Question Paper Quantitative Methods-II (132) : October 2005
| · Answer all questions. · Marks are indicated against each question. |

1. | If two fair dice are thrown simultaneously ten times then, what is the likelihood that a minimum of 3 can be simultaneously observed in both the dice, on at least nine throws? (a) 0.00376 (b) 0.00030 (c) 0.00406 (d) 0.99594 (e) 0.44444. (2 marks) | < Answer >
2. | A restaurant serves eight items of fish, twelve items of mutton and ten items of chicken. If customers select these items randomly then, what is the probability that two of the next four customers order either mutton or chicken items? (a) 0.7705 (b) 0.733 (c) 0.267 (d) 0.2295 (e) 0.5378. (1 mark) | < Answer >
3. | Western Garments Company is in the business of trading men’s garments. It purchases readymade garments from different manufacturers and sells them in the retail market. The company sells a particular brand of T-shirts named “Macho” every year during the summer season. It has to decide on the number of the XL size of these T-shirts to stock for this season. Each T-shirt costs Rs.150 and sells for Rs.200. The salesman at the counter is given a commission of Rs.10 on every T-shirt sold. The stock of this type of T-shirts is maintained for a period of 4 months. In the past the sales of this brand and size of T-shirts during the same period of time have averaged 400 with a standard deviation of 60. The T-shirts which remain unsold at the end of 4 months are sold at 45% discount from the selling price. Further, on each T-shirt sold at discount the salesman at the counter is given a commission of Rs.20. The sales of the T-shirts have been found to be normally distributed. What is the optimal number of “Macho” brand T-shirts of XL size which should be stocked by the company? (a) 60 (b) 15 (c) 385 (d) 400 (e) 785. (2 marks) | < Answer >
4. | A random variable, Y, has the following probability distribution:.
Y | 20 | 40 | 60 | 80 | 100
Probability | 0.15 | 0.20 | 0.30 | 0.25 | 0.10
15 values of Y have been observed. What is the probability that less than 12 observations are such that 40 £ Y £ 80? (a) 0.8441 (b) 0.1559 (c) 0.0668 (d) 0.4613 (e) 0.5387. (2 marks) | < Answer >
5. | In a normal distribution 30.85% of the items are under 45 and 8.08% are over 64. Find the standard deviation of the distribution. (a) 8 (b) 9 (c) 10 (d) 11 (e) 12. (1 mark) | < Answer >
6. | A workshop produces 2000 units per day. The average weight of units is 130 kg with a standard deviation of 10 kg. Assuming a normal distribution of the weights, approximately how many units are expected to weigh less than 142 kg? (a) 1470 (b) 1570 (c) 1670 (d) 1770 (e) 1870. (1 mark) | < Answer >
7. | In a particular examination 10% of candidates passed with distinction, 60% candidates passed without distinction while rest 30% candidates failed in the examination. It is known that a candidate fails in the examination if he/she obtains less than 40 marks (out of 100) while he/she must obtain at least 75 marks in order to pass with distinction. The mean marks is more than 40 and less than 75. Determine the mean of the distribution of marks, assuming this to be normal. (a) 46.13 (b) 47.14 (c) 48.15 (d) 49.16 (e) 50.16. (1 mark) | < Answer >
8. | A box contains 60 ballpoint pens out of which 10 pens are defective. 8 pens are randomly picked from the box. What is the probability that exactly three of the pens randomly picked are defective? (a) 0.0994 (b) 0.7913 (c) 0.0008 (d) 0.9992 (e) 0.00012. (1 mark) | < Answer >

9. | A businessman has to decide on investing in one of three types of assets. The net gains from the assets under three different states of nature are given below: Net gains (in Rs.)
Asset | State of nature
S1 | S2 | S3
A | 30,000 | 12,000 | 10,000
B | 40,000 | 37,000 | - 6,000
C | 50,000 | 25,000 | -30,000
The businessman is not in a position to assign probabilities of occurrence to the three states of nature. Further, he is perfectly optimistic about the future. Which asset should he invest in? (a) Asset A (b) Asset B (c) Asset C (d) Either asset A or C (e) Either asset A or B. (1 mark) | < Answer >
10. | In decision theory, which of the following conditions characterize decision-making under conditions of risk? I. The various states of nature are not known. II. The various states of nature are known. III. The decision maker has insufficient information to assign probabilities of occurrence to various states of nature. IV. The decision maker has sufficient information to assign probabilities of occurrence to various states of nature. (a) Only (I) above (b) Only (II) above (c) Only (III) above (d) Both (I) and (III) above (e) Both (II) and (IV) above. (1 mark) | < Answer >
11. | City Finance Ltd. offers various types of financing to its clients. Its clientele includes individuals as well as business firms. It offers leasing and hire purchase financing as well as direct loan to the business firms for a variety of asset purchases. Presently, there are 160 business firms among its clients; the remaining are individuals. It has been observed that the borrowers belonging to the business firm category tend to pay their repayment dues regularly. However, of late some irregularities have been observed in the payment habits of its individual borrowers. The individuals constitute 90 percent of all its borrowers. The managing director of the company ordered a random survey of all the accounts in the individual borrowers category. 144 accounts in the individual borrowers category were randomly surveyed and it was found that 108 of them were in excellent standing. An interval estimate for the absolute number of accounts belonging to the individual borrowers category that meet the requirements of excellent standing, has been constructed with a confidence level of 90 percent. Which of the following represents the interval estimate for the absolute number of accounts in the individual borrowers category, which are in excellent standing? (a) 108 ± 85.24 (b) 108 ± 80.78 (c) 144 ± 80.78 (d) 1080 ± 85.24 (e) 1080 ± 81.07. (2 marks) | < Answer >
12. | The following details are available with regard to a hypothesis test on difference between means of two populations: H0: H1: n1 = 16 n2 = 9 The samples are independently collected. It is assumed that the two populations are normally distributed and have the same variance. If the significance level is 0.01 then (a) The critical value is 2.500 and the null hypothesis is rejected (b) The critical value is 2.500 and the null hypothesis is accepted (c) The critical values are and the null hypothesis is rejected (d) The critical values are 2.069and the null hypothesis is rejected (e) The critical values are 1.714 and the null hypothesis is accepted. (2 marks) | < Answer >
13. | The following details are available with regard to a hypothesis test on a population mean: The significance level is 0.05. For which of the following values of the sample mean will the null hypothesis be rejected? (a) 8.50 (b) 7.50 (c) 12.00 (d) 13.15 (e) 10.50. (1 mark) | < Answer >

