Quantitative Methods (MB151) : April 2005

Quantitative Methods (MB151) : April 2005

• Answer all questions.
• Marks are indicated against each question.
1. The following index values pertain to a year:
Fisher’s ideal price index = 120.50
Laspeyres price index = 118.00
Paasche’s price index for the year is approximately
(a) 102.90 (b) 115.50 (c) 123.05 (d) 127.05 (e)
129.10.
(1 mark)
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2. The sum of 20 terms of the series 3, 6, 9, 12, 15…. is
(a) 440 (b) 510 (c) 560 (d) 590 (e)
630.
(1 mark)
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3. If a function is defined as y = f(x) = 2x + 5, then the inverse function f–1(x) can be denoted as
(a) (y – 5)/2 (b) (y – 2)/5 (c) 5y + 2 (d) 2y + 5 (e)
2x + 5.
(1 mark)
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4. If the quartiles, Q1 = 100.0 and Q3 = 160.0, then the coefficient of quartile deviation is equal to
(a) 0.11 (b) 0.23 (c) 0.80 (d) 1.25 (e)
4.33.
(1 mark)
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5. The expression
is equal to
(a) (b) (c) (d)
(e)
(2 marks)
2 log x + 2 log x2 + 2 log x3 +..... n terms (for x > 0)
n(n +1)
logx
2 n(n +1)logx
n(n 1)
logx
2
−
n(n −1) logx
n2 logx .
<
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6. 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 0.
(1 mark)
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7. How many committees of five members each can be formed from 8 official and 4 non-official members
when one particular non-official member would be always in the committee?
(a) 220 (b) 330 (c) 462 (d) 495 (e) 792.
(1 mark)
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8. In a single throw with two dice, the probability of throwing a total of 11 is
(a) 1/12 (b) 1/16 (c) 1/18 (d) 1/24 (e)
1/36.
(1 mark)
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9. Three mangoes and five apples are placed in a box. If two fruits are chosen at random, the probability <
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that one is a mango and the other is an apple is
(a) 3/8 (b) 5/8 (c) 2/7 (d) 15/28 (e)
17/28.
(1 mark)
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10. Which of the following is not a type of random sampling?
(a) Simple random sampling (b) Stratified sampling (c) Systematic samplin
(d) Convenience sampling (e) Cluster sampling.
(1 mark)
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11. In a G.P. sum of n terms is 255, the last term is 128 and the common ratio is 2. The value of n is
(a) 6 (b) 7 (c) 8 (d) 9 (e)
10.
(2 mark)
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12. In a football championship, 153 matches were played. Every two teams played one match with each
other. The number of teams participating in the championship is
(a) 15 (b) 16 (c) 17 (d) 18 (e)
19.
(2 mark)
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13. If three quantities a, b and c, are in harmonic progression then
(a) a − b = b − c (b) a(b − c) = c(a − b)
(c) ac = a + b (d) bc = a + c (e) ab = b + c.
(1 mark)
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14. A privately owned lake contains two types of game fish – bass and trout. The owner provides two types
of food, A and B, for these fish. Bass requires 2 units of food A and 4 units of food B, and trout requires
5 units of food A and 2 units of food B. If the owner has 800 units of each food, the maximum number
of fishes that the lake can support would be
(a) 100 (b) 150 (c) 250 (d) 300 (e)
350.
(2 mark)
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15. As the sample size increases
(a) The variation of the sample 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)
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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)
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17. Zaheda has 31 days to complete her quilt for the county fair. The blue squares in the quilt can be sewn at
a rate of 4 squares per day, and the white squares at a rate of 7 squares per day. The quilt can have up to
96 squares total. The blue fabric costs about Rs. 0.80 per square and the white fabric costs about Rs.
1.20 per square. Zaheda wants to keep costs at a minimum. Which of the following functions does
describe the cost of the quilt?
[Note: The number of blue and white squares are b and w respectively]
(a) f(b, w) = 4b + 7w (b) f(b, w) = 0.80b + 1.20w+ 96
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(c) f(b, w) = 1.20b + 0.80w + 96 (d) f(b, w) = 0.80b + 1.20w
(e) f(b, w) = 0.80b + 7w + 96.
(1 mark)
18.
If f(x) = x2 – 5x + 7; the value of is
(a) x + h – 3 (b) 2x + h – 5 (c)
x – h – 5
(d) 2x + 2h – 7 (e) x + 2h – 5.
(1 mark)
f (x h) f(x)
h
+ − <
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19. A speaks truth in 75 percent of the cases and B speaks truth in 80 percent of the cases. In what
percentage of the cases are they likely to contradict each other in stating the same fact?
(a) 15% (b) 20% (c) 25% (d) 30% (e)
35%.
(2 marks)
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20. 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)
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21. There are two groups of subjects one of which consists of 5 science subjects and 3 engineering subjects
and the other consists of 3 science and 5 engineering subjects. An unbiased die is cast. If number 3 or
number 5 turns up a subject is selected at random from the first group, otherwise the subject is selected
at random from the second group. Find the probability that an engineering subject is selected ultimately.
(a) (b) (c) (d)
(e) .
(2 mark)
5
8
5
16
7
18
11
21
13
24
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22. 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)
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23. In a test an examinee either guesses or copies or knows the answer to a multiple choice question with 4
choices. The probability that he makes a guess is and the probability that he copies the answer is .
The probability that his answer is correct given that he copied it is . Find the probability that he knew
the answer to the question given that he correctly answered it.
(a) (b) (c) (d)
(e) .
(2 mark)
1
3
1
6
1
8
1
9
2
11
3
19
13
25
24
29
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24. The probability that a boy will get a scholarship is 0.9 and that a girl will get a scholarship is 0.8. The
events, the boy gets the scholarship and the girl gets the scholarship, are independent of each other.
What is the probability that at least one of them will get the scholarship?
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(a) 92% (b) 94% (c) 96% (d) 98% (e)
100%.
(1 mark)
25. The personnel department of a company has records which show the following analysis of its 200
engineers:
If an engineer is selected at random from the company, what is the probability that he is under 30 given
that he has only a bachelor’s degree?
(a) 30% (b) 45% (c) 50% (d) 60% (e)
70%.
