY
20 22 18 26 30
Y−Yˆ –2 2 2 –1 2
(Y −Yˆ )2
4 4 4 1 4
Σ
(Y −Yˆ )2 = 17
∴ S.E. =
17
5 − 2 = 2.380.
< TOP >
74. Answer : (c)
Reason : Coefficient of determination =
RSS RSS
TSS ESS RSS
=
+
=
( )
( ) ( )
2
2 2
Yˆ Y
Y Yˆ Yˆ Y
Σ −
Σ − +Σ −
=
8750 0.7955
2250 8750
=
+
< TOP >
75. Answer : (a)
Reason : The slope of the regression line with X as the independent variable
=
( )( )
( )2
X X Y Y
X X
Σ − −
Σ − = 2 2
XY nXY 1200 0.48
X nX 2500
Σ −
= =
Σ −
< TOP >
76. Answer : (d)
Reason : Unweighted aggregates price index =
1
0
P
100
P
Σ
×
Σ = 125
Or
1
0
P
n 100
P
n
 Σ 
 
 ×
 Σ 
 
  = 125
< TOP >
29
Or
1
0
P
100
P
×
= 125
Or
1
0
P 100
P
125
×
=
=
75 100
125
×
= Rs.60.
77. Answer : (b)
Reason : Laspeyres price index =
1 0
0 0
PQ
100
P Q
Σ
×
Σ =
(320 600 700 380) 100
(160 500 560 380)
+ + +
×
+ + +
=
2000 100
1600
×
= 125.
< TOP >
78. Answer : (c)
Reason : Paasches price index =
1 1
0 1
PQ
100
P Q
Σ
×
Σ
For each good : P1Q1 = P1Q0 ×
1
0
Q
Q
P0Q1 = P0Q0 ×
1
0
Q
Q
∴ We can rewrite from the table:
Good P1Q0 P0Q0 1
0
Q
Q P1Q1 = P1Q0 ×
1
0
Q
Q P0Q1 = P0Q0 ×
1
0
Q
Q
A 640 320 3/4 480 240
B 1200 1000 1/2 600 500
C 1400 1120 5/7 1000 800
D 760 760 13/19 520 520
ΣP1Q1 = 2600 ΣP0Q1 = 2060
∴ Paasches price index =
1 1
0 1
PQ
100
P Q
Σ
×
Σ =
2600 100
2060
×
= 126.21.
< TOP >
79. Answer : (d)
Reason : The unweighted average of relatives price index for the given data is
1
0
P x100
P
n
 
 
 
Σ
i.e., [(30/15 x 100) + (14/8 x 100) + (17/6 x 100) + (16/10 x 100)] / 4 i.e., 204.58.
Hence from above discussion, we can infer that option (d) is correct.
< TOP >
80. Answer : (b)
Reason : Fisher’s ideal price index =
1 1 1 0
0 1 0 0
P Q P Q
100
P Q P Q
Σ Σ
× ×
Σ Σ
=
1 1 1 0
0 0 0 0
P Q P Q
100
P Q P Q
Σ Σ
× ×
Σ Σ
= 0.8125×1.9417 ×100 = 125.60.
< TOP >
81. Answer : (c)
Reason : We use the following table to calculate the correlation coefficient:
< TOP >
30
X x = X- X x2 Y y = Y- Y y2 xy
9 4 16 15 3 9 12
8 3 9 16 4 16 12
7 2 4 14 2 4 4
6 1 1 13 1 1 1
5 0 0 11 -1 1 0
4 -1 1 12 0 0 0
3 -2 4 10 –2 4 4
2 -3 9 8 -4 16 12
1 -4 16 9 -3 9 12
ΣX = 45 Σx= 0 Σ x2 = 60 ΣY= 108 Σy = 0 Σy2= 60 Σxy =57
We have X = 45 / 9 = 5; Y = 108/9 = 12 and r =
2 2
xy 57 0.95
x y 60 60
Σ
= =
Σ Σ ×
82. Answer : (c)
Reason : Let:
Age = X
Blood Pressure = Y
The following table is generated to calculate the regression equation:
X Y X2 Y2 XY
52 62 2704 3844 3224
63 53 3969 2809 3339
45 51 2025 2601 2295
36 25 1296 625 900
72 79 5184 6241 5688
65 43 4225 1849 2795
47 60 2209 3600 2820
25 33 625 1089 825
ΣX = 405 ΣY = 406 ΣX2 = 22237 ΣY2 = 22658 ΣXY= 21886
b = 2 2 2
n XY X Y 8 21886 405 406
n X ( X) 8 22237 (405)
Σ −Σ Σ × − ×
=
Σ − Σ × − = 0.7684
a = Y− bX = (406/8) – 0.7684 × (405/8) = 11.85
Therefore the equation of the regression line is,
ˆY
= 11.85 + 0.7684 X.
The expected blood pressure of a person aged 49 years is 49.502.
< TOP >
83. Answer : (c)
Reason : Let the mean be μ and the standard deviation σ. Since there are 30.85% of the items
are under 45, area under the normal curve to the left of X = 45 is 30.85%. The area
lying to the right of the ordinate at X = 45 and up to the mean is (0.50 – 0.3085) =
0.1915. The value of z corresponding to this area is 0.5
∴ z
X 45 0.5
− μ − μ
= = =−
σ σ or 45 = μ – 0.5σ …………(i)
Now, 8.08% of the items are above 64. Therefore, area to the right of the ordinate at
< TOP >
31
64 is 0.0808. Area to the left of the ordinate at X = 64 up to mean ordinate is (0.5 –
0.0808) = 0.4192 and the value corresponding to this area is 1.4.
∴ z
X 64 1.4
− μ − μ
= = =
σ σ or 64 = μ + 1.4σ …………….(ii)
From equation (i) and (ii), we get the values μ = 50 and σ = 10.
84. Answer : (b)
Reason : ˆY = a + bX = 6 + 3X
ˆM
= a′ + b′X M = Y + 2
b′ = 2 2
n XM X M
n X ( X)
Σ −Σ Σ
Σ − Σ
= 2 2
n X(Y 2) X (Y 2)
n X ( X)
Σ + −Σ Σ +
Σ − Σ
= 2 2
n XY n 2X X Y X 2
n X ( X)
Σ + Σ −Σ Σ −Σ Σ
Σ − Σ
= 2 2
n XY X Y 2n X 2n X
n X ( X)
Σ −Σ Σ + Σ − Σ
Σ − Σ
= 2 2
n XY X Y
b 3
n X ( X)
Σ −Σ Σ
= =
Σ − Σ
a ′ = M− b′X = (Y+ 2) − bX = (Y− bX) + 2
= a + 2 = 6 + 2 = 8
ˆM
∴ = 8 + 3X.
< TOP >
85. Answer : (d)
Reason : Yˆ = a + bX
e
S ESS
n 2
=
−
ESS =
( 2 ) ( 2 ) ( 2 )
2 2
1 r TSS 1 r RSS 1 0.70 161.21 167.79
r 0.70
− = − = − =
e
S 167.79 4.58.
8
= =
< TOP >
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