Quantitative Methods

Question Paper

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 (d) Pie chart

< Answer >

(b) Line chart (c) Pictogram (e) Scatter diagram. (1 mark)

2.

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Which of the following is not true about median? (a) (b) (c) (d) (e)

Median is not strongly affected by the extreme values It can also be calculated from grouped data It can also be calculated for qualitative data It can not be computed for grouped data having open-ended classes It involves a time consuming process for raw data as the data has to be arranged. (1 mark)

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 (d) Median

(b) Geometric Mean (e) Mode.

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(c) Arithmetic Mean (1 mark)

4.

If every item in the data set is increased by the same quantity, then the standard deviation of the data set (a) (b) (c) (d) (e)

< Answer >

Remains the same Increases by the same quantity by which every data item is increased Decreases by the same quantity by which every data item is increased Increases by the square root of the same quantity by which every data item is increased Decreases by the square root of the same quantity by which every data item is increased. (1 mark)

5.

< Answer >

The logarithm of a number (a) (b) (c) (d) (e)

Is always expressed with respect to base 10 Is always expressed with respect to base 1 Is always expressed with respect to base e Is equal to the base which must be raised to a given exponent in order to get the number Is equal to the exponent to which a given base must be raised in order to get the number. (1 mark)

6.

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Which of the following measures is based only on two observations in a data set? (a) Arithmetic mean (d) Mean absolute deviation

(b) Harmonic mean (c) Range (e) Standard deviation. (1 mark)

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 1

< Answer >

(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 (c) Either a positive or negative value (e) A multiple of the arithmetic mean.

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(b) A negative value (d) Equal to zero (1 mark)

9.

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Which of the following measures represents the scatter of the values in a data set? (a) Arithmetic mean (d) Median

(b) Geometric mean (c) Mode (e) Standard deviation. (1 mark)

10. The slope of the simple linear regression equation (X is the independent variable and Y is the dependent variable) represents the (a) (b) (c) (d) (e)

< Answer >

Mean value of Y when X = 0 Change in mean value of Y per unit change in X True value of Y for a fixed value of X Variance of the values of X 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

(d) 0 to ∞

(c) –1 and 0

(e) –∞ to ∞. (1 mark)

13. Which of the following is true with regard to a given coefficient of correlation and its corresponding coefficient of determination? (a) (b). (c). (d). (e).

< Answer >

The coefficient of determination is always greater than or equal to zero The coefficient of determination is always negative The coefficient of determination is always zero The coefficient of determination always has the same sign as the coefficient of correlation 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)

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 2

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(b) The sum of the squared values of the vertical distances from each plotted point to the line is minimum (c) The sum of the squared values of the horizontal distances from each plotted point to the line is maximum (d) The sum of the squared values of the horizontal distances from each plotted point to the line is minimum (e) The sum of the squared values of the vertical distances from each plotted point to the line is equal to zero. (1 mark) < Answer >

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) (b) (c) (d) (e)

Base year values are used as weights Current year values are used as weights Base year prices are used as weights Current year prices are used as weights Sum of base year and current year quantities are used as weights. (1 mark) < Answer >

19. The value index number measures the (a) (b) (c) (d) (e)

Change in prices of a basket of commodities from one period to another Change in quantities consumed of a basket of commodities over a period of time Change in the total monetary value of a basket of commodities over a period of time Change in the retail prices of various commodities Change in the general price level in a country. (1 mark)

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 (c) Fisher’s ideal price index (e) Weighted aggregates index.

< Answer >

(b) Paasches price index (d) Fixed weight aggregates price index (1 mark)

21. If all the terms of an arithmetic progression are multiplied by a constant quantity the resulting terms will always form (a) (b) (c) (d) (e)

< Answer >

A geometric progression A harmonic progression An arithmetic progression Either a geometric progression or a harmonic progression Either a geometric progression or an arithmetic progression. (1 mark)

22. Which of the following is true with regard to Fisher’s ideal price index? (a)

It does not consider the base year prices 3

< Answer >

(b) (c) (d) (e)

It does not consider the base year quantities It does not consider the current year prices It does not consider the current quantities It is the geometric mean of the Laspeyres’ and Paasche’s price indices. (1 mark) < Answer >

23. Two events are said to be mutually exclusive if (a) (b) (c) (d) (e)

The sum of their probabilities is less than 1.00 The sum of their probabilities is greater than 1.00 They contain every possible outcome of an experiment They cannot occur at the same time The sum of their probabilities is equal to zero. (1 mark)

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 (c) Dependent (e) Non-mutually exclusive.