14. | The following data are from a simple random sample:
5 | 10 | 8 | 10 | 7 | 14
What is the point estimate of the population mean? (a) 6 (b) 54 (c) 9 (d) 15 (e) 10. (1 mark) | < Answer >
15. | When a sample of size n with a variance of s2, is selected from a normal population with a variance of s2 the sampling distribution of the statistic has a (a) Normal distribution (b) t distribution with n –1 degrees of freedom © t distribution with n degrees of freedom (d) Chi-square distribution with n –1 degrees of freedom (e) Chi-square distribution with n degrees of freedom. (1 mark) | < Answer >
16. | The following details are available for a hypothesis test on population proportion: H0: p = 0.40 H1: p 0.40 Sample size = 360 Sample proportion = 0.34 Significance level = 0.05 It is later known that the true population proportion is 0.40. Which of the following can be said with regard to the test? (a) There is insufficient information for doing the test (b) The chi-square distribution should be used © 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 >
17. | The following details are available with regard to a hypothesis test on means of two populations: H0: H1: n1 = 64 n2 = 36 The samples collected from the two populations are independent. What is the value of the test statistic? (a) 0.707 (b) –1.789 (c) 0 (d) –0.354 (e) 1.789. (2 marks) | < Answer >
18. | A small sample has been taken from a population and the sample mean has been found to be 80. The population variance is not known. The sample size is 16 and the upper limit of a 95 percent confidence interval for the population mean is 101.31. What is the sample variance? (a) 100 (b) 16 (c) 1600 (d) 3200 (e) 800. (1 mark) | < Answer >
19. | A sample of size 96 has been taken from a population and the sample proportion has been found to be 0.48. The upper limit of a 95% confidence interval for population proportion is 0.531. What is the lower limit of the confidence interval? (a) 0.051 (b) 0.48 (c) 0.429 (d) 0.571 (e) 0.469. (1 mark) | < Answer >
20. | The following details are available with regard to a hypothesis test on a population mean: The null hypothesis is rejected if . What is the probability of committing a type I error? (a) 0.0025 (b) 0.025 (c) 0.25 (d) 0.50 (e) 0.475. (1 mark) | < Answer >

21. | A sample of size 96 has been taken from a population and a 95 percent confidence interval for the population proportion has been constructed. The upper and lower limits of the confidence interval are 0.4776 and 0.2424 respectively. What is the estimated standard error of proportion? (a) 0.2352 (b) 0.12 (c) 0.06 (d) 0.1176 (e) 0.72. (1 mark) | < Answer >
22. | A researcher has selected a random sample of 36 observations from an infinitely large population of known mean equal to 50 and standard deviation equal to 6. What is the probability that the sample mean will be greater than 51? (a) 0.3413 (b) 0.50 (c) 0.8413 (d) 0.1587 (e) 0.6826. (1 mark) | < Answer >
23. | In which of the following situations would be the appropriate formula for computing the standard error of mean? I. Sampling is done from an infinite population. II. Sampling is done from a finite population without replacement. III. Sampling is done from a finite population with replacement. (a) Only (I) above (b) Only (II) above (c) Only (III) above (d) Both (I) and (III) above (e) Both (II) and (III) above. (1 mark) | < Answer >
24. | The arithmetic mean of the upper and lower limits of the confidence interval for population mean is equal to the (a) Population mean (b) Sample mean (c) Population standard deviation (d) Sample standard deviation (e) Standard error of mean. (1 mark) | < Answer >
25. | If a population distribution is known to be normal, then which of the following logically follows? (a) The sample mean must equal the population mean (b) The sample mean must equal the population mean for large samples (c) The sample standard deviation must equal the population standard deviation (d) The sampling distribution of mean will not be a normal distribution (e) The sampling distribution of mean will be a normal distribution. (1 mark) | < Answer >
26. | The following details are available with regard to a hypothesis test on population mean: H0: H1: n = 64 Significance level = 0.05 It is later known that the true population mean is 11. Which of the following can be said with regard to the test? (a) There is insufficient information for doing the test (b) The t distribution should be used (c) The test does not lead to either type I or type II error (d) The test leads to a type I error (e) The test leads to a type II error. (1 mark) | < Answer >
27. | The following details are available with regard to a hypothesis test on means of two populations: n1 = 64 n2 = 36 The samples collected from the two populations are independent. What is the standard error of difference between means? (a) 2.828 (b) 4 (c) 8 (d) 16 (e)1.414. (1 mark) | < Answer >

28. | Which of the following is true? (a) The probability of making a Type I error is lesser in a one-tailed test than a two-tailed test for a given significance level (b) The probability of making a Type I error is greater in a one-tailed test than a two-tailed test for a given significance level © A two-tailed test has only one critical value (d) A two-tailed test has two critical values (e) A one tailed test has two critical value. (1 mark) | < Answer >
29. | Which of the following assumptions is/are not required for testing the hypothesis of the difference between two sample means when the samples are small and independent and the population variances are unknown? (1 Both populations are normally distributed II. Both populations have equal variances III. Both populations follow the standard normal distribution IV. The means of the two populations are equal (a) Only (I) above (b) Only (II) above © Both (I) and (II) above (d) Both (I) and (IV) (e) Both (III) and (IV) above. (1 mark) | < Answer >
30. | If the sample size is 2 or more, then the standard error of mean will (a) Always be equal to the standard deviation of the population (b) Always be less than the standard deviation of the population © Always be greater than the standard deviation of the population (d) Always be equal to the variance of the population (e) Always be greater than the variance of the population. (1 mark) | < Answer >
31. | The following details are available with regard to a basket of goods: Paasches price index = 128 Weighted average price of the goods in the current year using the base year quantities as weights = Rs.96 Weighted average price of the goods in the base year using the base year quantities as weights = Rs.64 What is Fisher’s ideal price index? (a) 72.17 (b) 192 (c) 150 (d) 128 (e) 138.56. (1 mark) | < Answer >
32. | The following details are available with regard to a basket of goods: Weighted average quantity consumed during the base year = 100 units Weighted average quantity consumed during the current year = 64.375 units Both the weighted averages are computed on the basis of the prices during the base year as weights. Which of the following is correct? (a) Laspeyers price index = 64.375 (b) Laspeyres quantity index = 64.375 © Paasches price index = 155.34 (d) Fishers price index = 155.34 (e) Paasches quantity index = 155.34. (1 mark) | < Answer >
33. | The following details are available with regard to a basket of commodities:
Commodity | A | B | C | D
P1Q1 (Rs.) | 2400 | 8000 | 8960 | 6000
Q1/Q0 | 0.75 | 0.50 | 0.625 | 0.60
What is Paasches quantity index for the basket of commodities? (a) 171.67 (b) 30 (c) 58.25 (d) 333.33 (e) 67. (1 mark) | < Answer >
34. | If we know the average quantity consumed during the base year and the average quantity consumed during the current year (where both the averages are simple averages), of a basket of goods then which of the following can be computed? (a) Paasches price index (b) Laspeyres price index © Laspeyres quantity index (d) Unweighted aggregates price index (e) Unweighted aggregates quantity index. (1 mark) | < Answer >

