Quantitative Methods-II (MB152): July 2006

1
Question Paper

Quantitative Methods-II (MB152): July 2006

· Answer all questions.
· Marks are indicated against each question.
1. If one card is drawn from a pack of well shuffled cards, what is the probability of getting a black
jack?
(a) 1/52 (b) 2/52 (c) 2/26 (d) 3/52 (e)
4/56.
(2 marks)
< Answer >
2. Which of the following is true provided P(A or B) = P(B)?
(a) A and B must be mutually exclusive
(b) A and B must be independent
(c) P(A) + P(B) is the joint probability of A and B
(d) P(A) = P(B)
(e) Occurrence of A implies the occurrence of B.
(1 mark)
< Answer >
3. The probability that a person will die in a street accident is 0.0025 while the same for an air
crash is 0.018. What will be the probability of the death of a person either in a street accident or
in an air crash?
(a) 0.0180 (b) 0.0045 (c) 0.03046
(d) 0.0205 (e) 0.0050.
(1 mark)
< Answer >
4. Let there is a 35% probability that the weight of a person will be 55 Kg, a 45% probability that
the weight of the person will be 45 Kg and there is a 20% probability that the weight of the
person will be 50 Kg, then the mean weight of the person is
(a) 30.5 Kg (b) 52.6 Kg (c) 49.5 Kg
(d) 36.5 Kg (e) 40.2 Kg.
(1 mark)
< Answer >
5. The probability of getting two heads from three tosses of a fair coin is
(a)
1
8 (b)
1
4 (c)
3
8 (d)
1
2 (e)
3
5 .
(1 mark)
< Answer >
6. P(A/B) = 0.60, P(B/A) = 0.50 and P(A and B) = 0.25, then value of P(A or B) is
(a) 0.33 (b) 0.42 (c) 0.56 (d) 0.67 (e)
0.73.
(2 marks)
< Answer >
2
7. Which of the following statements is true?
(a) In probability theory, the result of an experiment is known as activity
(b) The probability of two or more statistically independent events occurring together is equal
to the sum of their marginal probabilities
(c) Bayes’ theorem is normally used to develop probabilities according to the classical
approach
(d) The classical probability approach enables us to determine revised probabilities or posterior
probabilities
(e) The set of all possible outcomes of an experiment is called the sample space of the
experiment.
(1 mark)
< Answer >
8. Let A and B are two events (not mutually exclusive), connected with a random experiment. If
P(A) = 1/4, P(B) = 2/5 and P(A or B) = 1/2. What is the probability of events A and B taking
place together?
(a) 3/20 (b) 3/11 (c) 1/10 (d) 1/4 (e)
1/2.
(1 mark)
< Answer >
9. A man draws 3 balls at random without replacement from a bag containing 6 red and 5 white
balls. What is the chance of getting all the balls red?
(a) 4/33 (b) 3/11 (c) 1/6 (d) 1/4 (e)
1/2.
(1 mark)
< Answer >
10. If A and B are two mutually exclusive events and P(A) = 1/3, P(B) = 2/3, then the probability of
events A and B happening together is
(a) 0 (b) 1/3 (c) 2/3 (d) 1/2 (e) 1.
(1 mark)
< Answer >
11. Mr. Resham Jain constructed a portfolio with 70% stocks and 30% bonds. The rate of return on
stocks was 25% and on bonds was 20%. The standard deviation or risk of investing in stocks
was 20 and that in bonds was 5. The correlation between the returns on stocks and bonds is 0.2.
Find the covariance between returns of the two assets?
(a) 15 (b) 13 (c) 10 (d) 20 (e)
12.
(2 marks)
< Answer >
12. The speed of a race car while crossing the target is approximately normally distributed with
mean of 130 km per hour and a standard deviation of 15 km per hour. What will be the percent
of speed of car that is below 85 km per hour?
(a) 0.47% (b) 0.34% (c) 0.59% (d) 0.13% (e)
0.25%.
(2 marks)
< Answer >
13. The binomial distribution may be approximated to a Poisson distribution, if the probability of
success (p) is ________ and the number of trials (n) is _______.
(a) Large, small (b) Small, large (c) 1, finite
(d) Finite, infinite (e) 1, infinite.
(1 mark)
< Answer >
3
14. The persons X and Y together tosses a coin, then the probability of X’s success is 1/4. The
standard deviation of the number of success in a binomial distribution is given by 2.81 .
Find the probability of obtaining exactly 6 successes?
(a) 0.90 (b) 0.85 (c) 0.090 (d) 0.143 (e)
0.45.
(2 marks)
< Answer >
15. Six coins are tossed 128 times. Using the Poisson distribution, find the approximate probability
of getting six heads 4 times?
(a) 0.650 (b) 0.090 (c) 0.100 (d) 0.010 (e)
0.030.
(1 mark)
< Answer >
16. In Balaji chem. Ltd., there are 20 persons working in drug dispatch section. Out of them, 12
members were experienced and 8 were freshers. After completion of one year 10 members got
promoted. Find what is the probability that 4 of the promoted ones were freshers?
(a) 0.415 (b) 0.650 (c) 0.091 (d) 0.856 (e)
0.350.
(1 mark)
< Answer >
17. Assuming that the probability of the birth of a male child is 1/2, find the probability that in a
family of 5 children there will be at least one boy and one girl?
(a) 11/16 (b) 13/16 (c) 15/16 (d) 29/32 (e)
31/32.
(2 marks)
< Answer >
18. With usual notations, find p for a binomial variate X, if n = 5 and 10 P(X = 4) = P(X = 2).
(a) 0.3090 (b) 0.4780 (c) 0.2345 (d) 0.6784 (e) 1.
(2 marks)
< Answer >
19. In a Journal of 420 pages, there occurred 52 printing errors. Assuming Poisson law for the
number of errors per page, find the probability that a random sample of 21 pages will contain no
error.
(a) 0.740 (b) 0.250 (c) 0.074 (d) 0.025 (e)
0.650.
(2 marks)
< Answer >
20. If the mean of the hypergeometric distribution nM/N is said to be small enough, then the
hypergeometric distribution may be replaced by
(a) t-distribution (b) Poisson distribution
(c) Binomial distribution (d) Normal distribution
(e) Lognormal distribution.