< Answer >

(b) Independent (d) Mutually exclusive (1 mark) < Answer >

25. Bayes’ theorem helps the statistician to calculate (a) Subjective probability (c) Posterior probability

(b) Classical 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) (b) (c) (d) (e)

It assumes that the outcomes are not equally likely The probability of an event is determined after performing the experiment large number of times The probability of an event is determined before performing the experiment It assumes that all possible outcomes of the experiment are not known The classical approach cannot be used to find out the probability of mutually exclusive events. (1 mark)

27. In a binomial distribution the probability of getting zero or more number of successes is equal to (a) (b) (c) (d) (e)

< Answer >

0 The probability of getting zero success The probability of getting successes in all the trials 1 minus the probability of getting successes in all the trials 1. (1 mark) < Answer >

28. Events A and B are dependent. The joint probability of the events A and B is (a) (b) (c) (d) (e)

Equal to the product of the marginal probabilities of the events A and B Not equal to the product of the marginal probabilities of the events A and B Equal to the sum of the marginal probabilities of the events A and B Equal to the difference between the marginal probabilities of the events A and B Is always equal to 1. (1 mark)

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) (b) (c) (d) (e)

Joint probability of events A and B Conditional probability of event B given event A Marginal probability of event B Marginal probability of event A Zero. (1 mark) 4

< Answer >

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 (c) Histogram

(b) Frequency polygon (d) p chart

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(e) Ogive. (1 mark)

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 (d) Continuous distribution

< Answer >

(b) Uniform distribution (c) Normal distribution (e) Hypergeometric distribution. (1 mark) < Answer >

32. A continuous random variable is one which (a) (b) (c) (d) (e)

Can assume only integer values Can assume only even values Can assume only odd values Can assume any value within a given range Can assume only a limited number of values. (1 mark) < Answer >

33. Which of the following is true about binomial distribution? (a) (b) (c) (d) (e)

It is a continuous distribution The probability of the outcome of any trial varies over time Each trial will have more than two possible outcomes The outcomes of the trials are statistically independent of each other 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) (b) (c) (d) (e)

It is the geometric average of all possible outcomes of the variable It is the simple average of all possible outcomes of the variable It is the weighted average of all possible outcomes of the variable It is the outcome of the variable, which has the highest probability of occurrence 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) (b) (c) (d) (e)

Means of individual populations Observations within a population Observations within a sample Means of all possible samples of a specific size taken from a population 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) (b) (c) (d) (e)

It is less than the standard deviation of the population It decreases as the sample size increases It measures the variability of the mean from sample to sample It is the standard deviation of the sampling distribution of mean It is the standard deviation of the sample. (1 mark)

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 (d) Systematic sampling

(b) Cluster sampling (e) Judgmental sampling. 5

(c) Stratified sampling

< Answer >

(1 mark) 38. In which of the following conditions a Type I error is said to have occurred, in testing a hypothesis? (a) (b) (c) (d) (e)

< Answer >

The sample statistic is incorrectly calculated The sample variance is incorrectly calculated Null hypothesis is rejected though it is true Null hypothesis is accepted though it is false The sample size is small. (1 mark)

39. In the graphical method of solving linear programming problems if there is a unique optimal solution, then the optimal solution (a) (b) (c) (d) (e)

< Answer >

Is always found at the center of the feasible region Is always at the origin Lies outside the feasible region Is located at one of the corner points of the feasible region Always lies on one of the two axes. (1 mark)

40. In the graphical method of solving linear programming problems the feasible region is the set of all points (a) (b) (c) (d) (e)

< Answer >

Which do not satisfy any of the constraints Which satisfy exactly one of the constraints Which satisfy all the constraints At which the objective function has the same value At which the objective function is equal to zero. (1 mark)

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

< Answer >

(e) 5.0. (2 marks)

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

(b) 1

3

5

(c) 3

5

(d) 5

3

< Answer >

(e) 2 .