35. | We know about a basket of goods, the weighted average of the quantities consumed during the base year and the weighted average of the quantities consumed during the current year, where both the weighted averages have been computed on the basis of the base year prices as weights. Which of the following can be calculated? (a) Paasches price index (b) Paasches quantity index (c) Laspeyres price index (d) Laspeyres quantity index (e) Unweighted aggregates quantity index. (1 mark) | < Answer >
36. | The prices and quantities of fruits consumed in year 1992 and year 2002 are as follows:
Fruits | Year 1992 | Year 2002
Price (Rs. per kg.) | Quantity (Kgs.) | Price (Rs. per kg.) | Quantity (Kgs.)
Mangoes | 15 | 30 | 25 | 26
Oranges | 45 | 40 | 65 | 30
Apples | 60 | 25 | 65 | 25
Papaya | 8 | 50 | 12 | 45
Grapes | 18 | 30 | 25 | 28
Calculate Laspeyres price index for the year 2002 using 1992 as the base year (a) 134.86 (b) 132.74 (c) 130.62 (d) 128.48 (e) 126.72. (1 mark) | < Answer >
37. | Trinity Computers Ltd. manufactures and sells personal computers and accessories for office automation and the home users. The company is planning to expand its manufacturing facility if the sales of its computers increase beyond 150,000 units this year. It has compiled the sales figures of its computers across the country for the last five years. It feels that with the increase in the income level of the consumers and the advanced features coming in the personal computers, which enable it to be used not only as an educational tool but also for entertainment at home, will boost its sales in the household segment in the coming years.
Year | 1998 | 1999 | 2000 | 2001 | 2002
Number of PCs sold (in thousands) | 67 | 71 | 76 | 82 | 96
Fit a curvilinear estimating equation of type , where x =X-(X represents year). What is the value of the coefficient c in the estimating equation? (a) 0.5 (b) 1.5 (c) 2.5 (d) 3.5 (e) 4.5. (2 marks) | < Answer >
38. | Fit a straight line trend by the method of least squares to the following data and estimate the sales for the year 2004.
Year | 1995 | 1996 | 1997 | 1998 | 1999 | 2000
Sales of air conditioners (in ‘000 units) | 10 | 13 | 16 | 21 | 24 | 30
(a) 40,628 units (b) 41,856 units (c) 42,112 units (d) 43,224 units (e) 44,628 units. (2 marks) | < Answer >
39. | Pradeep Khanna, a second year MBA student, is doing a study of companies going public for the first time. He is curious to see whether or not there is a significant relationship between the sizes of the offering (in crores of rupees) and the price per share after the issue. The data are given below:
Size (in crores of rupees) | 108.00 | 39.00 | 68.40 | 51.00 | 10.40 | 4.40
Price (in Rs.) | 12.00 | 13.00 | 19.00 | 12.00 | 6.50 | 4.00
Calculate the coefficient of correlation between the issue size and the post-issue price (a) 0.6511 (b) 0.6725 (c) 0.7533 (d) 0.8044 (e) 0.8555. (2 marks) | < Answer >
40. | The following summaries are obtained with regard to 5 paired observations of the variables X and Y: åX = 5, åY = 5, åXY = –21, åX2 = 45. If the mean values of X and Y are 6 and 8 respectively, find the regression equation of Y on X with X as the independent variable. (a) = 11.9 – 0.65X (b) = 0.65 + 11.9X (c) = -11.9 + 0.65X (d) = 11.9 + 0.65X (e) = 0.65 –11.9 X. (1 mark) | < Answer >

41. | The following details pertain to a correlation analysis of variables X and Y: r = 0.80, å xy =60, sy = 2.5 and å x2 = 90. If x and y denote the deviations from the arithmetic means of the data in X series and Y series respectively. Find the number of items in each series. It is assumed that the correlation analysis is performed among the entire population of the values of X and Y respectively. (a) 8 (b) 9 (c) 10 (d) 11 (e) 12. (2 marks) | < Answer >
42. | The following data shows the number of personal loan applications processed per day by four employees of Oriental Finance Ltd. observed over a number of days:
Employee 1 | 11 | 14 | 12 | 15 | 13
Employee 2 | 15 | 18 | 13 | 16 |
Employee 3 | 13 | 15 | 17 | 12 | 13
Employee 4 | 14 | 18 | 17 | 15 |
We want to find out whether the mean number of personal loan applications processed per day is the same for the four employees. A test of ANOVA has to be performed. Use 0.05 level of significance. Find the estimated population variance from the variance within the samples (a) 1.50 (b) 2.50 (c) 3.50 (d) 4.50 (e) 5.50. (1 mark) | < Answer >
43. | A study compared the effects of four 1-month point-of-purchase promotions on sales. The unit sales for five stores using all four promotions in different months follow:
Free samples | 78 | 87 | 81 | 89 | 85
One-pack gift | 94 | 91 | 87 | 90 | 88
Rupees off | 74 | 78 | 70 | 82 | 76
Refund by mail | 79 | 83 | 78 | 69 | 81
A test of ANOVA has to be performed What is the grand mean for number of units sold through different schemes? (a) 77 (b) 79 (c) 82 (d) 83 (e) 85. (1 mark) | < Answer >
44. | A sample of 20 observations from a normal distribution has a mean of 37 and a variance of 12.2. We want to construct a 90 percent confidence interval for the true population variance. Find the upper limit of the confidence interval for the true population variance. (a) 19.61 (b) 20.71 (c) 21.81 (d) 22.91 (e) 23.01. (1 mark) | < Answer >
45. | The standard deviation of a normal population is hypothesized to be 50. An observed sample of 30 yields a sample standard deviation of 57. We want to test whether we should reject the null hypothesis that the population standard deviation is 50 at 5% level of significance, and conclude that the population standard deviation is not 50. What is the critical value in the left tail for the appropriate statistical test? (a) 16.047 (b) 16.791 (c) 17.708 (d) 18.493 (e) 19.768. (1 mark) | < Answer >
46. | Two random samples were drawn from two normal populations and their values are:
A: | 66 | 67 | 75 | 76 | 82 | 84 | 88 | 90 | 92 |  |
B: | 64 | 66 | 74 | 78 | 82 | 85 | 87 | 92 | 93 | 95 | 97
We want to test whether the population B has larger variance than the population A. What would be the critical value for the above test at a significance level of 5%? (a) 3.07 (b) 3.14 (c) 3.35 (d) 3.64 (e) 4.06. (1 mark) | < Answer >
47. | The number of units demanded for a particular spare part in a factory was found to vary from day to day. In a sample study the following information was obtained:
Day | Mon | Tue | Wed | Thurs | Fri | Sat | Total
No. of units demanded | 1124 | 1125 | 1110 | 1120 | 1126 | 1115 | 6720
It is to be tested whether the number of units demanded is uniformly distributed over the week. What is the critical value for the above test at a significance level of 5%? (a) 5.991 (b) 6.251 (c) 7.779 (d) 11.070 (e) 12.592. (1 mark) | < Answer >

48. | An automobile-manufacturing firm is bringing out a new model. In order to map out its advertising campaign, it wants to determine whether the model appeal depends on age group or not. The firm takes a random sample from persons attending a preview of the new model and obtained the following results:
Persons who | Under 20 | (20 – 40) | (40 – 50) | (50 and over) | Total
Liked the car | 146 | 78 | 48 | 28 | 300
Disliked the car | 54 | 52 | 32 | 62 | 200
Total | 200 | 130 | 80 | 90 | 500
What is the critical value for the appropriate statistical test at a significance level of 5%? (a) 5.991 (b) 7.815 (c) 9.488 (d) 11.070 (e) 12.592. (1 mark) | < Answer >
49. | The manager of a warehouse wants to construct a control chart for proportion of defectives obtained in repeated random samples of size 100 from a process, which is considered to be under control when the proportion of defectives, p, is equal to 0.20. Calculate the lower control limit for the p chart. (a) 0.16 (b) 0.14 (c) 0.12 (d) 0.10 (e) 0.08 (1 mark) | < Answer >
50. | The following data provides the values of sample mean, , and range, R, for samples of size 5 each.
Sample No. | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10
Mean () | 11.2 | 11.8 | 10.8 | 11.6 | 11.0 | 9.6 | 10.4 | 9.6 | 10.6 | 10.0
Range (R) | 7 | 4 | 8 | 5 | 7 | 4 | 8 | 4 | 7 | 9
Calculate the value of the upper control limit for the chart. (a) 14.2627 (b) 14.2738 (c) 14.2849 (d) 14.2951 (e) 14.3169. (1 mark) | < Answer >
51. | Which of the following is false for the multiple correlation coefficient between dependent variable Y and two independent variables X1 and X2? (a) It depends on the correlation coefficient between Y and X1 (b) It depends on the correlation coefficient between X1 and X2 (c) It depends on the correlation coefficient between Y and X2 (d) It will take values between 0 and 1 (e) It will take values between –1 and +1. (1 mark) | < Answer >
52. | The Lower Control Limit in an R chart if n = 7, and = 7.1, is (a) 0.338 (b) 0.438 (c) 0.538 (d) 0.658 (e) 0.748. (1 mark) | < Answer >
53. | If the coefficient of correlation between two variables lies between –1 and 0, then the covariance between them is (a) Positive (b) Negative (c) Zero (d) Greater in magnitude than the variance of each of the variables (e) Lesser in magnitude than the variance of each of the variables. (1 mark) | < Answer >
54. | Which of the following represents the proportion of variation in the dependent variable that is explained by the regression line? (a) Coefficient of determination (b) Coefficient of correlation (c) Coefficient of variation (d) Standard error of estimate (e) Standard deviation. (1 mark) | < Answer >
55. | In the linear regression model, the residuals are assumed to (a) Have a binomial distribution with a probability of success of 0.50 (b) Have a normal distribution with a mean of 0 (c) Have a chi-square distribution with a mean of 0 (d) Have a t distribution with a mean of 1.00 (e) Have a normal distribution with unknown mean. (1 mark) | < Answer >