(1 mark)
Age (Years) Bachelor’s Degree Master’s Degree Total
Under 30 90 10 100
30 to 40 20 30 50
Over 40 40 10 50
Total 150 50 200
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26.
The probability of X, Y and Z becoming managers are and respectively. The probabilities that
a bonus scheme will be introduced if X, Y and Z become managers are respectively. What
is the probability that the bonus scheme will be introduced?
(a) 35% (b) 43% (c) 51% (d) 59% (e)
64%.
(1 mark)
4 , 2
9 9
1
3
3 , 1 and 4
10 2 5
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27. Calculate arithmetic mean from the following data:
(a) 30 (b) 28 (c) 33 (d) 35 (e)
38.
(1 mark)
Marks 0 – 10 10 – 20 20 – 30 30 – 40 40 – 50 50 – 60
No. of students 5 10 25 30 20 10
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28. The mean of 100 observations is found to be 40. At the time of computation of the mean two
observations are wrongly taken as 30 and 27 instead of 3 and 72 respectively. What is the correct mean?
(a) 39.22 (b) 39.79 (c) 39.94 (d) 40.18 (e)
40.48.
(1 mark)
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29. Find the modal wage from the following data:
(a) 25 (b) 26.30 (c) 27.16 (d) 27.86 (e)
28.29.
(1 mark)
Wages in (Rs.) 10 – 15 15 – 20 20 – 25 25 – 30 30 – 35 35 – 40 40 – 45
No. of workers 60 140 110 150 120 100 90
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30. 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
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(e) Always lies on one of the two axes.
(1 mark)
31. 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)
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32. A data set contains the following observations:
10, 12, 8, 14, 15, 9, 17, 20
What is the range of the data set?
(a) 8 (b) 9 (c) 10 (d) 12 (e)
13.
(1 mark)
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33. Y is a binomial random variable. For 20 trials the expected value of Y is 4. What will be the standard
deviation of Y for 25 trials?
(a) 2 (b) 4 (c) 5 (d) 20 (e)
25.
(1 mark)
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34. There are 25 students in a class, which consist of 14 boys and 11 girls. 5 students of the class were
absent on a particular day. What is the probability that all of the absent students were boys?
(a) 0.03768 (b) 0.3768 (c) 0.9623 (d) 0.6232 (e)
0.2072.
(1 mark)
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35. A medical shop sells a variety of drugs and one of the drugs sold by the shop is Drug X. This drug has a
short shelf life and it is quite expensive. Each packet of Drug X costs Rs.300 and can be sold for Rs.400.
The manager of the shop has observed that the demand for this drug before its expiry has the following
probability distribution:
Any packet of this drug not sold before expiry has no value and has to be rejected. It is assumed that the
demand for this drug will be one of the aforementioned values only.
How many packets of Drug X should be stocked by the shop?
(a) 300 (b) 400 (c) 500 (d) 600 (e) 700.
(2 marks)
Demand level before expiry
(number of packets) 300 400 500 600 700
Probability 0.15 0.25 0.40 0.15 0.05
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36. Variable Y follows a normal distribution with variance 16.
The standardized value for Y = 18 is 2.5. What is the mean of Y?
(a) 10 (b) 18 (c) 8 (d) 4 (e)
2.50.
(1 mark)
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37. The life expectancy of a particular brand of tire is normally distributed with a mean of 40,000 miles and
a standard deviation of 5,000 miles. What is the probability that a randomly selected tire will have a life
of at least 30,000 miles?
(a) 0.4772 (b) 0.9772 (c) 0.0228 (d) 0.5 (e)
0.45.
(1 mark)
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38. The manager of the quality department for a tire manufacturing company wants to study the average
tensile strength of rubber used in making a certain brand of radial tire. The population is normally
distributed and the population standard deviation is known. She wants to test the null hypothesis that the
mean tensile strength is equal to 800 pounds per square inch against the alternative hypothesis that the
mean tensile strength is not equal to 800 pounds per square inch. She has taken a sample of 25
observations and the test statistic is calculated to be 1.995. If the significance level is 0.05, then
(a) The critical values are and the null hypothesis is rejected
(b) The critical values are and the null hypothesis is accepted
(c) The critical values are and the null hypothesis is accepted
(d) The critical values are and the null hypothesis is rejected
(e) The critical values are and the null hypothesis is rejected.
(1 mark)
±1.96
±1.96
±2.064
±2.064
±1.711
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39. If we reject H0 : μ = 10 in favor of H1 : μ 10 at a given level of significance with a positive value of
the test statistic, then a test with revised H0 : μ = 5 versus revised H1 : μ 5 using the same sample and
the same level of significance
(a) Will always accept the revised H0
(b) Will always reject revised H0
(c) May or may not accept revised H0
(d) Will certainly lead to a Type I error
(e) Will certainly lead to a Type II error.
(1 mark)
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40. The following information are available with regard to a sampling distribution of mean:
Sample size = 64
Probability that the sample mean is less than 65 = 0.0228
Population standard deviation = 20
It is assumed that the Central Limit Theorem will be applicable and the population mean is greater than
65.
What is the population mean?
(a) 60 (b) 65 (c) 70 (d) 75 (e)
5.
(2 marks)
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41. A magazine claims that 25% of its readers are college students. Of a random sample of 200 readers, 42
are college students. It is to be tested at a 0.10 level of significance whether the proportion of college
students among all the readers of the magazine is equal to 0.25. What is the conclusion?
(a) The proportion of college students among the readers of the magazine is 0.25
(b) The sample data are incorrect
(c) The proportion of college students among the readers of the magazine is less than 0.25
(d) The proportion of college students among the readers of the magazine is more than 0.25
(e) No conclusion can be drawn on the basis of the given information.
(2 marks)
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42. 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
(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)
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43. For a sample randomly collected from a population, the following details are available: <
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= 1840
= 160
n = 16
What is the estimated standard error of mean?
(a) 1 (b) 2 (c) 4 (d) 10 (e)
16.
(1 mark)
Σx2
Σx
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44. The time that it takes to find a taxi when leaving a restaurant follows a left skewed distribution with a