3

(1 mark) 43. The product of three numbers in geometric progression is 125 and the sum of their products taken in pairs is 87 (a) 5

1

< Answer >

. The highest of the three numbers is

2

(b) 7

(c) 10

(d) 12

(e) 15. (2 marks)

44. The pth term of an harmonic progression is qr and the qth term is rp. The rth term of the harmonic progression is

p (a) q

(b) pq

q (c) p

rp (d) q

< Answer >

rq (e) p . (2 marks)

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

< Answer >

(e) q. (2 marks)

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 6

< Answer >

first 15 terms of the series is (a) 110

(b) 144

(c) 196

(d) 225

(e) 289. (2 marks)

st

th

th

47. The 1 term of an harmonic progression is 5/7 and 4 term is 10/29. The 20 term of the harmonic progression is (a) 10

81

(b) 10

(c) 21

109

(d)

61

25 71

< Answer >

(e) 10 .

89

(1 mark) 48.

If

n

C10 = n C15

(a) 165

, the value of

27

< Answer >

Cn is

(b) 190

(c) 220

(d) 351

(e) 2925. (1 mark)

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

< Answer >

(e) 6480. (1 mark)

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) 1200.

252

(b) 450

(c) 780

< Answer >

(d) 920 (e) (1 mark)

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

< Answer >

(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)

53.

< Answer >

log a log b log c If = = ;the value of aabbcc would be (b - c) (c - a) (a - b) (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)

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

7

10

< Answer >

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%

< Answer >

(e) 15.6%. (2 marks)

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.

< Answer >

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)

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

< Answer >

(e) 6. (1 mark) < Answer >

59. The following distribution shows the ages of 100 persons in a group: Age group (in years) 20 - 25 25 - 30 30 - 35 35 - 40 40 - 45 45 - 50 50 - 55 55 - 60 60 - 65

Number of persons 3 16 22 18 14 10 7 6 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)

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 years.

(b) 9.3 years

(c) 10.2 years

(d) 10.9 years

< Answer >

(e) 11.5 (1 mark)

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

(d)

3

4 9

(e)

5. 9 (1 mark)

8

< Answer >

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

< Answer >

(e) 8/225. (1 mark)

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

< Answer >

(e) 0.55. (2 marks)

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 , 2 , 1 , 1 and 3 respectively. 5 3 6 2 8

< Answer >

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)

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%

< Answer >

(e) 50%. (1 mark)

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

< Answer >

(e) 0.6354. (1 mark)

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

< Answer >

(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) 69. The following details are available with regard to a test of hypothesis for the population mean: 9

< Answer >

H 0: µ = 60 x = 72 σ = 60

Test statistic = 1.20 What is the sample size? (a) 6

(b) 10

(c) 12

(d) 36

(e) 72. (1 mark) < Answer >

70. The following details are available with regard to a hypothesis test on population mean: H0: µ = 10 H1: µ > 10 n = 64

σ 2 = 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) (b) (c) (d) (e)

There is insufficient information for doing the test The t distribution should be used The test does not lead to either type I or type II error The test leads to a type I error 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: H0 : µ = 20 H1 : µ ≠ 20 σ2 = 81 n = 36

The null hypothesis is rejected if type I error? (a) 1.00.

x ≤ 17.06 or x ≥ 22.94 . What is the probability of committing a

0.5

(b) 0.05

(c) 0.025

(d) 0.005 (e) (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)

73. The following details are available with regard to a regression analysis (Y is the dependent variable) with 5 observations: Y

18 20

ˆ Y

24 22

20 18

25 26

32 30

What is the standard error of estimate? (a) 2.38

(b) 7.67

(c) 17

(d) 39.5

(e) 289. (1 mark)

10

< Answer >

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:

< Answer >

ˆ − Y ) = 8750 Σ(Y 2

ˆ ) = 2250 Σ(Y − Y 2

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

Σ (X − X)

2

=

2500

Σ (Y − Y)

2

=

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)

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

Q1 Q0

P0Q0

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

11

< Answer >

(e) 123.08. (1 mark)

< Answer >

79. The following data pertains to the consumption of materials by a bakery. 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) Inputs

Units

< Answer >

80. The following information are available with regard to a basket of goods:

ΣP Q 1 1 = 0.8125 ΣP Q 0 0

ΣP Q 1 0 = 1.9417 ΣP Q 0 1 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)

81. Calculate the coefficient of correlation between variables X and Y, from the following data:

(a) 0.91

X

9

8

7

6

5

4

3

2

1

Y

15

16

14

13

11

12

10

8

9

(b) 0.93

(c) 0.95

(d) 0.97

< Answer >

(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) 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

< Answer >

(e) 12. (1 mark)

84. The estimated regression relationship between variables X (X is the independent variable) and Y

< Answer >

ˆ

is: Y = 6+ 3X Let M = Y + 2 What will be the estimated regression relationship between M and X (X is the independent variable)?