56. | 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 >
57. | If two variables X and Y are perfectly positively correlated then their covariance will be equal to (a) 0 (b) 1 (c) The product of the standard deviations of X and Y (d) The sum of the standard deviations of X and Y (e) The difference between the standard deviations of X and Y. (1 mark) | < Answer >
58. | Which of the following is true when the slope of a regression line is positive? (a) The correlation coefficient between the dependent and independent variables is zero (b) There is a positive correlation between the dependent and independent variables (c) There is a negative correlation between the dependent and independent variables (d) The regression line is perpendicular to the horizontal axis (e) The regression line is parallel to the horizontal axis. (1 mark) | < Answer >
59. | Which of the following is not true with regard to the regression relationship, = a + bX? (a) The point always lies on the regression line (b) The expected value of the residuals is zero (c) The mean of the fitted values of Y is the same as the mean of the observed values of Y (d) There are always as many points above the fitted line as there are below it (e) The regression line minimizes the sum of the squared residuals. (1 mark) | < Answer >
60. | In a test involving ANOVA the F statistic is calculated on the basis of (a) The mean of the largest sample (b) The mean of the smallest sample (c) The standard deviation of the smallest sample (d) The estimated population variance based on the variance among the sample means and the estimated population variance based on the variance within the samples (e) The variance of the largest sample. (1 mark) | < Answer >
61. | A two-wheeler dealer has sold 30,000 two-wheelers during the year 2003-04. According to the secular trend equation, the estimated number for the year is 32,000. The cyclical variation for the year measured by the relative cyclical residual technique is (a) –6.25 (b) 0.94 (c) 6.25 (d) 6.67 (e) 106.67. (1 mark) | < Answer >
62. | Which of the following patterns of chart indicates that the variations have been reduced significantly? (a) Hugging the control limits (b) Hugging the center line (c) Cycles (d) Decreasing trend (e) Increasing trend. (1 mark) | < Answer >
63. | Which of the following is true with regard to the p chart in quality control? (a) The center line is drawn on the basis of overall sample mean (b) The lower control limit can only be greater than or equal to zero (c) The upper control limit is equal to the overall sample proportion (d) The lower control limit is equal to the overall sample proportion (e) The upper and the lower control limits are based on the overall sample mean. (1 mark) | < Answer >

64. | Which of the following is used to monitor the variability in a process? (a) charts (b) R chart © p chart (d) Scatter diagram (e) Frequency polygon. (1 mark) | < Answer >
65. | The value of chi-square statistic cannot be negative, because (a) The observed frequencies are always greater than the expected frequencies (b) The differences between the observed and expected frequencies are squared © The chi-square statistic is based upon the absolute value of the differences between the observed frequencies and expected frequencies (d) The chi-square statistic is the summation of squares of observed frequencies (e) The chi-square statistic is the summation of squares of expected frequencies. (1 mark) | < Answer >
66. | 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 © 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 >
67. | Which of the following is an underlying assumption in ANOVA? (a) The populations follow unknown distributions (b) The sample means are equal © The samples are dependent on each other (d) The population variances are equal (e) The population variances are not equal. (1 mark) | < Answer >
68. | Which of the following is the graphical plot of the values of the dependent and independent variables, in the context of simple regression analysis? (a) Scatter diagram (b) Frequency polygon (c) Histogram (d) chart (e) Ogive. (1 mark) | < Answer >
69. | Which of the following index numbers does not satisfy the time reversal test? (a) Fisher’s ideal index (b) Marshall-Edgeworth index © Fixed weights aggregate index (d) Laspeyres index (e) Time reversal test is not applicable to index numbers. (1 mark) | < Answer >
70. | Seasonal variations have minimal impact on (a) Price index (b) Quantity index © Weighted index (d) Unweighted index (e) Chain index. (1 mark) | < Answer >
71. | The weighted average of price relatives using base values as weights is same as the (a) Unweighted aggregates price index (b) Unweighted aggregates quantity index © Laspeyres price index (d) Laspeyres quantity index (e) Paasche’s price index. (1 mark) | < Answer >
72. | According to the factor reversal test if we interchange the price and quantity terms in an index number then the product of the original index number and the index number obtained by reversing the factors should be equal to (a) One (b) One hundred (c) The value index (d) The Fisher’s ideal index (e) The chain index number. (1 mark) | < Answer >

73. | Which of the following is true with regard to Fisher’s ideal price index? (a) It does not consider the base year prices (b) It does not consider the base year quantities (c) It does not consider the current year prices (d) It does not consider the current quantities (e) It is the geometric mean of the Laspeyre’s and Paasche’s price indices. (1 mark) | < Answer >
74. | Which of the following is a technique for studying seasonal variations? (a) Regression analysis (b) Residual method (c) Hypothesis testing (d) Ratio to moving average method (e) Linear programming. (1 mark) | < Answer >
75. | Which of the following is an example for cyclical variation in time series analysis? (a) Substantial increase in the demand for greeting cards in December of every year (b) Improvement in the quality and uses of the product due to the continuous development of the technology (c) Fall in the demand for products due to the depression in the economy (d) Increase in the growth of population due to improvement in medical facilities (e) Increase in the population in Andhra Pradesh resulting from the sudden migration of people from Orissa due to the occurrence of floods. (1 mark) | < Answer >
76. | Which of the following statements is false? (a) The expected value of the sum of two random variables is equal to the sum of the expected values of the two random variables (b) The variance of the sum of two random variables is equal to the sum of the variances of the two random variables if the random variables are independent (c) The expected value of the sum of two random variables is equal to zero if the expected value of each of the random variables is zero (d) The variance of the sum of two random variables is equal to zero if the random variables are independent (e) The covariance between two dependent random variables is not zero. (1 mark) | < Answer >
77. | For a simple linear regression equation =1024 and the standard error of estimate is 8. What is the number of observations? (a) 8 (b) 12 (c) 16 (d) 18 (e) 64. (1 mark) | < Answer >
78. | If the covariance of two random variables X and Y is equal to zero then it can be said that (a) The random variables are dependent on each other (b) The random variables have a positive correlation (c) The random variables have a negative correlation (d) The random variables are independent of each other (e) The variances of the two random variables are always zero. (1 mark) | < Answer >
79. | The variance of the sampling distribution of mean is (a) Always equal to the variance of the underlying population (b) Less than the variance of the underlying population if the sample size is more than one (c) More than the variance of the underlying population if the sample size is more than one (d) Equal to the standard deviation of the underlying population if the sample size is more than one (e) Not dependent on the sample size. (1 mark) | < Answer >
80. | The arithmetic mean of the upper and lower limits of the confidence interval for population proportion is equal to the (a) Population proportion (b) Sample proportion (c) Population variance (d) Sample variance (e) Standard error of proportion. (1 mark) | < Answer >