(1 mark)
< Answer >
21. An estimator which has a lower standard error than another is said to be
(a) A more efficient estimator (b) A more consistent estimator
(c) A more sufficient estimator (d) A more unbiased estimator
(e) An accurate estimator.
(1 mark)
< Answer >
4
22. For a sample randomly collected from a population, the following details are available:
Sx = 120 Sx2 = 1650
Number of observations = 12
What is the estimated standard error of mean?
(a) 1 (b) 0.672 (c) 1.864 (d) 1.052 (e)
1.650.
(2 marks)
< Answer >
23. Other things remaining same, as the sample size increases, the interval estimate for the
population mean at a given confidence level will
(a) Increase in width
(b) Remain the same
(c) Increase initially and then decrease in width
(d) Decrease in width
(e) Double in width for increase in sample size by one observation.
(1 mark)
< Answer >
24. A researcher is studying the number of trees that contain a certain type of locust in the Smoky
Mountains each year. He has found that in a year an average of 25% of the trees contain the
locust with a standard deviation of 4%. If 100 trees were randomly sampled, which of the
following statements about the sampling distribution mean is incorrect?
(a) The standard error of the distribution of sample mean is 4%
(b) The mean of the distribution of sample mean is 25%
(c) The standard error of the distribution of sample mean is 0.40%
(d) The shape of the distribution of sample mean is approximately normal
(e) The variance of the distribution of sample mean is 16%2.
(1 mark)
< Answer >
25. A local credit card agency reported to the government that the unpaid balance for their
customers is normally distributed with an average of Rs.5,500 with a standard deviation of
Rs.500. If an auditor randomly sampled 10 of the credit card company's customers, find the
probability that the mean of the sample would be less than Rs.5,700.
(a) 0.50 (b) 0.3971 (c) 0.1029
(d) 0.6029 (e) 0.8962.
(2 marks)
< Answer >
26. If the sampling distribution of a sample statistic has a mean equal to the population parameter
the statistic is intended to estimate, the statistic is said to be
(a) An unbiased estimator of the parameter
(b) Biased estimator of the parameter
(c) A point estimator of the parameter
(d) Having a low standard error
(e) Having a high standard error.
(1 mark)
< Answer >
27. A sample has been taken from a population and the sample mean has been found to be 100. The
upper limit of a 90 percent confidence interval for the population mean is 112. Which of the
following can be known on the basis of the given information?
(a) Population mean
(b) Population variance
(c) Sample variance
(d) Sample size
(e) Lower limit of the confidence interval.
(1 mark)
< Answer >
5
28. The following details are available with regard to a test of hypothesis for the population mean
Sample size = 24
Population size = 280
0
2
H : 42
x 58
1600
m =
=
s =
What is the value of the test statistic?
(a) 1.075 (b) 2.520 (c) 2.046 (d) 4.560 (e)
3.500.
(2 marks)
< Answer >
29. In which type of sampling, the sample is generated by picking up one element after a fixed
interval depending on the sampling ratio. Identify the type of sampling?
(a) Stratified sampling
(b) Cluster sampling
(c) Systematic sampling
(d) Sequential sampling
(e) Simple random sampling.
(1 mark)
< Answer >
30. Which of the following can be directly used as the test statistic in hypothesis tests on the basis
of non-standardized scale?
I. The sample mean, when the test involves the population mean.
II. The difference between two sample means, when the test involves the difference between
two population means.
III. The sample proportion, when the test is about the population proportion.
(a) Only (I) above (b) Only (II) above
(c) Only (III) above (d) Both (I) and (II) above
(e) All (I), (II) and (III) above.
(1 mark)
< Answer >
31. Which of the following is not true?
(a) The standard normal distribution is a normal distribution
(b) The standard normal distribution has a variance of 1
(c) The normal distribution is a continuous probability distribution
(d) The t-distribution remains the same for all sample sizes
(e) The t-distribution approximates the normal distribution as the sample size increases.
(1 mark)
< Answer >
32. Which of the following statements is false?
(a) An interval estimate is a range of values within which the actual value of the population
parameter will always fall
(b) An interval estimate indicates the probability that the true population parameter will fall
within the given range of values
(c) A point estimate may be either correct or incorrect
(d) A point estimate is a single number that is used to estimate an unknown population
parameter
(e) An interval estimate is a range of values within which the estimated value of the population
mean will always fall.
(1 mark)
< Answer >
6
33. A normally distributed population has a known standard deviation of 1.0. What is the width of a
95% confidence interval for the population mean?
(a) 1.00 (b) 1.96 (c)
0.98
(d) 3.92 (e) Cannot be determined from the given information.
(2 marks)
< Answer >
34. Tyres manufactured by a company are designed to provide a mean life of 28,000 miles. Tests
with 30 tyres show a sample mean of 27,500 miles and a sample standard deviation of 1000
miles. It is to be tested whether the mean life is less than 28,000 miles. Which of the following
can be inferred at a 0.05 level of significance? It is assumed that the life of the tyres
manufactured by the company follow a normal distribution.
(a) The average life of the tyres is 28000 miles
(b) The average life of the tyres is less than 28000 miles
(c) The average life of the tyres is more than 28000 miles
(d) The sample mean is incorrect
(e) No conclusion can be drawn on the basis of the given information.
(3 marks)
< Answer >
35. 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 not 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.
(3 marks)
< Answer >
36. The following details are available for a hypothesis test on population proportion:
H0: p = 0.40
H1: p ¹ 0.40
Sample size = 360
Sample proportion = 0.43
Significance level = 0.10
It is later known that the true population proportion is 0.47.
Which of the following can be said with regard to the test?
(a) There is insufficient information for doing the test
(b) The chi-square distribution should be used
(c) The test does not lead to either type I or type II error
(d) The test leads to a type I error
(e) The test leads to a type II error.
(2 marks)
< Answer >
37. Which of the following is/are not required for the two-sample test of means for independent
samples when both samples contain less than 30 observations?
I. Both the populations are normally distributed.
II. Both the populations have equal standard deviations.
III. Both the samples have equal means.
IV. Both the samples are of equal sizes.
(a) Only (I) above (b) Only (II) above
(c) Only (III) above (d) Only (IV) above
(e) Both (III) and (IV) above.