ˆ = 6 + 2X (a) M

ˆ = 8 + 3X (b) M

ˆ = 6 + 3X (c) M

ˆ = 8 + X (e) M ˆ = 6 + X. (d) M (1 mark)

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)? 12

< Answer >

(a) 4.096

(b) 16.779

(c) 20.974

(d) 4.58

13

(e) 4.32. (1 mark)

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)

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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).

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3.

Answer : (c)

< TOP >

Sum of all items n Reason : A =

∴ Sum of all items = nA.

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.

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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.

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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.

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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.

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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.

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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.

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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 >

14

d. e.

Y for a fixed value of X. The slope of the simple regression equation does not represent the variance of the values of X. 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).

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12.

Answer : (b) Reason : The coefficient of correlation can take values in the range of –1 to 1. Therefore the correct answer is (b).

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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)

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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.

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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).

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16.

Answer : (c) Reason : a. not be 1. b.

< TOP >

c. d. e.

When the slope of a regression line is negative the correlation coefficient need 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. When the slope of a regression line is negative, there is a negative correlation between the variables. When the slope of a regression line is zero, the regression line will be parallel to the horizontal axis. When the y-intercept of a regression line is zero, the regression line passes through the intersection of horizontal and vertical axes

17.

Answer : (c) Reason : Any multiple regression equation consists of only one dependent variable and more than one independent variables.

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18.

Answer : (a)

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15

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).

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20.

Answer : (b) Reason : Paasche price index uses the quantity measures for the current period as weights. Therefore the correct answer is (b)

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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. b. c. d. e.

< TOP >

Fisher’s ideal price index considers base year prices. Fisher’s ideal price index considers base year quantities. Fisher’s ideal price index considers current year prices. Fisher’s ideal price index considers current year prices. Fisher’s ideal price index is the geometric mean of the Laspeyres and Paasche’s price indices

23.

Answer : (d) Reason : If two events cannot occur at the same time then they are called mutually exclusive events.

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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.

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25.

Answer : (c) Reason : Baye’s theorem helps the statistician to calculate posterior probability.

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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.

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27.

Answer : (e) Reason : In a binomial distribution the probability of getting zero or more number of

< TOP >

16

= 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.

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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.

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30.

Answer : (a) Reason : a.

< TOP >

b.

c. d. e. 31.

Answer : (e) Reason : a. b. c. d. e.

The graphical plot of the values of the dependent and independent variables, in the context of regression analysis, is called scatter diagram. 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. A histogram is a graphical representation of a frequency distribution. A p chart is a quality control chart. An ogive is a graphical plot of a cumulative frequency distribution < TOP >

The data may follow a binomial distribution if the probability of success is constant from trial to trial. The data may follow a uniform distribution if the probabilities of the outcomes of the trial are equal. The data may follow a normal distribution if the probability of success is constant from trial to trial. The data may follow a continuous distribution if the outcome is any value within a given range of values. 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.

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.

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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).

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34.

Answer : (c) Reason : a.

< TOP >

b. c. d. e.

The expected value of a discrete random variable is not a geometric average of the outcomes of the variable. The expected value of a discrete random variable is not a simple average of the outcomes of the variable. The expected value of a discrete random variable is a weighted average of the outcomes of the variable. The expected value of a discrete random variable is not the outcome, which has the highest frequency. The expected value of a discrete random variable is not the highest probability 17

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.

< TOP >

b. c. d. e.

The standard error of mean is less than the population standard deviation(σ) because it is equal to σ/√n. From above we can see that it will decrease as the sample size increases. The standard error is a measure of the variability of the mean across various samples of the same size taken from the population. 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. From above we can see that the standard error of mean is not the standard deviation of the sample.

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. d.

< TOP >

A type I error occurs if the null hypothesis is rejected though it is true. 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.

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.

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40.

Answer : (c) Reason : The feasible region is the set off all points which satisfy all the constraints in the LPP.