mean of 20 minutes and a variance of 100 minutes. If 64 restaurant patrons are randomly sampled and
the average time that it takes for them to find a taxi is calculated, then what is the probability that the
sample mean will be between 18 and 23 minutes?
(a) 0.4918 (b) 0.4452 (c) 0.9918 (d) 0.9452 (e)
0.9370.
(1 mark)
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45. The following details are available with regard to a test of hypothesis for the population mean:
What is the value of the test statistic?
(a) 1.20 (b) 1.44 (c) 2.40 (d) 1.00 (e)
0.83.
(1 mark)
0
2
H : 18
x 24
n 36
x 52236
μ =
=
=
Σ =
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46. The following details are available for a hypothesis test on population proportion:
H0: p = 0.60
H1: p 0.60
The standard error of proportion is 0.03098. What is the sample size? (Round off your answer to the
nearest integer)
(a) 100 (b) 250 (c) 19 (d) 625 (e)
417.
(1 mark)
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47. A financial analyst has collected the following data on sales and net profits of some companies
manufacturing TVs:
What percentage of variations in net profit is explained by the variations in sales? (Round off your
answer to the nearest integer)
(a) 98% (b) 95% (c) 92% (d) 85% (e)
81%.
(2 marks)
Sales (Rs. in crores) 42 45 52 34 60
Net profit (Rs. in crores) 2.5 2.45 3.25 1.75 4.00
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48. A simple regression relationship was developed between two variables X and Y, with X as the
independent variable. The following details are available:
ΣY = 792 ΣY2 = 79144 = 200 n = 8
Σ(Y−Yˆ )2
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The coefficient of determination is
(a) 0.2717 (b) 0.7283 (c) 0.3731 (d) 0.2137 (e)
0.7863.
(1 mark)
49. 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)
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50. The covariance between the variables X and Y is 10, and the variances of X and Y are 16 and 64
respectively. What is the coefficient of correlation between X and Y?
(a) 0.3125 (b) 0.625 (c) 0.15625 (d) 0.40 (e)
0.16.
(1 mark)
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51. Consider the following information:
The weighted average of relatives price index is
(a) 118.42 (b) 130.77 (c) 134.54 (d) 144.85 (e)
139.20.
(1 mark)
Commodity
100
P
P
0
1 × Pn Qn
Rice 118.42 10,000
Wheat 130.77 4,050
Salt 160 1,300
Sugar 145 9,600
Pulses 137.78 13250
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52. The following data are collected with regard to some commodities:
What is the unweighted aggregates price index for the year 2004 considering the year 1994 as the base
year?
(a) 83.57 (b) 114.77 (c) 116.67 (d) 119.67 (e)
87.13.
(1 mark)
Commodity Quantity (Units) Price (Rs./Unit)
1994 2004 1994 2004
A 15 20 17 21
B 20 20 190 220
C 25 30 30 31
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53. The average price of a group of six commodities in the base year is Rs.30. The unweighted aggregates
price index for the group for the current year is 150. What is the average price of the commodities in the
current year?
(a) Rs.30 (b) Rs.45 (c) Rs.60 (d) Rs.75 (e)
Rs.90.
(1 mark)
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54. If events A and B are mutually exclusive then which of the following is true?
(a) P(A and B) = 0 (b) P(A) = 0 (c) P
(B) = 0
(d) P(A or B) = 0 (e) P(A or B) = 1.
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(1 mark)
55. A card is drawn from a pack of cards randomly. What is the probability that the card drawn is a Queen
of Hearts?
(a) 1/4 (b) 1/13 (c) 1/51 (d) 17/52 (e)
1/52.
(1 mark)
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56. In which of the following conditions will a binomial distribution be symmetrical?
(a) Number of trials is 30 or more
(b) Number of trials is 100 or more
(c) Probability of success in any trial is less than 0.5
(d) Probability of failure in any trial is less than the probability of success
(e) Probability of failure in any trial is equal to 0.5.
(1 mark)
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57. 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)
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58. Which of the following is considered to be unknown when the t statistic is calculated for a sample
mean?
(a) Hypothesized population mean (b) Population variance (c) Sample variance
(d) Sample size (e) Sample mean.
(1 mark)
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59. The logarithm of 1 with respect to any base is equal to
(a) –10 (b) –1 (c) 1 (d) 0 (e)
10.
(1 mark)
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60. For a given hypothesis test, if we do not reject the null hypothesis, and the null hypothesis is true, 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)
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61. Which of the following is true with regard to the standard error of a sample statistic?
(a) It is equal to the standard deviation of the population
(b) It is equal to the standard deviation of a sample taken from the population
(c) It indicates the extent of non-sampling error
(d) It indicates the variability arising out of sampling error due to chance
(e) It is the difference between the value of the sample statistic and the true value of the population
parameter.
(1 mark)
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62. 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)
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63. 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)
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64. 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)
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65. Which of the following represents the proportion of variation in the dependent variable that is explained
by the regression line?
(a) Coefficient of determination (b) Coefficient of correlation
(c) Coefficient of variation (d) Standard error of estimate
(e) Standard deviation.
(1 mark)
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66. 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)
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67. Which of the following is not true with regard to the regression relationship, = a + bX?
(a) The point always lies on the regression line
(b) The expected value of the residuals is zero
(c) The mean of the fitted values of Y is the same as the mean of the observed values of Y
(d) There are always as many points above the fitted line as there are below it
(e) The regression line minimizes the sum of the squared residuals.
(1 mark)
ˆY
(X, Y)
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68. In which of the following conditions will the problem of multicollinearity arise in a multiple regression
relationship given as : = 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)
Yˆ