81. | In which of the following conditions will a binomial distribution be symmetrical? (a) Number of trials is 30 or more (b) Number of trials is 100 or more (c) Probability of success in any trial is less than 0.5 (d) Probability of failure in any trial is less than the probability of success (e) Probability of failure in any trial is equal to 0.5. (1 mark) | < Answer >
82. | If a z-value is to the left of the mean in the standard normal distribution, then its value (a) Is negative (b) Is positive (c) Lies between 0 and (d) Is zero (e) Is 1. (1 mark) | < Answer >
83 | When decisions are made under uncertain conditions, which of the following criteria for decision making is based on the opportunity costs of making particular decisions? (a) Maximin criterion (b) Maximax criterion (c) Hurwicz criterion (d) Regret criterion (e) Expected value. (1 mark) | < Answer >
84. | Which of the following refers to the random (chance) occurrences that can affect the outcome of an individual's decision? (a) Payoffs (b) States of nature (c) Decision alternatives (d) Probabilities (e) Regret criterion. (1 mark) | < Answer >
85. | The finite population correction factor (a) Increases as the sample size increases (b) Decreases as the sample size increases (c) Remains constant for all possible sample sizes (d) Is equal to 1 for sample sizes of 2 and above (e) Is more than 1 for sample sizes of 2 and above. (1 mark) | < Answer >
86. | If the regression coefficient of the independent variable in a simple regression equation is negative then, which of the following statements is correct? (a) The coefficient of correlation between the variables is zero (b) The coefficient of correlation between the variables is the positive square root of the coefficient of determination (c) The coefficient of correlation between the variables is the negative square root of the coefficient of determination (d) The coefficient of correlation between the variables is 1 (e) The coefficient of correlation between the variables is between zero and 1. (1 mark) | < Answer >
87. | If the relative cyclical residual measure for a year in a time series is less than zero, it indicates that (a) The actual time-series value lies below the trend line and the percent of trend is more than 100 (b) The actual time-series value lies below the trend line and the percent of trend is equal to 100 (c) The actual time-series value lies above the trend line and the percent of trend is less than 100 (d) The actual time-series value lies above the trend line and the percent of trend is more than 100 (e) The actual time-series value lies below the trend line and the percent of trend is less than 100. (1 mark) | < Answer >
88. | A continuous uniform probability distribution is described by (a) The expected value of the distribution (b) The variance of the distribution (c) The upper and lower limits of range of values of the distribution (d) The mid-point of the range of values of the distribution (e) Only the lower limit of the range of values of the distribution. (1 mark) | < Answer >
89. | In which of the following ways can binomial trials be performed on a finite population with specific numbers of successes and failures? (a) By sampling without replacement (b) By sampling with replacement (c) By adding a constant to the number of successes obtained in a given number of trials (d) By deducting a constant from the number of successes obtained in a given number of trials (e) By multiplying a constant to the number of successes obtained in a given number of trials. (1 mark) | < Answer >