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18

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

< TOP >

(m - 2 d) 2 + (m - d) 2 + (m + d) 2 + (m + 2 d) 2 = 276 Or, m 2 - 6 md + 9 d 2 + m 2 - 2 md + d 2 + m 2 + 2 md + d 2 + m 2 + 6 md + 9 d 2 = 276 Or, 4 m 2 + 20 d 2 = 276 Putting m = 8 we get 4 × 64 + 20 d 2 = 276 20 d 2 = 276 - 256 = 20

Or, Or,

d2 = 1

Or,

d = ±1

So the lowest number would be 8 - 3 ×1= 5 42.

< TOP >

Answer : (c) Reason :

a ( r 3 − 1) S3 S8

=

( r − 1)

a ( r − 1) 6

=

a ( r 3 − 1) a ( r − 1) 6

=

( r 3 − 1)

=

125

( r + 1)( r − 1) 152 3

3

( r − 1) Or ,

1 ( r + 1) 3

=

125 152

Or , (125r + 125) = 152 3

Or , 125r = 27 3

Or , r =

27

3

Or , r =

43.

125 3 5 < TOP >

Answer : (c) Reason :

19

m

Let the numbers be

, m, mr.

r

The product of the numbers is m = 125; or, m = 5. 3

The sum of the products in pair is m r

.m + m.mr+ mr.

1



m r

175

=

2

175

25  + r + 1  = r  2 2

Or, 2r + 2r + 2 = 7r 2

Or, 2r - 5r + 2 = 0 2

Or, 2r - 4r - r + 2 = 0 Or, 2r(r - 2) -1(r - 2) = 0 Or, (r - 2)(2r - 1) = 0 So, r = 2 or,

1 2

So the highest number is 5 × 2 = 10

44.

< TOP >

Answer : (b) 1

1

Reason : The pth term of the A.P. is qr i.e. A + (p – 1)d = qr 1

1

The qth term of the A.P. is pr i.e. A + (q – 1)d = pr By subtracting the two equations we get 1

1

(p - q)

(p – q)d = qr – pr =

rpq

1

Or, d = rpq (r - p)

The rth term would exceed the pth term by (r – p)d. i.e. 1

+

(r - p)

rpq So the value of rth term is rq So the rth term of the H.P. is pq.

45.

=

1 rq

+

1 pq

+

1 rq

rpq =

pq < TOP >

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 bc ⇒ q2 = bc

q is the G.M. between b and c ⇒ q = ∴ p 2 + q2 = or p2 + q2 = or p2 + q2 = or p2 + q2 =

1

ab + bc b(a + c) b(2b) 2b2

20

p2 + q 2 2

or b2 =

∴ b2 is the A.M. between p2 and q2. 46.

< TOP >

Answer : (d) Reason :

S8 = ( 2 a + 7 d ) 4 = 64 Or, 2 a + 7 d = 16 ..................(i) S19 = ( 2 a + 18 d )

19

= 361 2 Or, 2 a + 18 d = 38 .................(ii) By (ii) - (i) we get

11d = 38 -16 = 22 Or, d = 2 So, the common difference is 2. Putting the value of d in equation (i) we get

( 2 a + 7 × 2 ) 4 = 64 Or, a =1 So, the sum of first15 terms is

( 2 +14 × 2 ) 47.

15 2

=

30 ×15 2

= 225

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 3d =

29 10

7

−

5

=

(29 - 14) 10

So the value of d =

=

< TOP >

3 2

1 2

7 5 7

So the 20th term in the A.P. is 5

+ +

1

× (20 − 1)

2

19 2

=

(14 + 95) 109 = 10 10 10

So the corresponding term in H.P would be 109 48.

< TOP >

Answer : (d) Reason : n

n

We know c r = c ( n-r) n

n

So, when c10 = c15 That indicates if r = 10 ; (n-r) = 15 i.e. (n - 10) = 15 ∴ n = 10+ 15 = 25 So,

27

cn =

27

c 25 =

27 × 26 1× 2

= 351 21

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

< TOP >

3

given word in P2 ways. Remaining six positions are to be filled with six letters in 6! ways. So, total number of word is P2 .6! = 2 × 3 × 720 = 4320 . 3

50.