(1 mark)
< Answer >
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38. The following details are available with regard to a hypothesis test on means of two populations:
H0: m1 =m2 +1
H1: 1 2 m ¹m +1
n1 = 64
2
1 s = 256 1 åx= 320
n2 = 36
22
s =144 2 x å =144
The samples collected from the two populations are independent.
What is the value of the test statistic?
(a) 2.828 (b) 1 (c) 0.354 (d) 0.125 (e) 0.
(2 marks)
< Answer >
39. The sample mean is an unbiased point estimator of
(a) The population variance (b) The population mean
(c) The population proportion (d) The standard error of
mean
(e) The standard error of proportion.
(1 mark)
< Answer >
40. The following details are available with regard to a hypothesis test on means of two populations:
H0: 1 2 m -m = 0
H1: 1 2 m -m ¹ 0
n1 = 84 1 x å = 504
n2 = 48 2 x 192 å = 22
s = 144
The value of the test statistic is 0.353553.
The samples collected from the two populations are independent. What is the standard deviation
of the first population? Round off your answer to the nearest integer.
(a) 19 (b) 27 (c) 49 (d) 160 (e)
256.
(2 marks)
< Answer >
41. When the chi-square test is used as a test of independence, the number of degrees of freedom is
determined by
(a) The sample size
(b) The ratio of sample size and the population
(c) Number of rows in the contingency table, only
(d) Number of columns in the contingency table, only
(e) Both the number of rows and the number of columns in the contingency table.
(1 mark)
< Answer >
42.The return on a portfolio is equal to the ______ of the returns on individual assets in the
portfolio.
(a) Median
(b) Mode
(c) Weighted average
(d) Mean
(e) Standard deviation.
(1 mark)
< Answer >
8
43. Let dx and dy represent the deviations from assumed means of the data of variables X and Y
respectively. The data of X and Y are paired. The following results are available:
Sdxdy = 3,044
Number of pairs of observations = 10
Sdx = -170 Sdy = -20
2x
Sd = 8,288
2
Sdy = 2,264
Find out the coefficient of correlation between X and Y variables.
(a) 0.6701 (b) – 0.5602 (c) 0.4503 (d) 0.7804 (e) –
0.4025.
(3 marks)
< Answer >
44. 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 >
45. Which of the following is true with regard to a given coefficient of correlation and its
corresponding coefficient of determination?
(a) The coefficient of determination is always greater than or equal to zero, and less than or
equal to 1
(b) The coefficient of determination is always less than zero
(c) The coefficient of determination is always equal to 1
(d) The coefficient of determination always has the same sign as the coefficient of correlation
(e) The magnitude of coefficient of determination is always higher than the magnitude of
coefficient of correlation.
(1 mark)
< Answer >
46. The following regression relationship between two variables, X and Y, has been obtained:
Yˆ
= 1,236 – 104X
Where X is the independent variable and Y is the dependent variable.
The following details are also available:
SY2 = 19,00,400
Y = 612
SXY = 18,100
Number of observations = 5
What is the approximate 95 percent prediction interval for the value of Y if X = 8?
(a) 1,236 ± 404 (b) 404 ± 46.49 (c) 1,236 ± 104
(d) 404 ± 104 (e) 1,236 ± 46.49.
(2 marks)
< Answer >
9
47. The Excise Department of Patna has gathered information on the number of drinkers and the
income on liquor in the city over a weekend. According to them, there is a close relationship
between the number of drinkers and the income on liquor during weekends.
Number of drinkers(in ’000) 18 28 8 10 20 23 32
Income on liquor(in lakhs) 5 8 3 4 6 7 10
Assuming that there is a linear relation between these two variables, what would be your
prediction about the income on liquor on a weekend during which the number of drinkers rises
to 40 that took place in Patna?
(a) 6.13 (b) 7.68 (c) 8.50 (d) 10.45 (e)
11.56.
(2 marks)
< Answer >
48. If r is the coefficient of correlation between two variables X and Y, then the variation in the
observed Y values which is not explained by the regression line is equal to
(a) r2 (b) 1- r2 (c) r (d) 1 – r (e) 1
+ r2.
(1 mark)
< Answer >
49. The method of least squares indicates the smallest possible
(a) Sum of the squared values of the vertical distances from the regression line to the X-axis at
the corresponding ‘x’ values of the plotted points
(b) Sum of the squared values of the horizontal distances from the regression line to the Y-axis
at the corresponding ‘y’ values of the plotted points
(c) Sum of the squared values of the vertical distance from each plotted point to the regression
line
(d) Sum of the squared values of the horizontal distance from each plotted point to the
regression line
(e) Sum of the values of the horizontal distance from each plotted point to the regression line.
(1 mark)
< Answer >
50. The standard error of estimate can be decreased for a regression equation by
(a) Increasing the number of observations
(b) Using ‘t’ distribution
(c) Using the two tails of a ‘t’ distribution
(d) Choosing an observation point close to the estimate required
(e) Increasing the number of independent variables.
(1 mark)
< Answer >
51. In regression analysis, the graphical plot of the numerical values of the dependent and
independent variables is called a
(a) Histogram (b) Frequency polygon
(c) Scatter diagram (d) x chart (e)
Ogive.
(1 mark)
< Answer >
52. The regression line of ‘return on security’ on ‘return on market index’ is given by
Y = 5.6+1.67X. What is the regression coefficient of the line?
(a) 7.27 (b) 5.6 (c) 1.67 (d) 3.93 (e)
3.34.
(1 mark)
< Answer >
10
53. The maximax criterion is used to make decision under
(a) Conditions of certainty irrespective of the nature of the decision maker
(b) Conditions of risk irrespective of the nature of the decision maker
(c) Conditions of uncertainty, the decision maker being perfectly optimistic
(d) Conditions of uncertainty, the decision maker being perfectly pessimistic
(e) Conditions of uncertainty, the decision maker being neither perfectly optimistic nor
perfectly pessimistic.
(1 mark)
< Answer >
54. The regret criterion for decision making suggests that
(a) The largest regret should be maximized
(b) The smallest regret should be maximized
(c) The largest regret should be minimized
(d) The smallest regret should be minimized
(e) The average regret should be minimized.