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69. Which of the following is true with regard to the simplex method of solving a linear programming
problem (LPP) on profit maximization?
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(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)
70. Which of the following conditions indicates the existence of multiple optimal solutions when a linear
programming problem is solved by the graphical method?
(a) One of the constraints is parallel to the horizontal axis
(b) The objective function is parallel to the vertical axis
(c) The objective function is parallel to one of the edges of the feasible region which is in the
direction of optimal movement of the objective function
(d) If two or more constraints are parallel to each other
(e) If there is a redundant constraint present in the problem.
(1 mark)
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71. A graphical method of representing states of nature and courses of action involved in decision making is
referred to as a
(a) Decision tree (b) Histogram (c) Scatter
diagram
(d) Frequency distribution (e) Probability
distribution.
(1 mark)
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72. If the covariance between two variables is positive then, the coefficient of correlation between them will
be in the range of
(a) 0 to –1.00 (b) 0 to –0.50 (c) 0 to –0.25 (d) 0 to 2.00 (e) 0
to 1.00.
(1 mark)
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73. Event A is dependent on event B. 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
>
74. 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) > (b) >
(c) > (d) > (e) > .
(1 mark)
P(A) P(B) P(A) P(B)
P(A) P(B) P(B) P(A) P(B | C) P(A | C)
<
Answer
>
75. The events S and T are independent and are equiprobable such that P(S and T) = p, where p > 0, then
P(S) = ?
(a) (b) p (c) p2 (d) 2p
(e) p/2.
(1 mark)
p
<
Answer
>
76. 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
<
Answer
>
(e) The mode is always uniquely defined.
(1 mark)
77. 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
>
78. The coefficient of variation cannot be meaningfully used to compare the variability of two or more sets
of data, when
(a) The standard deviations equal for all 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 means are equal for all the sets of data.
(1 mark)
<
Answer
>
79. 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
>
80. The middle value of a series arranged either in ascending or descending order is called ________.
(a) Arithmetic mean (b) Geometric mean (c) Range
(d) Median (e) Mode.
(1 mark)
<
Answer
>
81. 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
>
82. What are the possible numbers of permutations of 5 different things taking 2 things at a time?
(a) 10 (b) 20 (c) 30 (d) 40 (e)
60.
(1 mark)
<
Answer
>
83 If every item in a data set is increased by a constant C, then the arithmetic mean of the resulting data set
will be equal to
(a) The mean of the original data set
(b) C + Mean of the original data set
(c) C – Mean of the original data set
(d) Mean of the original data set ÷ C
(e) Mean of the original data set × C.
(1 mark)
<
Answer
>
84. The logarithm of a number
(a) Is always expressed with respect to base 10
(b) Is always expressed with respect to base 1
<
Answer
>
Suggested Answers
Quantitative Methods (MB151) : April 2005
(c) Is always expressed with respect to base 100
(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)
85. Arithmetic, geometric and harmonic means of two numbers, X and Y, are A, G and H respectively.
Which of the following is false?
(a) AH = XY
(b) X – A = A – Y
(c)
(d)
(e) G = AH.
(1 mark)
Y
G
G
X =
1 1 1 1
X H H Y
− = −
<
Answer
>
86. Which of the following measures of central tendency is based on the most frequently occurring value in
a set of observations?
(a) Arithmetic mean (b) Mode
(c) Geometric mean (d) Median (e) Harmonic
mean.
(1 mark)
<
Answer
>
87. 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
>
88. If the first term in a geometric progression is greater than 1 and the common ratio is less than 1, then
(a) The consecutive terms will be in increasing order
(b) The consecutive terms will be in decreasing order
(c) The consecutive terms will be equal
(d) All the consecutive terms will be less than the common ratio
(e) All the consecutive terms will be less than 1.
(1 mark)
<
Answer
>
89. is equal to
(a) × r (b) × r! (c) × n
(d) ÷ r! (e) × n!.
(1 mark)
nCr
nPr nPr nPr
nPr nPr
<
Answer
>
1. Answer : (c)
Reason : Fisher’s ideal price index =
Laspeyres price index × Paasches price index
< TOP >
Paasches price index =
= = 123.05
(Fisher 's ideal price index)2
Laspeyres price index
(120.50)2
118
2. Answer : (e)
Reason : The given series is an arithmetic progression with first term as 3 and common difference as 3. The
sum of n terms of an arithmetic progression is
Sn= (n/2){2a+(n-1)d}
∴Sn = (20/2){2×3+(20-1)×3}
= 630
Therefore the correct answer is (e).
< TOP >
3. Answer : (a)
Reason : If the given function y = 2 x + 5 then, the inverse function f-1(x) = (y − 5) / 2.
Therefore the correct answer is (a).
< TOP >
4. Answer : (b)
Reason : Coefficient of quartile deviation = =
Therefore the correct answer is (b)
3 1
3 1
Q Q
Q Q
−
+
160 100 60 0.23
160 100 260
−
= ≈
+
< TOP >
5. Answer : (b)
Reason : The given sum is
2 3
2 3
2 3
(1 2 3 ....... )
( 1)
2
2 log 2 log 2 log ..... terms
2[log log log ..... log ]
2[log . . ........ ]
2[log ]
2 log
( 1)log
+ + + +
+
= + + + ∞
= + + + +
=
=
=
= +
n
n
n
n
n n
S x x x n
x x x x
x x x x
x
x
n n x
< TOP >
6. Answer : (b)
Reason : The coefficient of correlation can take values in the range of –1 to 1. Therefore the correct answer is
(b).
< TOP >
7. Answer : (b)
Reason : As one non-official member is to be included in all the committees , so effectively four members of
the committee are to be selected from 12 – 1 = 11 members. So, the total number of committees
could be formed is .
11
4
11 10 9 8
C 330
1 2 3 4
× × ×
= =
× × ×
< TOP >
8. Answer : (c)
Reason : The total possible set of numbers = n(S) = 36
The set of favorable event = E = {6, 5), (5, 6)}
The number of favorable event = n (E) = 2
Therefore the required probability = P (E) = .
( ) 2 1
( ) 36 18
= =
n E
n S
< TOP >
9. Answer : (d) < TOP >
Reason : Total number of ways two fruits can be picked from 8 fruits = n (S) =
The total number of ways one mango and one apple is picked is n(E) =
Therefore the probability of the favorable event = .
8
2 C
3 5
C1 × C1
( ) 3 5 2 15
( ) 8 7 28
× ×
= =
×
n E
n S
10. Answer : (d)
Reason : This method is based on the convenience of the researcher. The researcher uses the sources available
to him to come to a conclusion. This is a non random sampling technique.
Therefore the correct answer is (d).
< TOP >
11. Answer : (c)
Reason : Let the first term be a
So, the nth term is a x 2n-1 = 128 Or, a = 128/2n-1
The sum is
Putting the value of a we get
( ) ( ) 2 1
2 1
2 1
−
= −
−
n
n a
a
( ) n-1
n-1
n-1
n-1 7
128
2 1 255
2
1
, 128 2 255
2
128
, 128 2 255 1
2
, 2 128= 2
, 1 7
, 7 1 8
− =
− =
= × − =
=
− =
= + =
 