Suggested Answers
Quantitative Methods II (132) – October 2005

1. | Answer : (c) Reason : | < TOP >
2. | Answer : (d) Reason : | < TOP >
3. | Answer : (c) Reason : Marginal profit (MP) = Selling price – Cost – Commission = 200 – 150 – 10 = Rs.40 Marginal loss (ML): Discounted price of the T-shirt = 200 (1 – 0.45)= Rs.110 Less: Commission payable = Rs. 20 Amount receivable from discount sale = Rs. 90 Marginal loss (ML) = Cost of the T-shirt – Amount receivable from the discount sale = 150 – 90 = Rs.60 Minimum required probability of selling an additional T-shirt (p*) = = = 0.60 = 60%. Since 0.50 of the area under the normal distribution is located between the mean and the right tail 0.10 of the shaded area in the diagram must be to the left of the mean, (0.60 – 0.50 = 0.10). From the standard normal distribution table we find that the nearest value to 0.10 is 0.0987 which corresponds to a Z-value of 0.25. Let the value of the variable be x for which we get a Z-value of 0.25. \ = –0.25 (–ve because it is on the left side of the mean). Given, m = 400, s = 60 \ = –0.25 or x = 400 + (–0.25) (60) = 385. \ The company should stock 385 T-shirts of XL size of the brand. | < TOP >
4. | Answer : (e) Reason : The event 40 £ Y £ 80 occurs when Y = 40 or 60 or 80 \ P(40 £ Y £ 80 ) = 0.20 + 0.30 + 0.25 = 0.75 Let Z donote the number of observations of Y which satisfy the condition 40 £ Y £ 80. \ Z follows a binomial distribution with number of trials = 15 and probability of success = 0.75. \ P (Z <12) = P (Z £11) = 1–P(Z ³ 12) = 1– [P(Z=12)+ P(Z = 13) + P(Z=14) + P(Z=15)] = 1– = 1– [0.2252 + 0. 1559 + 0.0668 + 0.0134] = 1 – 0.4613 = 0.5387. | < TOP >
5. | Answer : (c) Reason : Let the mean be m and the standard deviation s. 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 are lying to the right of the ordinate at X = 45 and up to the mean is (0.50 – 0.3085) = 0.1915. The value of z corresponding to this area is 0.5 \ z or 45 = m – 0.5s …………(i) Now, 8.08% of the items are above 64. Therefore, area to the right of the ordinate at 64 is 0.0808. Area to the left of the ordinate at X = 64 up to man ordinate is (0.5 – 0.08) = 0.4192 and the value corresponding to this area is 1.4. \ z or 64 = m + 1.4s …………….(ii) From equation (i) and (ii), we get the values m = 50 and s = 10. | < TOP >
6. | Answer : (d) Reason : Standard normal variate is z . Here X = 142, m = 130, s = 10, N = 2,000. Now the corresponding z value is z Now area between the ordinates z = 0 and z = 1.2 is 0.3849. Probability of units having weight less than 142 kg = 0.5000 + 0.3849 = 0.8849. \The expected number of units weighing less than 142 kg = 2,000 ´ 0.8849 = 1769.8 or 1770 units. | < TOP >
7. | Answer : (e) Reason : Since 30% students get less than 40 marks then 20% students get more than 40 marks and less than mean marks. From standard normal table, z value corresponding to 0.2 (20% of area) = – 0.524 Hence (using interpolation) Þ 40 = m – 0.524s ………. (i) Also 10% students get distinction marks, i.e. 75 or more. The z value corresponding to 0.4 (40 % of area) is 1.282 (using interpolation) Hence ………………(ii) Þ 75 = m + 1.282s Solving the two equations (i) and (ii), we get the mean, m = 50.16. | < TOP >
8. | Answer : (a) Reason : The number of defective pens out of the pens randomly picked from the box follows a hypergeometric distribution. The following details are available: N = No. of elements in the population = 60 r = No. of elements in the population labelled success = No. of defective pens = 10 n = No. of trials = 8 x = Desired number of successes = 3 P(X = 3) = | < TOP >
9. | Answer : (c) Reason : Since the businessman is perfectly optimistic the appropriate decision criterion is Maximax criterion. The decision will be made as follows: Net gains (Rs.)
Asset | State of nature | Maximum gains
S1 | S2 | S3
A | 30,000 | 12,000 | 10,000 | 30,000
B | 40,000 | 37,000 | –6,000 | 40,000
C | 50,000 | 25,000 | –30,000 | 50,000
The maximum of the maximum gains is offered by asset C. Hence he should invest in asset C. | < TOP >
10. | Answer : (e) Reason : In decision theory, the following conditions characterize decision-making under conditions of risk: (II) The various states of nature are known. (IV) The decision maker has sufficient information to assign probabilities of occurrence to various states of nature. Hence answer is (e). | < TOP >
11. | Answer : (e) Reason : Total number of individual borrower accounts = ´ 0.90 = 1440 Number of individual borrower accounts surveyed = 144 Sampling fraction = = = 0.10 > 0.05 Hence this can be treated as sampling done from a finite population. \ Proportion of excellent borrowers in the sample, = = 0.75 = 1– = 1–0.75 = 0.25 Estimated standard error of proportion, = = = 0.03608 ´ 0.9490 = 0.0342 Since both n and n are greater than 5, the sample can be treated as a large sample and normal distribution can be used. For a confidence level of 90%, Z = ± 1.645 1.645 = 1.645 ´ 0.0342 = 0.0563 Confidence interval for the proportion of accounts in the individual borrower category who are in the excellent standing = 0.75 ± 0.0563 \ Confidence interval for the absolute number of accounts in that category = (0.75 ´ 1440) ± (0.0563 ´ 1440) = 1080 ± 81.07 | < TOP >
12. | Answer : (b) Reason : Since both the samples are small and the population variances are unknown the appropriate distribution is t distribution with 16 + 9 – 2 = 23 d.o.f. This is a right-tailed test. At a significance level of 0.01 the critical value is 2.500. The observed test statistic falls well within the acceptance region. Hence we accept the null hypothesis. | < TOP >
13. | Answer : (d) Reason : Under the null hypothesis, the non-standardized critical values for this test are = Hence the null hypothesis will be rejected for sample means less than 7.06 and more than 12.94 and accepted for the values in between. | < TOP >
14. | Answer : (c) Reason : The point estimate of population mean = sample mean = | < TOP >
15. | Answer : (d) Reason : When a sample of size n with a variance of s2, is selected from a normal population with a variance of the sampling distribution of the statistic has a Chi-square distribution with n –1 degrees of freedom. | < TOP >
16. | Answer : (d) Reason : H0: p = 0.40 H1: p 0.40 This is a large sample test of proportion. So we can use the normal approximation to the binomial. = = = 0.02582. \ z = = –2.32 At a = 0.05, the critical values are ±1.96. The test statistic is less than the left tail critical value. So it falls in the rejection region. \ We reject H0. The true proportion is 0.40. So H0 is true. Hence the test leads to a type I error. | < TOP >
17. | Answer : (c) Reason : Estimated standard error of difference between means: Value of the test statistic = | < TOP >
18. | Answer : (c) Reason : Since the sample is a small and the population variance is not known the distribution to be used is tn–1 (or t15) \ UCL = or = = 101.31 – 80 = 21.31 Or = = 10 Or = 10 Or Sample variance, s2 = 100n = 100 ´ 16 = 1600 | < TOP >
19. | Answer : (c) Reason : UCL = or = UCL – = 0.531 – 0.48 = 0.051 \LCL = = 0.48 – 0.051 = 0.429. | < TOP >
20. | 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 is normally distributed with mean = and variance = Hence probability of rejecting the null hypothesis when it is true = | < TOP >
21. | Answer : (c) Reason : The distribution used for large sample interval estimation for population proportion is normal distribution. \ UCL = = 0.4776 LCL = = 0.2424 UCL – LCL = 3.92 Or 0.2352 = 3.92 Or = = 0.06. | < TOP >
22. | Answer : (d) Reason : By Central Limit Theorem for large samples the sample mean is approximately normally distributed with mean = population mean and standard deviation = m = 50 = = = 1 = = P(z > 1.00) = 0.50 – P(0 < z < 1) = 0.50 – 0.3413 = 0.1587. | < TOP >
23. | Answer : (d) Reason : is the appropriate formula for computing the standard error of mean when sampling is done from an infinite population or sampling is done from a finite population with replacement | < TOP >
24. | Answer : (b) Reason : The arithmetic mean of the upper and lower limits of the confidence interval for population mean is equal to the sample mean. | < TOP >
25. | Answer : (e) Reason : If a population distribution is known to be normal, then then the sampling distribution of mean will be normally distributed with mean equal to the population mean. | < TOP >
26. | Answer : (e) Reason : H0 : m = 9 H1 : m ¹ 9 The sample is large. So we shall use the normal distribution. = = = 1.125 z = = = 1.78. At a = 0.05, Critical values are ± 1.96 The test statistic is less than the right tail critical value. So it falls in the acceptance region. \ H0 is accepted. But H0 is false because m = 11. So the test leads to a type II error. | < TOP >
27. | Answer : (a) Reason : | < TOP >
28. | Answer : (d) Reason : A two-tailed test still has two critical values | < TOP >
29. | Answer : (e) Reason : For testing the hypothesis of the difference between two sample means when the samples are small and independent and the population variances are unknown, the required assumptions are: (I) Both populations have normal distributions and (II) Both populations have equal variances. Hence assumptions (III) and (IV) are not required. | < TOP >