(1 mark)
< Answer >
55. Which of the following criteria for decision-making is based upon weighted profits?
(a) Maximax criterion (b) Maximin criterion
(c) Hurwicz criterion (d) Regret criterion
(e) Minmax criterion.
(1 mark)
< Answer >
56. While calculating decision node, one should write down the cost of each alterative solution
along the decision line, then
(a) The cost is multiplied by the value of the outcome that is already calculated
(b) The cost is subtracted from the value of the outcome that is already calculated
(c) The cost is added to the value of the outcome that is already calculated
(d) The cost is divided by the value of the outcome that is already calculated
(e) The cost is squared and multiplied by the value of the outcome that is already calculated.
(1 mark)
< Answer >
57. Which of the following is not true about decision tree analysis?
(a) Decision tree is a diagrammatic representation of a decision process
(b) It is an excellent tool for making financial or number based decisions
(c) Alternative decisions and the implications of taking those decisions can be laid and
evaluated
(d) Decision tree cannot help in assessing the risk in making decisions concerning investment
(e) Decision tree analysis can be used with the help of computer.
(1 mark)
< Answer >
58. In a business research report, which of the following presents all the relevant data but makes no
attempt to draw any inferences?
(a) Conclusions
(b) Recommendations
(c) Appendices
(d) Findings
(e) Interpretations.
(1 mark)
< Answer >
11
59. What difficulties are there for the business researcher in ensuring the informed consent of the
participants?
I. It is a time consuming process.
II. It increases the issue of reactivity.
III. It is impossible to present participants with all the information they need.
IV. It has no relevance to a study’s findings.
(a) Only (I) above (b) Only (II) above
(c) Only (III) above (d) Only (IV) above
(e) All (I), (II), (III) and (IV) above.
(1 mark)
< Answer >
60. What should the business researcher do if their multi-strategy approach produces inconsistent
results?
(a) Start the whole research project again
(b) Treat one set of results as definitive
(c) Extend the number of research methods used
(d) Re-analyze the data in an attempt to locate more consistent results
(e) To cancel the whole research project.
(1 mark)
< Answer >
61. Which of the following is a way in which qualitative research can facilitate quantitative
research?
I. By using knowledge of context to inform survey design.
II. By allowing for more representative sampling.
III. By negating the need to pilot a questionnaire.
(a) Only (I) above (b) Only (II) above
(c) Only (III) above (d) Both (I) and (II) above
(e) All (I), (II) and (III) above.
(1 mark)
< Answer >
62. What is the benefit of subjecting qualitative data to secondary analysis?
(a) It overcomes ethical concerns about informed consent
(b) It enables the data to be analyzed without the hindrance of a contextual understanding
(c) It is more likely that the findings will be published
(d) It can exploit the large volume of qualitative data that is under-explored
(e) Because it is the better way of analysis.
(1 mark)
< Answer >
63. Among the following the characteristic of the scientific method is:
(a) Conclusions are tentative and subject to change
(b) The methods used can be understood only by experts
(c) Any researcher can replicate the study
(d) Findings frequently contradict personal experience
(e) The results of the research should be accurate.
(1 mark)
< Answer >
64. The group(s) in which the independent variable remains constant is
(a) A treatment group
(b) An experimental group
(c) A control group
(d) Research group
(e) Both (a) and (b) above.
(1 mark)
< Answer >
12
65. A researcher is interested in knowing how accurately something is done and they want to obtain
a more holistic picture of what is happening in a specific setting or situation, then what form of
research is suitable?
(a) Qualitative research (b) Experimental research
(c) Survey research (d) Content analysis research
(e) Quantitative research.
(1 mark)
< Answer >
66. A combination of quantitative and qualitative approaches is known as
(a) Descriptive research (b) Explanatory research
(c) Causal comparative research (d) Mixed methods research (e) Oral
research.
(1 mark)
< Answer >
67. In order for a question to qualify as a research question the researcher must be able to
(a) Make observations and then analyze them to find an answer
(b) Propose several answers to the question that are equally acceptable
(c) Identify value statements in the question that reflect current societal norms
(d) Develop metaphysical answers based on concise statements of belief
(e) Provide the sufficient data to the researcher that supports the research.
(1 mark)
< Answer >
68. Some of the features that the Good research questions must posses are?
(a) Feasible, clear and ethical
(b) Feasible, clear, significant and ethical
(c) Feasible, clear, significant and includes a hypothesis
(d) Clear, significant and ethical
(e) Insignificant and moral.
(1 mark)
< Answer >
69. If a researcher selected six groups at random and then interviewed each member of those six
groups, the researcher uses
(a) Simple random sampling (b) Systematic sampling
(c) Stratified random sampling (d) Two-stage random sampling
(e) Cluster random sampling.
(1 mark)
< Answer >
70. Sample errors or variations among sample statistics are due to
(a) Differences between each sample of the population means
(b) Differences between each sample of the population and among several samples
(c) Differences between one sample of the population and several sample means
(d) Variations among all the sample means
(e) Variations among all the population means.
(1 mark)
< Answer >
71. Which of the following sampling, on proper application can significantly increase the statistical
efficiency of sampling?
(a) Stratified sampling (b) Simple random sampling
(c) Systematic sampling (d) Cluster sampling
(e) Judgment sampling.
(1 mark)
< Answer >
72. The statistical measures like mean, median, mode and standard deviation which are used to
describe the characteristics of a sample, are known as
(a) Parameters (b) Constants
(c) Measurements (d) Statistics (e) Both (a) and (c) above.
(1 mark)
< Answer >
13
73. Which of the following sampling method is most susceptible to subjectivity in selection?
(a) Simple random sampling (b) Stratified sampling
(c) Cluster sampling (d) Judgmental sampling
(e) Systematic sampling.
(1 mark)
< Answer >
74. A person performing a market research will waste resources by taking larger samples, if the
estimator is not
(a) Unbiased (b) Efficient (c) Consistent (d) Sufficient (e)
Inconsistent.