 
 
n
Or
Or
Or
Or n
Or n
< TOP >
12. Answer : (d)
Reason : Let there be n number of teams. Number of matches would be . In this case .
Or,
Or,
Or,
Or,
Or, n = 18. (We ignore the negative value of n , i.e. n = −17)
n
2 C n
2 C =153
n(n - 1)
153
2
=
2 n - n = 306
2 n - 18n +17n − 306 = 0
n(n - 18) +17(n − 18) = 0
(n - 18) (n+17) = 0
< TOP >
13. Answer : (b)
Reason : If a, b and c are in harmonic progression, then or,
or, 2ac = ab + bc or, ac – bc = ab – ac
or, ab – ac = ac – bc
or, a (b – c) = c (a – b)
2 1 1
b a c
= + 2 a c
b ac
+
=
< TOP >
14. Answer : (c) < TOP >
Reason : Let there are x numbers of bass and y numbers of trout.
Objective is to maximize Z = x + y.
Subject to:
and the terminal line is ……………….(i)
and the terminal line is ……………....(ii)
We obtain the feasible region as OABC for the given problem
Evaluating the value of objective function at the corners of feasible region, we get
We see that the maximum value is obtained at B(150, 100). This is the optimum point
The maximum number of fish, that the lake can support is 250.
So, the optimal point is (150, 100) and the lake can contain at the most 250 fishes.
2x + 5y ≤800 2x + 5y =800
4x + 2y ≤800 4x + 2y =800
Corner Point Z = x + y
O (0, 0) 0
A (0, 160) 160
B (150, 100) 250
C (200, 0) 200
15. Answer : (b)
Reason : a. As the sample size increases the variation of the sample mean from the population mean
becomes smaller.
b. From above we can see that (b) is incorrect.
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.
< 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.
< TOP >
e. When the y-intercept of a regression line is zero, the regression line passes through the
intersection of horizontal and vertical axes.
17. Answer : (d)
Reason : The blue fabric b costs about Rs. 0.80 per square and the white fabric w costs about Rs. 1.20 per
square. So, when number of blue fabric is b and number of white square is w, the cost function would
be 0.80b + 1.20w.
< TOP >
18. Answer : (b)
Reason : f(x) = x2 – 5x + 7
substituting (x + h) for x,
f(x + h) = x2 + (2h – 5)x + h2 – 5h + 7
=
=
= = 2x + h – 5.
f (x h) f (x)
h
+ − x2 (2h 5)x h2 5h 7 (x2 5x 7)
h
+ − + − + − − +
x2 2hx 5x h2 5h 7 x2 5x 7
h
+ − + − + − + −
2hx h2 5h
h
+ −
< TOP >
19. Answer : (e)
Reason : If E1 denotes the event that A speaks the truth, then is the event that A does not speak the truth.
Similarly, we define the events E2 and for B.
Let E be the event that A and B contradict each other.
Then according to the question, we have
P(E1) =
P(E2) =
They will contradict each other if one speaks the truth and the other does not.
Hence E =
Now = the probability that A speaks the truth and B tells a lie
=
Similarly
[Note: The E1 and are independent events and so are and ]
Since are mutually exclusive events,] we have
P(E) = =
= i.e., 35%
Hence in 35% cases A and B will contradict each other.
1 E
2 E
( ) 1
75 3 and P E 1 3 1
100 4 4 4
= = − =
( ) 2
80 4 and P E 1 4 1
100 5 5 5
= = − =
1 2 1 2 E and E + E and E
( ) 1 2 P E and E
( ) ( ) 1 2
P E P E 3 1 3
4 5 20
× = × =
( ) 1 2
P E and E 1 4 1
4 5 5
= × =
2 E 1 E 2 E
1 2 1 2 (E and E ) and (E and E )
1 2 1 2 P (E and E ) or (E and E ) 1 2 1 2 P(E and E ) + P(E and E )
3 1 7 35
20 5 20 100
+ = =
< TOP >
20. 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 >
21. Answer : (e) < TOP >
Reason : Let E1 be the event that a subject is selected from first group
E2 be the event that a subject is selected from second group
E be the event that an engineering subject is selected.
It is given that a subject is selected from first group if the die shows 3 or 5, otherwise it is selected
from the second group.
Now the probability that the die shows 3 or 5 = =
∴ P(E1) = and P(E2) = = = =
Now P(E | E1) = Probability of choosing an engineering subject from first group
= =
Similarly =
Hence P(E) =
= = .
2
6
1
3
1
3 P (E1 ) 1− P(E1 )
1 1
3
− 2
3
3
1
8
1
C
C
3
8
( ) P E E2
5
1
8
1
C 5
C 8
=
( ) ( ) ( ) 1 1 2 2 P E .P E E + P(E ) P E E
1 3 2 5
3 8 3 8
× + × 13
24
22. 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 extreme of
values.
e. Median is middle value of a data array. So it is not affected by the presence of the extreme
values.
< TOP >
23. Answer : (e)
Reason : Let A1 be the event that the examine guesses the answer; A2 be the event that he copies the answer
and A3 be the event that he knows the answer. Also let A be the event that he answers correctly.
Then as given we have
; ;
[The events A1; A2 and A3 are mutually exclusive and collectively exhaustive]
Now ; (as given).
Again it is reasonable to take the probability of answering correctly given that he knows the answer
as 1, that is
We have to find .
By Baye’s theorem, we have
1
P(A ) 1
3
= ( ) 2
P A 1
6
= 3
P(A ) 1 1 1 1
3 6 2
= − − =
( ) 1
P A A 1
4
= ( ) 2
P A A 1
8
=
( ) 3 P A A = 1
3 A
P
A
 