< TOP >

Answer : (e) 5

Reason : 2 professors can be selected from 5 professors in

C2 =

5× 4 1× 2

= 10

ways. On the 10 × 9 × 8 10 C3 = = 120 1× 2 × 3 other hand, 3 students can be selected from 10 students in ways. So, the 120 × 10 = 1200 ways this committee can be formed in the given condition.

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 10 × 11 × 12 × 13 13 C9 = 1× 2 × 3 × 4 or 715 times.

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52.

Answer : (d)

< TOP > 4

P3 × 5 P3 × 6 P3

= 24 × 60 × 120 = 172800

Reason : 53.

< TOP >

Answer : (d) Reason : Let

log a

=

(b-c)

log b (c-a)

=

log c (a-b)

=k

∴ 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 a

b

c

So, a b c = 1 a

b

c

54.

Answer : (b) Reason : log3 (2x – 1) = 4 implies 2x-1 = 34 2x = 81 + 1 = 82 x = 41.

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55.

Answer : (c)

< TOP >

 (N + 1) / 2 − (F + 1)  Lm +  W fm   Reason : Median = th

500 + 1 N +1 2 2 The position of the median is the item i.e., = 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 22

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

 501/ 2 − (170 + 1)    × 50 160  ∴ Median = 350 +  = Rs.374.84. 56.

< TOP >

Answer : (b) Reason : Let the average and standard deviation of the original data let be x and s. Σ( x − 5) Σx Σ5 n5 − x− n = n = x – 5. n = n Average of all items ‘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).

s × 100 Given: x − 5 = 10 10 x – 50

or

100s =

or

10 x – 100s

=

50 …… ………

(A)

Σ( x + 5) Σx Σ5 n5 + x+ n = n = x +5 n = n Average of all items ‘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 or

:

100s =

s × 100 x+5 =6 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

=

or x =

80 4 = 20

80

∴ 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

=

150 100 = 1.50

∴s = 1.50.

s Coefficient of variation of the original data set = x × 100 1.50 × 100 = 20 = 7.5%.

23

57.

< TOP >

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 ∴ X 2 : Average salary of sales men with 10 years experience. By the question –

N1 × 7000 + N 2 X 2 = 9000 N1 + N 2

N1 N2 (7000) + X 2 = 9000 N1 + N 2 or N1 + N 2

or (0.6 × 7000) + (1– 0.6) X 2 = 9000 or 0.4 X 2 = 9000 – 4200 = 4800 or 58.

X2 =

4800 = Rs.12000. 0.4 < TOP >

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.

59.

< TOP >

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

Σf × m 3930 = 100 = 39.3 years Σf ∴ Mean = 60.

< TOP >

Answer : (c) Reason : Subgroup A : NA

2 × 150 = 2+5+3 = 30

x A = 8 years

24

Subgroup B : NB

5 × 150 2 + 5+3 = = 75

x B = 10 years

Subgroup C : NC

3 × 150 = 2+5+3 = 45

x C = 12 years

( N A .x A ) + ( N B .x B ) + ( N C .x C ) NA + NB + NC

∴ Average age of the entire group =

(30 × 8) + (75 × 10) + (45 × 12) 1530 30 + 75 + 45 = = 150 = 10.20 years.

61.

Answer : (b) Reason : Probability that the first dice gives an even number less than 5 and the second dice 2 2 1 × 6 6 9 gives an odd number less then 4 = =

< TOP >

Probability that the second dice gives an even number less than 5 and the first dice 2 2 1 × 9 gives an odd number less than 4 = 6 6 = 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 1 2 9+9 = 9 62.

Answer : (c) Reason : If each ball is not replaced, then R, W, and B are dependent events and

< TOP >

P{RWB} = P{R} P{W I R} P{B I WR} = (6/15)(4/14)(5/13) = 4/91. 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:

< TOP >

(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. 5

C1 × C1 3

8

Therefore, P(B/A1) = 4

Similarly, P(B/A2) =

=

C2

C1 × C1 5

9

C2

=

5× 3× 2 8× 7

5× 4× 2 8× 9

=

=

15 28 .

5 9.