(1 mark)
< Answer >
75. In SPSS, what is the 'Data Viewer'?
(a) A table summarizing the frequencies of data for one variable
(b) A spreadsheet into which data can be entered
(c) A dialog box that allows you to choose a statistical test
(d) A screen in which variables can be defined and labeled
(e) A table where output can be saved.
(1 mark)
< Answer >
76. Which of the following is true with regard to SPSS?
(a) This is just advanced to MS Excel
(b) This is a software language helps to analyze the research results
(c) This is a package which can perform highly complex data analysis using statistical and
mathematical functions
(d) We can input the data into SPSS only from MS Excel
(e) This is a package which cannot be used in business research.
(1 mark)
< Answer >
77. What does the word SPSS stands for
(a) Software plan for system schedule
(b) Statistical package for software solutions
(c) Software package for the social sciences
(d) Statistical package for the social sciences
(e) Systematic project scheduling services.
(1 mark)
< Answer >
14
Suggested Answers
Quantitative Methods-II (MB152): July 2006
1. Answer : (b)
Reason : The probability of getting a black card is 52
26
and the probability of getting a jack
is 52
4
. Therefore, the probability of getting a black jack can be given as 52
26
52
4
´
= 52
2
Alternatively, there are two black jacks in a pack of 52 cards. Therefore the probability of
getting a black jack can be given as 52
2
.
< TOP >
2. Answer : (e)
Reason : From the addition theorem of probability, P(A or B) = P(A) + P(B) – P(AB),
substituting P(A or B) = P(B) in this equation, we get P(A) = P(AB) which means that
occurrence of A implies the occurrence of B.
< TOP >
3. Answer : (d)
Reason : Let S be the event of death of a person in a street accident while A be the event of
death in an air crash. The probability of the death of a person either in a street accident
or an air crash = P(A) + P(B) = 0.0205 (i.e. 0.018+0.0025) as A and B are mutually
exclusive.
< TOP >
4. Answer : (c)
Reason : The expected weight of a person = (0.35) (55) + (0.45) (45) + (0.20) (50) = 49.5 Kg
< TOP >
5. Answer : (c)
Reason : Two heads in three tosses of a fair coin can be obtained in any one of the following
three combinations of outcomes – (Head, Head, Tail) or (Head, Tail, Head) or (Tail,
Head, Head). The probability of getting any one of these combinations
=
÷ø
ö
çè
+æ ´ ´ ÷
ø
ö
çè
+ æ ´ ´ ÷
ø
ö
çè
æ ´ ´
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
= 8
3
.
< TOP >
6. Answer : (d)
Reason : P (A/B) = 0.60 or P(B) = 0.60
P(A and B)
= 0.60
0.25
= 0.4167.
P(B/A) = 0.50 or P(A) = 0.50
P(A and B)
= 0.50
0.25
= 0.50.
P(A or B) = P(A) + P(B) – P(A and B)
= 0.4167 + 0.5000 – 0.2500 = 0.6667 » 0.67.
< TOP >
7. Answer : (e)
Reason : The first four alternatives are clearly fals e. The fifth alternative is true. (Self
explanatory.)
< TOP >
15
8. Answer : (a)
Reason : P (A or B) = P(A) + P(B) – P(A and B)
\ Probability of both event A and event B taking place together
P(A and B) = P(A) + P(B) – P(A or B)
= 2
1
5
2
4
1 + -
= 20
5 +8 -10
= 20
3
.
< TOP >
9. Answer : (a)
Reason : No. of Ways of drawing three red balls at random from 6 red balls are
1x2x3
6 4x5x6
C3 =
= 20
Ways of drawing three balls at random from 11 balls are
1x2x3
11 11x10x9
C3 =
= 165
\ Probability of all three drawn balls being red is 165
20
= 33
4
.
< TOP >
10. Answer : (a)
Reason : Two mutually exclusive events cannot take place together. So the probability of two
mutually exclusive events happening together is 0 (zero).
< TOP >
11. Answer : (d)
Reason : Covariance between returns of the two assets = r12s1s2
where s1 represents the standard deviation of risk of stocks = 20
s2 represents the standard deviation of risk of bonds = 5
r12 represents the correlation between returns = 0.2
r12s1s2 = 0.2*20*5 = 20
< TOP >
12. Answer: (d)
Reason: P (x<85) =
P z 85 130 P(z 3.0) 1 P(z 3.0)
15
1 P(z 3.0)
1 (0.5 0.4987)
1 0.9987
0.0013 i.e.0.13%.
æ < - ö = < - = - > - çè ÷ø
= - <
= - +
= -
=
< TOP >
13. Answer: (b)
Reason: This situation occurs when the Poisson distribution is limiting case of binomial
distribution. Where the probability of success (p) is infinitesimally small and the
number of trials (n) is so large that the product ‘np’ equals l, a finite constant.
< TOP >
16
14. Answer : (c)
Reason : Probability of success p = 1/4.
Then q = 1-p = 3/4.
To find the value of n, put the values of p and q in variance.
i.e. variance = npq
2.81 =
1 3
n
4 4
´ ´ Þn = 14.986 @ 15.
n = 15
6 9
15
6
1 3
p(r 6) 0.090
4 4
= = c æç ö÷ æç ö÷ = è ø è ø .
< TOP >
15. Answer : (b)
Reason : The probability of obtaining six heads in one throw of six coins(a single trial), is
P = (1/2)6, assuming that head and tail are equally probable.
l = np = 128 ´ (1/2)6 = 2.
Hence using Poisson probability law, the required probability of getting six heads 4
times is given by:
e . r e 2.(2)4 P(x 4) 0.090
r! 4!
-l l - = = = =
< TOP >
16. Answer : (e)
Reason : The number of pro moted members who are freshers follows
hypergeometric distribution.
Given details are:
N = no. of persons working in Balaji chemical’ s Ltd.= 20 .
r = no. of fresher members labeled as success = 8
x = desired success = 4
n = no. of promoted members = no. of trials = 10
r (N r ) 8 (20 8 ) 12
x (n x ) 4 (10 4) 6
N 20 20
n 10 10
70
P(x 4) 0.350
c c c c c
c c c
- -
- - ´
= = = = =
< TOP >
17. Answer : (c)
Reason : In a family of 5 children, at least one boy and one girl can occur in the following
mutually exclusive forms:
{(1B, 4G), (2B, 3G), (3B, 2G), (4B, 1G)}. The probability of such combination can be
found as:
=
1 4 2 3 3 2 4 1
5 5 5 5
1 2 3 4
1 1 1 1 1 1 1 1
2 2 2 2 2 2 2 2
C æç ö÷ æç ö÷ + C æç ö÷ æç ö÷ + C æç ö÷ æç ö÷ + C æç ö÷ æç ö÷ è ø è ø è ø èø è ø è ø è ø è ø
=
5 10 10 5 30 15 .