 
 
< TOP >
=
= .
( ) 3 P A A
( )
( ) ( ) ( )
3 3
1 1 2 2 3 3
P(A ) P A A
P(A ) P A A + P(A ) P A A + P(A ) P A A
1 1
2 24
1 1 1 1 1 1 29
3 4 6 8 2
 ×  
  =
 ×  +  ×  +  ×       
     
24. Answer : (d)
Reason : The probability that a boy will get a scholarship = 0.9
The probability that a girl will get a scholarship = 0.8
The probability at least one of them will get the scholarship is P(A or B) =
P(A) + P(B) – P(A and B)
= 0.9 + 0.8 – (0.9 × 0.8) = 1.7 – 0.72 = 0.98 i.e 98%.
< TOP >
25. Answer : (d)
Reason : Let A : an engineer has a bachelor degree only
B : an engineer is under 30 years of age
∴ P(B | A) =
90
P(Band A) 200 0.60 i.e. 60%
P(A) 150
200
= =
< TOP >
26. Answer : (c)
Reason : Let P(X) = Probability that x become manager
P(Y) = Probability that y become manager
P(Z) = Probability that z become manager
Let P(B | X) = Probability that bonus scheme is introduced when x become manager.
Similarly we can define P(B | Y) and P(B | Z)
We are given P(X) = ; P(Y) = ; P(Z) = .
P(B | X) = P(B | Y) = ; P(B | Z) = .
∴ P(B) = P(B and X) + P(B and Y) + P(B and Z)
=
= ≈ 51%.
4
9
2
9
1
3
3 ;
10
1
2
4
5
4 3 2 1 1 4
9 10 9 2 3 5
 ×  +  ×  +  ×       
     
46 23
90 45
=
< TOP >
27. Answer : (c)
Reason : Calculation of Arithmetic mean
Marks Mid-point
(m)
No. of student
(f) fm
0 – 10 5 5 25
10 – 20 15 10 150
20 – 30 25 25 625
30 – 40 35 30 1050
40 – 50 45 20 900
50 – 60 55 10 550
N = 100 Σfm = 3,300
< TOP >
=
x
fx 3, 300 33
N 100
Σ
= =
28. Answer : (d)
Reason : = or Σx =
Here = 40, n = 100
∴ Correct mean =
x
x
n
Σ
n.x
x
∴Σx = 100 × 40 4000
Less: Incorrect figures 57
3943
Add: Correct figures 75
Correct total 4018
4018 40.18
100
=
< TOP >
29. Answer : (d)
Reason : Mode lies in the class (25 – 30)
Where L = 25; Δ1 = (150 – 110) = 40; Δ2 = (150 – 120) = 30; i = 5
= 25 + 2.86 = 27.86.
1
0
1 2
M L i
Δ
= + ×
Δ + Δ
0
M 25 40 5
40 30
= + ×
+
< TOP >
30. 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 >
31. 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 >
32. Answer : (d)
Reason : Range = Highest value – Lowest value = 20 – 8 = 12
< TOP >
33. Answer : (a)
Reason :
< TOP >
For 20 trials E(Y) = n p = 20 p = 4
p = 4 = 0.20
20
For 25 trials V(Y) = n p q
= 25 0.20 (1 - 0.20)
= 4
For 25 trials st
× ×
∴
∴ × ×
× ×
∴ andard deviation of Y = V(Y)
= 4 = 2
34. Answer : (a)
Reason : The number of absentees who are boys follows a hypergeometric distribution. The following details
are available:
N = No. of elements in the population = 25
r = No. of elements in the population labelled success = No. of boys = 14
n = No. of trials = No. of absent students = 5
x = Desired number of successes = 5
P(x = 5) =
r (N r) 14 (25 14) 11
x (n x) 5 (5 5) 0
N 25
n 5
C C C C 2002 C
0.03768
C C 53130
− −
− − ×
= = =
< TOP >
35. Answer : (b)
Reason : Profit per packet = Rs. 400 – Rs. 300 = Rs. 100
Conditional profit table for Drug X (in Rs.):
Since the expected conditional profit is maximum ( Rs. 34,000) at a stock level of 400 packets the
shop should stock 400 packets.
Stock Demand Expected
300 400 500 600 700 Profit
(0.15) (0.25) (0.40) (0.15) (0.05)
300 30,000 30,000 30,000 30,000 30,000 30,000
400 0 40,000 40,000 40,000 40,000 34,000
500 (30,000) 10,000 50,000 50,000 50,000 28,000
600 (60,000) (20,000) 20,000 60,000 60,000 6,000
700 (90,000) (50,000) (10,000) 30,000 70,000 (22,000)
< TOP >
36. Answer : (c)
Reason :
Y - μ 18 -μ Standardised value for (Y = 18) = = = 2.50
σ 16
or μ = 18 - 2.50 × 4 = 8
< TOP >
37. Answer : (b)
Reason : P(X ³ 30000) = = P(z ³ –2) = 0.5 + 0.4772 = 0.9772
30000 40000 P z
5000
 −   ≥ 
 