Therefore, the required probability will be P(B) = P(A1) × P(B/A1) + P(A2) × 1 15 1 5 275 × + × P(B/A2) = 2 28 2 9 = 504 = 0.5456 ≅ 0.55 (approx.). 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. 25

< TOP >

Given: 2 2 1 3 1 P(S) = 2 P(T) = 8 P(P) = 5 P(Q) = 3 P(R) = 6 Probability that none of the students will be able to solve the problem

=

3 1 5 1 5 5  2  2  1  1  3  1 − 1 − 1 −  1 −  1 −  = × × × × = 5 3 6 2 8 96  5  3  6  2  8 

5 91 96 96 ∴ Probability that the problem will be solved = 1 – = = 0.948 i.e. 94.8%.

65.

Answer : (a) Reason : Percentage of the time will the station be shut down = P(x > 45)

< TOP >

45 − 25   Pz >  7  = P(z > 2.86) = 1 - (0.5 + 0.4979) = 0.0021 i.e. 0.21% = 

66.

< TOP >

Answer : (b) Reason : Given that p= 0.60 , q= 0.40, n = 4, n r n −r P(r) = Cr p q 4 3 4 −3 P (3) = C3 × (0.60) × (0.40) = 0.3456

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) =

< TOP >

4 2 = 6 3

Let Y denote the number of throws in which success ( X ≥ 3 ) happens. Y follows a binomial 2 distribution with p = 3 and number of trials = 20

P(Y ≤ 17) = 1 - [P(Y = 18) + P(Y = 19) + P(Y = 20)] = 1 - [ 20 C18 p18 (1 - p) 2 + 18

20

C19 p19 (1 - p) +

2

19

20

C20 p20 ] 20

 2 1  2 1  2 = 1 - [190 ×     + 20 ×     +   ]  3  3  3  3  3  = 1 - [ 0.01428 + 0.00301 + 0.0003] = 0.98241 68.

< TOP >

Answer : (d) Reason : H0

:

µ

=

9

µ ≠ 9 H1 : 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.

s σx

z

= =

256 25

n = x −µ σx

=

=

3.20

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 26

region. ∴ We reject H0. But the true mean is 9. So H0 is true. Hence the test leads to a type I error. 69.

< TOP >

Answer : (d) Reason : H0 : µ = 60

x

=

72

Test statistic

Or

1.20

=

x −µ σx

=

72 − 60  60     n

=

72 − 60 1.20

60 n

Or Or

70.

n

=

 60     10 

=

10

=

36.

2

< TOP >

Answer : (c) Reason : H0

:

µ

=

10

H1 : µ > 10 The sample is large. So we shall use the normal distribution.

σ σx

z

= =

256 64

n = x −µ σx

=

=

2

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. 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 = σ2 n Hence Probability of rejecting the null hypothesis when it is true =

P(x ≤ 17.06 or x ≥ 22.94) = P(x ≤ 17.06) + P(x ≥ 22.94)         17.06 − 20 22.94 − 20  + Pz ≥  = Pz ≤ 2 2     σ σ     n n             2.94 − 2.94  + Pz ≥  = Pz ≤   81  81      36  36    = P(z ≤ −1.96) + P(z ≥ 1.96) = 0.025 + 0.025 = 0.05

27

< TOP >

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

P ( x > 51)

=

0.1587

=

=

< TOP >

σ n

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

1

x −µ σx

=

or

1

=

σ   1

σ ∴ n 73.

=

1

or

n

=

51 − 50 σx

or

σx

σ2

=

36.

1 1

=

2

=

< TOP >

Answer : (a) Reason :

=

yˆ represents the estimated value of the dependent variable(y)

Se =

ˆ 2 Σ(Y − Y) n−2

Y

18

24

20

25

32

ˆ Y

20

22

18

26

30

ˆ Y−Y

–2

2

2

–1

2

ˆ 2 (Y − Y)

4

4

4

1

4

ˆ 2 Σ (Y − Y) = 17 ∴ S.E. = 74.

17 5 − 2 = 2.380. < TOP >

Answer : (c) Reason : Coefficient of determination

=

RSS RSS = TSS ESS + RSS

ˆ − Y) Σ (Y

75.

=

ˆ ) + Σ (Y ˆ − Y) Σ (Y − Y

=

8750 = 0.7955 2250 + 8750

2

2

Answer : (a) Reason : The slope of the regression line with X as the independent variable Σ (X − X) (Y − Y)

= 76.

2

2 Σ (X − X)

=

< TOP >

ΣXY − nXY 1200 = = 0.48 2500 ΣX 2 − nX 2 < TOP >

Answer : (d) Reason : Unweighted aggregates price index =

Or

ΣP1 × 100 ΣP0

 ΣP1     n  × 100  ΣP0     n 

28

=

=

125

125

P1

Or Or 77.