32 32 32 32 32 16
+ + + = =
< TOP >
17
18. Answer : (a)
Reason : For the binomial variable X with parameters n=5 and p, the probability function is:
P(X=r) =
r 5 r
4 1 2 3
2 2 2 2 2 2
2 2 2
5
pq ;r 0,1,2,.....5
r
5 5
wearegiven:10P(X 4) P(X 2) 10 pq p q
4 2
50p 10q 5p q 0 5p (1 p) 0
5p (1 2p p ) 0 4p 2p 1 0 p 0.3090and 0.8090
æ ö -
= ç ÷ =
è ø
= = = Þ ´æ ö = æ ö ç÷ ç ÷
èø è ø
= Þ - = Þ - - =
Þ - - + = Þ + - = Þ = -
Since the probability cannot be negative value, so the probability p = 0.3090.
< TOP >
19. Answer: (c)
Reason: The average number of typographical errors per page in the book is given by:
52 0.124
420
l= =
errors per page, then the probability is given by:
e . x e 0.124 .(0.124)x P(X x) ;x 0,1,2,......
x! x!
-l l - = = = =
The required probability that a random sample of 21 pages will contain no error is
given by:
[P(X=0)]21 = (e–0.124)21 = e–2.604 =0.074.
< TOP >
20. Answer : (c)
Reason : If the mean of the hypergeometric distribution nM/N is said to be small enough, then
the hypergoemetric distribution may be replaced by the binomial distribution.
< TOP >
21. Answer : (a)
Reason : An estimator which has lower standard error than another is said to be a more efficient
estimator.
< TOP >
22. Answer : (c)
Reason : Estimated standard error of mean =
Sample standard deviation
n
Sample standard deviation,
x2 nx2 s
n 1 n 1
= -
- -
å
x 120
x 10
n 12
= = = å
\ s =
( )2 1650 12 10 6.396
11 11
- =
\ Estimated standard error of mean =
6.396
1.846
12
=
< TOP >
23. Answer : (d)
Reason : Other things remaining same, as the sample size increases, the interval estimate for the
population mean at a given confidence level will decrease in width.
< TOP >
18
24. Answer : (a)
Reason : E (x) = m = 25%
x s = n
s
=
4
100 = 0.40%
The shape of the distribution of sample means is approximately normal. (a) is incorrect
because x s ¹ 4.0%.
< TOP >
25. Answer : (e)
Reason : Because the sample size is small we cannot apply the CLT. However, since the
population is normally distributed the sampling distribution of mean will also be
normally distributed with mean = population mean and standard deviation n
s
=
E (x) = m = 5,500
x s = n
s
=
500
10 = 158.11
\P (x <5700) =
P z 5700 5500
158.11
æ < - ö çè ÷ø = P(z < 1.265)
= 0.5000 +0.3962
= 0.8962.
< TOP >
26. Answer : (a)
Reason : If the sampling distribution of a sample statistic has a mean equal to the population
parameter the statistic is intended to estimate, the statistic is said to be an unbiased
estimator of the parameter.
< TOP >
27. Answer : (e)
Reason : The population mean or population variance or sample variance or sample size cannot
be known. However the lower limit of the confidence interval can be mathematically
derived as = x - (UL - x )
< TOP >
28. Answer : (c)
Reason : This is a case of sampling from finite population, because
n
N =
24
280 = 0.085 > 0.05.
\ x s =
N n
n N 1
s -
- =
1600 280 24
24 280 1
-
- = 7.82.
\Test statistic = x
x- m
s =
58 42
7.82
-
= 2.046.
< TOP >
29. Answer : (c)
Reason : Systematic sampling proceeds by picking up one element after a fixed interval
depending on the sampling ratio.
< TOP >
30. Answer : (e)
Reason : In hypothesis tests using the non-standardised scale:
The sample mean can be directly used as the test statistic when the test involves the
population mean. The difference between two sample means can be directly used as
the test statistic when the test involves the difference between two population means.
The sample proportion can be directly used as the test statistic when the test is about
the population proportion.
< TOP >
19
31. Answer : (d)
Reason : t-distribution depends on the degrees of freedom (df). For each ‘df’ there will be a
particular t-distribution. ‘df’ is dependent on the sample size. We know that as sample
size increases the t-distribution approaches the normal distribution. Alternatives (a),
(b), (c) and (e) are true. Alternative (d) is false.
< TOP >
32. Answer : (a)
Reason : An interval estimate is a range of values within which the actual value of the
population parameter may (or may not) fall.
< TOP >
33. Answer : (e)
Reason : Width of confidence interval for mean = UCL – LCL
= (x +zsx ) -(x - zsx )
= 2z x s
x s cannot be known without knowing the sample size, which is not given.
< TOP >
34. Answer : (b)
Reason : H0 : m = 28000
H1 : m < 28000
Standard error of mean =
1000
182.57
30
=
Test statistic value =
=
s
-m
n
x 27500 28000 2.74
182.57
- = -
The sample size can be considered small because samples of size greater than 30 are
considered large. The population standard deviation is not known. It is assumed that
the life of the tires manufactured by the company follow a normal distribution. Hence
the appropriate distribution is the t distribution with 29 degrees of freedom. The test
will be a left tailed test.
\ Critical value = -1.699
The value of the test statistic is less than the critical value. Hence it falls in the
rejection region. \ We reject H0. We can infer that the mean life is less than 28000
miles.
< TOP >
35. Answer : (a)
Reason : H0 : p0 = 0.25
H1 : p0 ¹ 0.25
Standard error of proportion =
0 0
p
p (1-p ) 0.25(1 0.25) s 0.03062
n 200
= = - =
Sample proportion of success n
p
p =
=
42 0.21
200
=
Test statistic: z =
0
p
p p 0.21 0.25
1.31
0.03062
- -
= = -
s
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.