< TOP >
38. Answer : (a)
Reason : The population is normally distributed and the population standard deviation is known. So the
standard normal distribution should be used. The test is two tailed and the significance level is 0.05.
So the critical values for the test are . Since the test statistic (1.995) is more than the right tail
critical value it falls in the critical region and we reject the null hypothesis.
±1.96
< TOP >
39. Answer : (b)
Reason : If we reject H0:μ = 10 in favor of H1:μ ≠ 10 at a given level of significance with a positive value of
< TOP >
the test statistic, then a test with revised H0:μ = 5 versus revised H1:μ > 5 using the same sample and the same
level of significance will always reject H0 because the value of the test statistic will be higher than
before and hence higher than the critical value.
40. Answer : (c)
Reason : By Central Limit Theorem for large samples the sample mean is approximately normally distributed
with mean = population mean and standard deviation
= μ
= 0.0228
= 0.50 – 0.0228 = 0.4772 = P(k < z < 0)
From tables are find that k = –2.0
∴z = –2 = or – 2 = =
or – 2 × 2.50 = 65 – μ
or μ = 65 + 5 = 70.
=
n
σ
E(x)
P(x < 65)
∴P (65< x <μ)
x
x − μ
σ
65
20
64
− μ
 
 
 
65
2.50
− μ
< TOP >
41. Answer : (a)
Reason : H0 : p = 0.25
H1 : p ¹ 0.25
Standard error of proportion =
Sample proportion =
Test statistic: z =
This is a large sample test of proportion. The standard normal distribution will be used to
approximate the sampling distribution of proportion. The test will be a two-tailed test. From the
standard normal distribution table and using interpolation the critical values are –1.645 and 1.645.
We can see that the test statistic does not fall below the lower tail critical value. Hence the test
statistic falls in the acceptance region. We accept H0 and conclude that proportion of college students
among the readers is 0.25.
(1 ) 0.25(1 0.25)
0.03062
200
− −
= = p p
n
42 0.21
200
=
0.21 0.25 1.31
0.03062
−
= −
< TOP >
42. 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 >
43. Answer : (a)
Reason : Estimated standard error of mean =
Sample standard deviation,
Sample standard deviation
n
x2 nx2 s
n 1 n 1
= −
− −
Σ
x 160 x 10
n 16
= = = Σ
< TOP >
∴ s =
∴ Estimated standard error of mean =
1840 16(10)2 4
15 15
− =
4 4 1
16 4
= =
44. Answer : (e)
Reason : By Central Limit Theorem for large samples the sample mean is approximately normally distributed
with mean = population mean and standard deviation
∴ = μ = 20
= = = = = 1.25
∴ = = P(–1.60 < z < 2.40)
= P (− 1.60 < z < 0) + P( 0 < z < 2.40)
= 0.4452 + 0.4918 = 0.9370.
n
σ
=
E(x)
x σ n
σ 2
n
σ 100
64
10
8
P (18< x < 23)
P 18 20 z 23 20
1.25 1.25
 − −   < < 
 
< TOP >
45. Answer : (a)
Reason : Since the sample size is large we can approximate the sampling distribution of mean by the normal
distribution (by CLT). Since the variance of the population is unknown, the sample variance will be
used for calculating the estimated standard error and the test statistic.
Test statistic =
=
s = = = 30
∴Test statistic = = = 1.20.
x
x
ˆ
− μ
σ
x ˆ σ
s
n
1
1 x2 nx2 2
n 1
 Σ −      − 
1
1 52236 36 242 2
35
  − ×      
24 18
30
36
−
6
5
< TOP >
46. Answer : (b)
Reason : = = 0.03098
or = 0.03098
or n = = 250.
p σ
( ) 0 0 p 1 p
n
−
0.60(1 0.60)
n
−
( )2
0.60 0.40
0.03098
×
< TOP >
47. Answer : (a)
Reason : Let the following notations be used.
X : Sales (Rs. in crores)
Y : Net profit (Rs. in crores)
Coefficient of correlation, r =
(X X)2 (Y Y)2
(X X)(Y Y)
Σ − −
Σ − −
< TOP >
= = = 46.6 = = = 2.79
∴ r = = 0.99 (approx.)
Percentage of variations in net profit that is explained by the variations in sales
= coefficient of determination = r2 = (0.99) 2 = 0.9801
∴ 98.01% of the variations in net profits is explained by the variations in sales.
A. X 42 45 52 34 60 ΣX =233
B. Y 2.5 2.45 3.25 1.75 4.00 ΣY = 13.95
C. X – X -4.6 -1.6 5.4 -12.6 13.4
D. Y –
Y -0.29 -0.34 0.46 -1.04 1.21
E. C × D 1.334 0.544 2.484 13.104 16.214
Σ(X – ) (Y– )
= 33.68
X Y
F. C2 21.16 2.56 29.16 158.76 179.56
Σ(X– ) 2 =
391.2
X
G. D2 0.0841 0.1156 0.2116 1.0816 1.4641
Σ(Y– )2 =
2.957
Y
X
X
n
Σ 233
5 Y n
ΣY 13.95
5
33.68
391.2 × 2.957
48. Answer : (b)
Reason : Coefficient of determination, r2 = 1–
Y denotes the observed values and denotes the estimated values.
= 99
= 79144 – 8 × 992 = 736
∴ Coefficient of determination, r2 = = 0.7283
( )
( )
2
2
Y Yˆ
Y Y
Σ −
Σ −
ˆY
Y Y 792
n 8
Σ
= =
Σ(Y − Y)2 = ΣY2 − nY2
1 200
736
−
< TOP >
49. Answer : (a)
Reason : (a) The probability of the occurrence of an event is expressed as a number which lies between 0
and 1.
< TOP >
50. Answer : (a)
Reason : .
X Y
Coeffient of correlation = Cov (X,Y) 10 = 10 = 5 0.3125.
. 16 64 4 8 16
=
σ σ × ×
< TOP >
51. Answer : (c)
Reason : Weighted average of relatives price index
n n
n n
0
1
P Q
100 P Q
P
P
Σ
 

 

 

 

Σ ×
Commodity
100
P
P
0
1 × Pn Qn
n n
0
1 100 P Q
P
P
 

 

×
< TOP >
= 51,39,403.50
Σ PnQn = 38,200.
∴ Weighted average of relatives price index for year is
= = 134.54.
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
Σ = 38,200 Σ 51,39,403.50
( ) 