30. | Answer : (b) Reason : If the sample size is 2 or more, than the standard error of mean will always be less than the standard deviation of population | < TOP >
31. | Answer : (e) Reason : Fisher’s ideal price index = = = | < TOP >
32. | Answer : (b) Reason : Laspeyres quantity index = = | < TOP >
33. | Answer : (c) Reason : Paasches quantity index = = = | < TOP >
34. | Answer : (e) Reason : Unweighted aggregates quantity index = = = (Where ‘n’ represents the no. of goods in the basket). | < TOP >
35. | Answer : (d) Reason : Laspeyres quantity index = = | < TOP >
36. | Answer : (a) Reason : For Fishers price index we do the following calculations: å P0 Q0 = 4690 å P1 Q0 = 6325 Laspeyres Price Index = | < TOP >
37. | Answer : (b) Reason : For a curvilinear trend we fit the equation of type Y = a + bX + cX2. The corresponding estimating equation is . Where x represents the coded time variable. The calculation is shown below: Let X represents the Year and Y represents the number of PCs sold in thousands.
Data | X | Y | x =X- | x2 | x4 | xY | x2Y
1 | 1998 | 67 | -2 | 4 | 16 | -134 | 268
2 | 1999 | 71 | -1 | 1 | 1 | -71 | 71
3 | 2000 | 76 | 0 | 0 | 0 | 0 | 0
4 | 2001 | 82 | 1 | 1 | 1 | 82 | 82
5 | 2002 | 96 | 2 | 4 | 16 | 192 | 384
Total | 10000 | 392 | 0 | 10 | 34 | 69 | 805
Mean of X, = We have , , And b = Substituting the values in these equations, we have 392 = 5 a + 10 c ………………. (i) 805 = 10 a +34 c ………………. (ii) b = ………………… (iii) Multiplying equation (i) by two and subtracting from equation (ii) we get, (805 = 10 a + 34 c) – ( 784 = 10 a + 20 c) Þ 21 = 14 c \c = = 1.5. | < TOP >
38. | Answer : (e) Reason : The estimating equation can be calculated as below:
Year (X) | Sales (Y) | Z = X- | x = 2Z | xY | x2
1995 | 10 | -2.5 | -5 | -50 | 25
1996 | 13 | -1.5 | -3 | -39 | 9
1997 | 16 | -0.5 | -1 | -16 | 1
1998 | 21 | 0.5 | 1 | 21 | 1
1999 | 24 | 1.5 | 3 | 72 | 9
2000 | 30 | 2.5 | 5 | 150 | 25
Total | 114 |  | 0 | 138 | 70
= 1997.5 The equation of the straight line trend is Yc = a + bx Since Sx = 0, Thus the equation of the trend line is = 19 + 1.9714x Therefore = 19 + 1.9714 ´ 2 (2004 – 1997.5) = 19 + 1.9714 ´ 13 = 44.6282 (in thousands) » 44,628 numbers of air conditioners. | < TOP >
39. | Answer : (b) Reason : To compute the coefficient of correlation between the two variables we tabulate them as below:
Size (X) | Price (Y) |  |
108 | 12 | 61.13 | 0.92
39 | 13 | -7.87 | 1.92
68.4 | 19 | 21.53 | 7.92
51.0 | 12 | 4.13 | 0.92
10.4 | 6.5 | -36.47 | -4.58
4.40 | 4.0 | -42.47 | -7.08
| x2 | y2 | xy
| 3736.8769 | 0.8464 | 56.2396
| 61.9369 | 3.6864 | 15.1104
| 463.5409 | 62.7264 | 170.5176
| 17.0569 | 0.8464 | 3.7996
| 1330.0609 | 20.9764 | 167.0326
| 1803.7009 | 50.1264 | 300.6876
Total = | 7413.1734 | 139.2084 | 683.1666
The mean values We know that the coefficient of correlation r = \ r = = 0.6725 | < TOP >
40. | Answer : (a) Reason : = - 0.65 a = = 8 – (– 0.65) ´ 6 = 11.9 The regression equation is = 11.9 – 0.65 X. | < TOP >
41. | Answer : (c) Reason : Given that: r = 0.8, å xy = 60, sy = 2.5 and å x2 = 90. Now we know that | < TOP >
42. | Answer : (c) Reason : This problem requires the application of ANOVA. Calculating variance within the samples: Second estimate of population variance using variance within the samples = S = = 13 == = = 15.5 == = = 14 == = = 16 == \ Variance within the samples = = = = 3.50. | < TOP >
43. | Answer : (c) Reason : We first find the value of the grand mean:
Samples | A | B | C | D
1 | 78 | 94 | 74 | 79
2 | 87 | 91 | 78 | 83
3 |  |  |  |
81 | 87 | 70 | 78 |
4 | 89 | 90 | 82 | 69
5 | 85 | 88 | 76 | 81
Total | 420 | 450 | 380 | 390
The value of the grand mean | < TOP >
44. | Answer : (d) Reason : The upper confidence limit for the estimated population variance is given by . The value of Chi square statistic from the table with 0.05 area in the left tail and (20 – 1) = 19 degrees of freedom is 10.117. \ The upper confidence limit for the estimated population variance is . | < TOP >
45. | Answer : (a) Reason : H0 : s = 50 or s2 = 502 H1 : s ¹ 50 or s2 ¹ 502 The value of Chi square statistic from the table with 0.025 area in the left tail and (30 – 1) = 29 degrees of freedom is 16.047. | < TOP >
46. | Answer : (c) Reason : The critical value can be found by looking at the F table with: Keeping the larger sample variance in the numerator Numbers of degrees of freedom in numerator as (n2 - 1) = 11 – 1=10 Numbers of degrees of freedom in denominator as (n1 - 1) = (9 – 1) =8 The critical value from the table is F(10, 8, 0.05) = 3.35. | < TOP >
47. | Answer : (d) Reason : This is a chi-square goodness of fit test There are 6 elements in this table therefore the number of degrees of freedom = (6 – 1) = 5. Therefore from the table we obtain the critical value of chi square distribution as 11.070 at 5% level of significance. | < TOP >
48. | Answer : (b) Reason : This is a chi-square test of independence The number of degrees of freedom of the given contingency table is equal to (r – 1) (c – 1) = (2 – 1)(4 – 1) = 3. The chi square value from the table at 5% level of significance and 3 degrees of freedom = 7.815. | < TOP >
49. | Answer : (e) Reason : Lower Control Limit for p chart = | < TOP >
50. | Answer : (d) Reason : The central line of the chart is = Mean of the sample means () Mean of the sample means or grand mean, = 106.6 /10 = 10.66 Mean of sample range = = åR / n = 63 /10 = 6.3 Upper Control Limit of chart = + A2 = 10.66 + 0.577 ´ 6.3 = 14.2951 (The value of A2 for sample of size 5 is 0.577 from control chart factors table) | < TOP >
51. | Answer : (e) Reason : (a) This is true for multiple correlation coefficient. It depends on the correlation coefficient between Y and X1. (b) This is true for multiple correlation coefficient. It depends on the correlation coefficient between X1 and X2. (c) This is true for multiple correlation coefficient. It depends on the correlation coefficient between Y and X2. (d) This is true for multiple correlation coefficient. It will take values between 0 and 1. (e) This is false for multiple correlation coefficient. It will take values between 0 and 1. It does not take negative values. | < TOP >
52. | Answer: (c) Reason: The Lower Control Limit in an R Chart = - = 7.1 - = 0.538. Hence from above discussion, we can infer that option (c) is correct. | < TOP >
53. | Answer: (b) Reason: If the coefficient of correlation between two variables lies between –1 and 0, then the covariance between them is negative. | < TOP >
54. | Answer: (a) Reason: The proportion of variations that is explained by the regression line is represented by the coefficient of determination. | < TOP >
55. | Answer : (b) Reason : In the linear model, the residuals are assumed to have a normal distribution with a mean of zero. | < TOP >
56. | Answer : (a) Reason : The coefficient of determination is the square of coefficient of correlation (r). Hence it will always be 0 and 1. From above we can see that the coefficient of determination cannot be less than zero. 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. The coefficient of determination is always positive; The coefficient of correlation may be negative. 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 >
57. | Answer : (c) Reason : r = r = +1 Þ Cov (x, y) = sx . sy. | < TOP >
58. | Answer : (b) Reason : When the slope of a regression line is positive there is a positive correlation between the dependent and independent variables | < TOP >
59. | Answer : (d) Reason : The statement that there are always as many points above the fitted line as there are below it is wrong. | < TOP >
60. | Answer : (d) Reason : a. The F statistic is not calculated on the basis of the mean of the largest sample. b. The F statistic is not calculated on the basis of the mean of the smallest sample. c. The F statistic is not calculated on the basis of the standard deviation of the smallest sample. d. The F statistic is calculated on the basis of two estimates of the population variance viz., the estimated population variance based on the variance among the sample means and the estimated population variance based on the variance within the samples. e. The F statistic is not calculated on the basis of the variance of the largest sample | < TOP >
61. | Answer : (a) Reason : The required cyclical variation for the given year by relative cyclical residual technique will be = = –6.25 | < TOP >
62. | Answer : (b) Reason : a. ‘Hugging the control limits’ indicates that two different populations are being observed. b. ‘Hugging the center line’ indicates that the variations have been reduced significantly. c. Cycles indicate the possibility of presence of random variations in the process. d. Increasing trend indicates that the process mean is increasing. e. Decreasing trend indicates that the process mean is decreasing | < TOP >