× 100

=

P0 =

P1 ×100 125 = =

125 75 × 100 125

Rs.60. < TOP >

Answer : (b) Reason : Laspeyres price index

ΣP1 Q 0 × 100 = ΣP0 Q 0

= 78.

P0

2000 × 100 1600

=

(320 + 600 + 700 + 380) × 100 (160 + 500 + 560 + 380)

=

125. < TOP >

Answer : (c) Reason : Paasches price index = For each good : P1Q1 P0Q1

ΣP1 Q1 × 100 ΣP0 Q1

=

Q1 Q P1Q0 × 0

=

Q1 P0Q0 × Q 0

∴ We can rewrite from the table: Good

P1Q0

P0Q0

Q1 Q0

Q1 P1Q1 = P1Q0 × Q 0

Q1 P0Q1 = P0Q0 × Q 0

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

ΣP1 Q1 × 100 ∴ Paasches price index = ΣP0 Q1 =

79.

2600 × 100 2060 = 126.21. < TOP >

Answer : (d)  P1

 x100   0  n Reason : The unweighted average of relatives price index for the given data is 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.

∑ P

80.

81.

< TOP >

Answer : (b)

Reason : Fisher’s ideal price index =

ΣP1 Q1 ΣP1 Q 0 × × 100 ΣP0 Q1 ΣP0 Q 0

=

ΣP1 Q1 ΣP1 Q0 × × 100 ΣP0 Q 0 ΣP0 Q 0

=

0.8125 × 1.9417 × 100 = 125.60.

Answer : (c) Reason : We use the following table to calculate the correlation coefficient: 29

< TOP >

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

∑X = 45

∑x= 0

∑ x2 = 60 ∑Y= 108 ∑y = 0

∑y2= 60 ∑xy =57 ∑ xy

We have X = 45 / 9 = 5; Y = 108/9 = 12 and r = 82.

∑x ∑y 2

2

=

57 60 × 60

= 0.95 < TOP >

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

∑X = 405

∑Y = 406 ∑X2 = 22237 ∑Y2 = 22658 ∑XY= 21886

n ∑ XY − ∑ X ∑ Y 8 × 21886 − 405 × 406 = 2 2 8 × 22237 − (405) 2 = 0.7684 b = n ∑ X − (∑ X)

a = Y − bX = (406/8) – 0.7684 × (405/8) = 11.85 Therefore the equation of the regression line is,

ˆ = 11.85 + 0.7684 X. Y The expected blood pressure of a person aged 49 years is 49.502. 83.

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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 =

X − µ 45 − µ = = − 0.5 σ σ or 45 = µ – 0.5σ …………(i)

∴z Now, 8.08% of the items are above 64. Therefore, area to the right of the ordinate at 30

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 − µ 64 − µ = = 1.4 σ σ or 64 = µ + 1.4σ …………….(ii)

=

From equation (i) and (ii), we get the values µ = 50 and σ = 10. 84.

< TOP >

Answer : (b)

ˆ Reason : Y

=

a + bX = 6 + 3X

ˆ M

=

a ′ + b′X

M=Y+2

nΣ XM − ΣX ΣM nΣX 2 − (ΣX) 2

b′

=

=

nΣ X(Y + 2) − ΣX Σ(Y + 2) nΣX 2 − (ΣX) 2

nΣXY + nΣ 2X − ΣX ΣY − ΣX Σ 2 nΣX 2 − (ΣX) 2

=

nΣXY − ΣX ΣY + 2nΣX − 2nΣX nΣX 2 − (ΣX) 2

=

nΣXY − ΣX ΣY nΣX 2 − (ΣX) 2

= a′

= =

ˆ = ∴M 85.

= b =3

M − b′X = ( Y + 2 ) − bX = ( Y − bX ) + 2 a+2

=

6+2 = 8

8 + 3X. < TOP >

Answer : (d) ˆ = a + bX Reason : Y Se =

ESS n−2

RSS 161.21 1 − r ) TSS = (1 − r ) = (1 − 0.70 ) = 167.79 ( r 0.70 ESS = 2

2

2

2

Se =

2

167.79 = 4.58. 8 < TOP OF THE DOCUMENT >

31