Σ × n n
0
1 100 P Q
P
P
38,200
51,39,403.50
52. Answer : (b)
Reason : Unweighted aggregates price index =
= = 114.77.
1
0
P
100
P
Σ
×
Σ
(21 220 31) 100
(17 190 30)
+ +
×
+ +
< TOP >
53. Answer : (b)
Reason : Unweighted aggregates price index =
Or = 150
Or = 150
Or = = 30 × 1.50 = Rs.45.
1
0
P
100 150
P
Σ
× =
Σ
1
0
P
n 100
P
n
 Σ 
 
 ×
 Σ 
 
 
1
0
P
100
P
×
1 P
0 P 150
100
×
< TOP >
54. Answer : (a)
Reason : (a) If events A and B are mutually exclusive then P(A and B) = 0 because A and B can not occur at
the same time.
(b) and (c) Mutual exclusiveness of two events A and B does not imply that both of them
have zero probabilities of occurrence.
(d) and (e) Mutual exclusiveness does not mean that the probability of any one of them
occurring is either 0 or 1.
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55. Answer : (e)
Reason : There is only one card in the pack of card which is a queen of hearts. Therefore the required
probability is equal to P (Queen of Hearts) = 1/52.
Therefore the correct answer is (e).
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56. Answer : (e)
Reason : A binomial distribution is symmetrical if the probability of success and the probability of failure are
equal. Since p + q = 1, this means that the probability of failure is equal to 0.5.
< TOP >
57. Answer : (d)
Reason : In a hypergeometric distribution the composition of the population changes from trial to trial.
< TOP >
58. Answer : (b)
Reason : The population variance is considered to be unknown when the t-statistic is calculated for a sample.
< TOP >
59. 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.
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60. 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.
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61. Answer : (d)
Reason : Standard error of a sample statistic indicates the sampling error due to chance. Standard error is not
the standard deviation of the population; nor is it the standard deviation of the sample. Standard error
does not indicate the extent of non-sampling error.
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62. 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 >
63. 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 >
64. Answer : (b)
Reason : In the coefficient of correlation between two variables lies between –1 and 0 then the covariance
between them is negative.
< TOP >
65. Answer : (a)
Reason : Coefficient of determination represents the proportion of variation in the dependent variable that is
explained by the regression line.
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66. 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 >
67. Answer : (d)
Reason : The statement that there are always as many points above the fitted line as there are below it is
wrong.
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68. 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 >
69. 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 >
70. Answer : (c)
Reason : (a) and (b) are not the indicators of multioptimality.
(c) In the graphical method of solving a LPP the situation of multioptimality arises when the
objective function is parallel to one of the edges of the feasible region which is in the direction
of the optimal movement of the objective function.
(d) and (e) are not correct because they only assume non-negative values.
< TOP >
71. Answer : (a)
Reason : a. A decision tree is a graphical method which represents states of nature and courses of action.
b. A histogram is a graphical representation of a frequency distribution.
c. A scatter diagram shows the distribution of data points in regression analysis.
d. A frequency distribution is a distribution of data along with their frequencies.
< TOP >
e. A probability distribution is a distribution of values of a random variable along with their
respective probabilities
72. Answer : (e)
Reason : The coefficient of correlation is defined as
Now, cov(x, y) > 0 and the standard deviation is the positive square root of variance. ∴ r > 0 and
hence 0 < r ≤ 1 (because r cannot be more than 1).
( )
X y
cov x, y
r =
σ ⋅σ
< TOP >
73. 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.
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.
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.
For two dependent events A and B, the joint probability of the events A and B is not always equal to
1.
< TOP >
74. Answer : (a)
Reason : If P(B) > P(A) then – P(A) > – P(B) i.e. 1 – P(A) > 1– P(B) i.e. > Hence, the
answer.
P(A) P(B)
< TOP >
75. Answer : (a)
Reason : We have P(S) = P(T) = x (say) and P(S and T) = p while the events are independent. Therefore, x2
= p and hence, we get x = . Hence, the option (a) is correct. p
< TOP >
76. 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 >
77. 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 >
78. Answer : (c)
Reason : Coefficient of variation = (standard deviation / mean) × 100
(a) If the standard deviation is equal for all sets of data, we can use coefficient of variation as a
measure of variability meaningfully.
(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 means are equal for all sets of data, the c.v. can be meaningfully used for comparison
of variability.
< TOP >
79. Answer : (e)
Reason : Bienayme-Chebyshev theorem is applicable to any type of distribution regardless of its shape.
< TOP >
80. Answer : (d)
Reason : The middle value of a series arranged in any of the orders of magnitude (ascending or descending) is
< TOP >
called Median.
81. 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 >
82. Answer: (b)
Reason: Number of permutations of five things taking 2 things at a time = = 20
P2
5
< TOP >
83. Answer : (b)
Reason : If every item in a data set is increased by a constant C, then the arithmetic mean of the resulting data
set will be equal to the mean of the original data set plus C, i.e., C + mean of the original data set.
< TOP >
84. 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 >
85. Answer : (e)
Reason : The relationship between arithmetic mean (A), geometric mean (G) and harmonic mean (H) of two
numbers if given by
G2 = AH.
Also
And
From the above we can derive the identities stated in the alternatives.
Hence the answer is (e).
G2 = XY
1 1 1 1
- = -
H X Y H
< TOP >
86. Answer : (b)
Reason : Mode represents the most frequently occurring item in a data set.
< TOP >
87. 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 >
88. Answer : (b)
Reason : a. The consecutive terms of the G.P. will be in increasing order if the first term in a geometric
progression is greater than one and the common ratio is more than 1.
b. The consecutive terms of the G.P. will be in decreasing order if the first term in a geometric
progression is greater than one and the common ratio is less than 1.
c. If the first term in a geometric progression is greater than one and the common ratio is less than
1, then the consecutive terms will not be the same.
d. All the consecutive terms will be less than the common ratio, if the first term is less than or
equal to the common ratio and the common ratio is less than one.
e. All the consecutive terms will be less than 1 if the first term as well as the common ratio is less
than 1.
< TOP >
89. Answer: (d)
Reason: = =
Cr n
n!
n − r!r! P r n
n!
n − r!
< TOP >
< TOP OF THE DOCUMENT >
∴ = ÷ r!
Cr n P r n