63. | Answer : (b) Reason : a. In a p chart, the center line is drawn on the basis of overall sample proportion. b. In a p chart, the lower control limit can only be greater than or equal to zero. c d & e. The upper and lower control limits are fixed by adding and subtracting the estimated standard error of proportion from the overall sample proportion | < TOP >
64. | Answer: (b) Reason: (a) This is a wrong answer. The charts are constructed using the mean value as the central limit and three standard deviations from the mean as upper and lower control limits. (b) This is the right answer. R charts are used to monitor the variability in the individual characteristics of the product. The control limit is the mean range and the upper and lower control limits are constructed with the standard deviation in the range values. (c) This is the wrong answer. p charts deal with the attributes that are qualitative in nature. The measure that is applied here is the proportion of the data points satisfying the given criteria and the proportion of the data points that does not satisfy the criteria. (d) This is the wrong answer. (e) This is the wrong answer. | < TOP >
65. | Answer: (b) Reason: a. The observed frequencies may be less than the expected frequencies. Hence this is not the correct reason. b. The differences between the observed and expected frequencies are squared. Hence the chi-square statistic cannot be negative. c. The chi-square statistic is not based upon the absolute value of the difference between the observed frequencies and expected frequencies. d & e. The chi-square statistic is not based upon the summation of the squares of the observed or expected frequencies | < TOP >
66. | 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 >
67. | Answer : (d) Reason : One of the assumptions in ANOVA is that the population variances are equal. | < TOP >
68. | Answer : (a) Reason : 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. An chart is a quality control chart. An ogive is a graphical plot of a cumulative frequency distribution | < TOP >
69. | Answer: (d) Reason: Laspeyres index does not satisfy the time reversal test | < TOP >
70. | Answer: (e) Reason: In Chain Index, the base year is not fixed but changes from year to year. Here, new relatives are constructed for each year with the previous year as base. Thus, seasonal variations have minimal impact on Chain Index. Hence from above discussion, we can infer that option (e) is correct. | < TOP >
71. | Answer : (c) Reason : (a) This is the wrong answer. The weighted average of price relatives using base values as weights is not same as the unweighted aggregates price index. (b) This is the wrong answer. The weighted average of price relatives using base values as weights is not same as the unweighted aggregates quantity index. (c) This is the right answer. The weighted average of price relatives using base values as weights is same as the Laspeyres price index. (d) This is the wrong answer. The weighted average of price relatives using base values as weights is not same as the Laspeyres quantity index. (e) This is the wrong answer. The weighted average of price relatives using base values as weights is not same as the Paasche’s price index. | < TOP >
72. | Answer : (c) Reason : (a) This is the wrong answer. According to the factor reversal test the product of the original index and the index obtained through reversing the factors is not equal to one. (b) This is the wrong answer. According to the factor reversal test the product of the original index and the index obtained through reversing the factors is not equal to one hundred. (c) This is the right answer. According to the factor reversal test if we interchange the price and quantity terms in an index number then the product of the original index number and the index number obtained by reversing the factors should be equal to the value index. (d) This is the wrong answer. According to the factor reversal test the product of the original index and the index obtained through reversing the factors is not equal to the Fisher’s ideal index. (e) This is the wrong answer. According to the factor reversal test the product of the original index and the index obtained through reversing the factors is not equal to the chain index number. | < TOP >
73. | Answer : (e) Reason : a. Fisher’s ideal price index considers base year prices. b. Fisher’s ideal price index considers base year quantities. c. Fisher’s ideal price index considers current year prices. d. Fisher’s ideal price index considers current year prices. e. Fisher’s ideal price index is the geometric mean of the Laspeyres and Paasche’s price indices | < TOP >
74. | Answer: (d) Reason: Ratio-to-moving-average method is a technique for studying seasonal variations. | < TOP >
75. | Answer : (c) Reason : Substantial increase in the demand for greeting cards in December of every year is an example for seasonal variation. Improvement in the quality and uses of the product due to the continuous development of the technology is an example for secular trend. Fall in the demand for products due to the depression in the economy is an example for cyclical variation. Increase in the growth of population due to improvement in medical facilities is an example for secular trend. Increase in the population in Andhra Pradesh resulting from the sudden migration of people from Orissa due to the occurrence of floods is an example for irregular variation. | < TOP >
76. | Answer : (d) Reason : The covariance of the sum of two random variables is equal to zero if the random variables are independent. In such a case the variance of the sum will equal to the sum of the individual variances and hence not equal to zero. | < TOP >
77. | Answer : (d) Reason : \ or or n = 16 + 2 = 18 | < TOP >
78. | Answer : (d) Reason : (a) This is a wrong answer. If the covariance of two random variables X and Y is equal to zero then it can be said that the random variables are independent of each other. If the variables are dependent on each other then the covariance will assume any non-zero value. (b) This is a wrong answer. If the coefficient of correlation is positive then we say that there is a positive correlation between two variables. (c) This is a wrong answer. If the coefficient of correlation is negative then we say that there is a negative correlation between two variables. (d) This is the correct answer. If the covariance of two random variables X and Y is equal to zero then it can be said that the random variables are independent of each other. (e) This is a wrong answer. If the covariance of the two random variables is zero it does not mean that the variances of the individual variables will be zero. | < TOP >
79. | Answer : (b) Reason : The variance of the sampling distribution of mean is less than the variance of the underlying population if the sample size is more than one. | < TOP >
80. | Answer : (b) Reason : The arithmetic mean of the upper and lower limits of the confidence interval for population proportion is equal to the sample proportion. UCL = LCL = | < TOP >
81. | Answer : (e) Reason : A binomial distribution is symmetrical if the probability of success and the probability of failure are equal. Since p + q = 1, this means that the probability of failure is equal to 0.5. | < TOP >
82. | Answer : (a) Reason : If a z value is to the left of the mean in the standard normal distribution, then its value is negative. | < TOP >
83. | Answer : (d) Reason : In decision making under uncertainty the regret criterion is based on opportunity costs of decisions. | < TOP >
84. | Answer : (b) Reason : a. The payoffs represent the outcomes of certain decisions. b. The states of nature refer to the chance occurrences that can affect the outcome of an individual’s decision c. The decision alternative courses of action that are available to the decision maker. d. The probabilities refer to the likelihood of the chance occurrences or the states of nature. e. Regret criterion is a measure of magnitude of loss incurred by not selecting the best alternative. | < TOP >
85. | Answer : (b) Reason : The finite population correction factor is \ As the sample size n increases, the finite population correction factor decreases. | < TOP >
86. | Answer: (c) Reason: ‘b’ is the regression coefficient of the independent variable. b < 0 implies that the variables x and y are negatively correlated. \ Coefficient of correlation = Negative square root of coefficient of determination. | < TOP >
87. | Answer : (e) Reason : If the relative cyclical residual measure for a year in a time series is less than zero, it indicates that the actual time-series value lies below the trend line and the percent of trend is less than 100. | < TOP >
88. | Answer : (c) Reason : A continuous uniform probability distribution is described by the upper and lower limits of the interval. | < TOP >
89. | Answer : (b) Reason : By sampling with replacement binomial trials can be performed on a finite population with specific number of successes and failures. | < TOP >

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