< TOP >
20
36. Answer : (e)
Reason : H0: p = 0.40
H1: p ¹ 0.40
This is a large sample test of proportion. So we can use the normal approximation to
the binomial.
p s =
( ) 0 0 p 1 p
n
-
=
0.40 0.60
360
´
= 0.02582.
\ z =
0
p
p - p
s =
p 0.40 0.43 0.40
0.02582 0.02582
- = -
= 1.162
At a = 0.10, the critical values are ±1.645.
The test statistic is less than the right tail critical value. So it falls in the acceptance
region.
\ We accept H0. The true proportion is 0.47. So H0 is false. Hence the test leads to a
type II error.
< TOP >
37. Answer : (e)
Reason : The populations must follow the normal distribution and have equal standard
deviations, but the sample sizes do not have to be the same or the sample means need
not be equal.
< TOP >
38. Answer : (e)
Reason :
Value of the test statistic =
( 1 2 ) ( 1 2 )
2 2
1 2
1 2
320 144 1 x x 64 36 1 1 0
256 144 2.828
n n 64 36
æ - ö - - - m -m çè ÷ø - = = =
s + s +
< TOP >
39. Answer : (b)
Reason : The sample mean is an unbiased estimator of the population mean.
< TOP >
40. Answer : (c)
Reason :
( ) ( )
1 2 1 2 1 2
1 2
1 2
1 2 1 2
x x x x x x
x x
2 2 2
2 1 2 2 1
x x
1 2
2 2
1
504 192 0 x x 84 48 2
Value of the test statistic = 0.353553
or 2 5.6568
0.353553
144
or 5.6568
n n 84 48
or 5.6568 144 84 (32 3)
48
- - -
-
-
æ - ö - - - m -m ç ÷ = è ø = =
s s s
s = =
s s s
\s = + = +
s = æ - ö´ = - çè ÷ø
1
84
2436 49.35 49
´
\ s= = @
< TOP >
21
41. Answer : (e)
Reason : (a) This is a wrong answer. The number of degrees of freedom does not depend on
the sample size.
(b) This is the wrong answer. The number of degrees of freedom does not depend on
the ratio of sample size and the population.
(c) This is the wrong answer. The number of degrees of freedom does not depend on
the number of rows in the contingency table only.
(d) This is the wrong answer. The number of degrees of freedom does not depend on
the number of columns in the contingency table only.
(e) This is the right answer. When the chi-square distribution is used as a test of
independence, the number of degrees of freedom is related to both the number of
rows and the number of columns in the contingency table.
< TOP >
42. Answer : (c)
Reason : The return on a portfolio is equal to the weighted average of the returns on individual
assets in the portfolio.
< TOP >
43. Answer : (d)
Reason : Given that,
å dx = -170, y å d = -20,
2
x åd = 8288,
2y
å d = 2264
x y å d d = 3044, n = 10.
x y x y
2 2 2 2
x x y y
2 2
n dd d d
r
n d ( d ) n d ( d )
10 3044 ( 170)( 20)
10 8288 ( 170) 10 2264 ( 20)
0.7804(approx.)
å -å å
=
å -å å - å
´ -- -
=
´ - - ´ - -
=
< TOP >
44. Answer : (c)
Reason : a. When the slope of a regression line is negative the correlation coefficient need
not be 1.
b. When the slope of a regression line is negative, there is a negative correlation
between the variables; hence the correlation coefficient lies between –1 and 0.
c. When the slope of a regression line is negative, there is a negative correlation
between the variables.
d. When the slope of a regression line is zero, the regression line will be parallel to
the horizontal axis.
e. When the y-intercept of a regression line is zero, the regression line passes
through the intersection of horizontal and vertical axes
< TOP >
45. Answer : (a)
Reason : a. The coefficient of determination is the square of coefficient of correlation (r).
Hence it will always be ³ 0 and £ 1.
b. From above we can see that the coefficient of determination cannot be less than
zero.
c. The coefficient of determination is will be equal to 1, only if the coefficient of
correlation is equal to –1 or 1; in other cases it will be > 0 and < 1.
d. The coefficient of determination is always positive; The coefficient of correlation
may be negative.
e. Since -1 £ r £ 1, the coefficient of determination which is the square of
coefficient of correlation (r), will always have magnitude less than or equal to r
< TOP >
22
46. Answer: (b)
Reason:
Standard error of estimate, se = n 2
Y2 a Y b XY
-
S - S - S
SY2 = 19,00,400 (given)
SY = Y ´ n = 612 ´ 5 = 3060
SXY= 18,100 (given)
Yˆ
= 1,236 – 104X
\ a = 1,236 and b = -104
\ se = 5 2
19,00,400 (1236) (3060 ) ( 104) (18,100)
-
- - -
= 14.61.
The sample size is less than 30 and the standard deviation of the population is not
known. Hence the appropriate distribution to be used for the prediction interval is the
t-distribution.
Degrees of freedom = n – 2 = 5 – 2 = 3.
The appropriate t–value is 3.182 (from the t–table).
The limits of the prediction interval are given below:
For, x = 7.50, Yˆ = 1236 – 104 (8.00) = 404
The prediction interval is: Yˆ ± t (se) = 404 ± (3.182´ 14.61) = 404 ± 46.49
passengers
< TOP >
47. Answer : (e)
Reason : Let X be the number of drinkers and Y be the income on liqour during weekends.
åX = 139, åY= 43, n = 7
åXY = 979,
åX2 = 3225,
X = 19.85 , Y = 6.14
The coefficient of regression b = ( )2 2
n XY X Y
n X X
å -å å
å - å =
( ) ( )
( ) ( 2 )
7 979 139 43
7 3225 139
´ - ´
´ -
=
876
3254 =
0.269
The Y intercept a = Y - b X = 6.14 – (0.269 ´ 19.85) = 0.800
Therefore the estimating equation is ˆY= 0.800+ 0.269 X
The estimate of Y for X = 40, is ˆY = 0.800 + 0.269 ´ 40 = 11.56 (Approx)
< TOP >
48. Answer : (b)
Reason : (a) This is the wrong answer. The coefficient of determination r2 measures how good
the fit is and gives the variation explained by the model.
(b) This is the right answer. If r be the coefficient of correlation between two
variables X and Y, then the variation that is not explained by the regression line
is equal to 1 - r2.
(c) This is the wrong answer. ‘r’ is the coefficient of correlation and it signifies the
degree of association between two variables.
(d) This is the wrong answer. ‘1- r ’ does not give the variation that is not explained
by the model.
(e) This is the wrong answer. ‘1+r2’ does not have any meaning associated with it.
< TOP >
23
49. Answer : (c)
Reason : The method of least squares indicates the smallest possible sum of the squared values
of the vertical distance from each plotted point to the regression line. Other options are
incorrect.
< TOP >
50. Answer : (a)
Reason : Standard error of estimate = (n 2)
) Yˆ (Y 2
-
S -
We see that number of observations in the denominator, i.e. as the value of ‘n’
increases, the value of standard error decreases.
< TOP >
51. Answer : (c)
Reason : In regression analysis the graphical plot of the numerical values of the dependent and
independent variables is called a scatter diagram.
< TOP >
52. Answer : (c)
Reason : The return from a given security are regressed with the return from the marketindex.
The regression line or the line of best fit for the observations is called as the
characteristic line. The slope of the line is the regression coefficient of the line.
< TOP >
53. Answer : (c)
Reason : Maximax criterion is used to make decision under conditions of uncertainty, when the
< TOP >
54. Answer : (c)
Reason : In case of regret criterion, the largest regret is minimized.
< TOP >
55. Answer : (c)
Reason : The decision maker lies in between two extremes, i.e. (neither complete optimistic nor
complete pessimistic) chooses the maximum of the weighted profits. This approach is
known as Hurwicz criterion.
< TOP >
56. Answer : (b)
Reason : While calculating decision node, one should write down the cost of each alterative
solution along the decision line, and then the cost is subtracted by the value of the
outcome that is already calculated.
< TOP >
57. Answer : (d)
Reason : Decision tree helps in assessing risk in decision concerning investment. All other
statements are true for decision tree analysis.
Therefore the option (d) is not true.
< TOP >
58. Answer : (d)
Reason : Findings in a business research report include all the relevant data but do not make any
attempt to draw any inferences.
< TOP >
59. Answer : (c)
Reason : It is impossible to present participants with all the information they need. Relatively
minor transgressions in obtaining informed consent are likely to be part of all business
research projects because it is not feasible to give a participant all of the information
about the research. Nevertheless, the researcher must still make every effort to obtain
informed consent if they wish to act ethically.
< TOP >
60. Answer : (b)
Reason : Treat one set of results as definitive.
Unfortunately, if the selection of one set of data over another is entirely arbitrary, this
is not an ideal approach to overcoming the difficulty of contradiction within a research
project that has adopted a multi-strategy approach. Instead, researchers may wish to
either explore reasons for the contrasting findings, or establish a clearly reasoned
argument for the selection of one type of data as definitive.
< TOP >
24
61. Answer : (a)
Reason : By using knowledge of context to in form survey design.
Qualitative research can give the researcher an in-depth understanding of context,
which can then mean that a survey instrument can be designed to more effectively
explore this context. Qualitative research can also provide hypotheses that can then be
tested using quantitative methods.
< TOP >
62. Answer : (d)
Reason : It can exploit the large volume of qualitative data that is under-explored.
Where a number of significant problems remain with the secondary analysis of
qualitative data, there are opportunities for researchers to exploit because of the
volume of data that is cast aside as part of the analysis.
< TOP >
63. Answer : (a)
Reason : The one of the characteristics of the scientific method is that the conclusions are
tentative and subject to change.
< TOP >
64. Answer : (c)
Reason : While conducting experimental hypothesis testing research, if the group is studied
under usual conditions then it is called a control group. A control group is one in
which the independent variable is kept constant.
< TOP >
65. Answer : (a)
Reason : Qualitative research places a greater emphasis on describing in detail all of what goes
on in a particular situation, rather than, on comparing the effects of a particular
treatment or on describing the behaviors of people.
< TOP >
66. Answer : (d )
Reason : A combination of quantitative and qualitative methods defines mixed methods
research.
< TOP >
67. Answer : (a)
Reason : Researcher must be able to make observations and then analyze them to find an
answer
The defining characteristic of a research question is that it can be answered by
collecting data that pertain to the question. Research questions are not metaphysical,
philosophical, or questions of value.
< TOP >
68. Answer : (b)
Reason : Questions should be feasible, should be clear, should be significant, and should be
ethical.
< TOP >
69. Answer : (e)
Reason : Cluster random sampling is when a researcher selects intact groups of individuals to
participate in the study. The difference between cluster and two-stage sampling is that
with two-stage sampling the clusters are selected first, then individuals are selected
from each cluster. In this exercise, all the individuals in each cluster participated in the
study.
< TOP >
70. Answer : (b)
Reason : Samples errors or variations among sample statistics are due to differences between
each sample of the population and among several samples.
< TOP >
71. Answer : (a)
Reason : Stratified sampling on proper application can significantly increase the statistical
efficiency of sampling.
< TOP >
72. Answer : (d)
Reason : The statistical measures like mean, median, mode and standard deviation which are
used to describe the characteristics of a sample are known as statistics.
< TOP >
25
73. Answer : (d)
Reason : Judgmental sampling is not a type of random sampling. In judgmental sampling the
sample is selected according to the judgment of the investigators. Hence, there is a
degree of subjectivity in the selection.
< TOP >
74. Answer : (c)
Reason : A person performing a market research will waste resources by taking larger samples,
if the estimator is not consistent.
< TOP >
75. Answer : (b )
Reason : It is a spreadsheet into which data can be entered.
The Data Viewer is one of the two screens that compromise the Data Editor in SPSS,
the other being the Variable Viewer. The Data Viewer is a spreadsheet grid into which
you can enter your data for analysis.
< TOP >
76. Answer : (c)
Reason : This is a package can perform highly complex data analysis using statistical and
mathematical functions.
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77. Answer : (d)
Reason : SPSS stands for Statistical package for the social sciences.