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

1. | A student takes a test containing ten multiple choice type questions. Each multiple choice type question has four alternative solutions only one of which is the correct solution. The student is not prepared for the test and he is expected to answer the questions by randomly selecting one alternative from the given choices. It is assumed that each alternative is equally likely to be selected by the student. What is the probability that the student will answer four questions correctly? (Round off your answer). (a) 14.6% (b) 0.39% (c) 85.4% (d) 17.8% (e) 82.2%. (1 mark) | < Answer >
2. | For a binomial distribution the mean is 8. The probability of failure in any trial is known to be 0.80. What is the standard deviation of the binomial distribution? (a) 8 (b) 6.40 (c) (d) 3.20 (e) 40.96. (1 mark) | < Answer >
3. | The wind speed at a wind energy station is approximately normally distributed with a mean of 25 miles per hour and a standard deviation of 7 miles per hour. When wind speeds exceed 45 miles per hour, the station produces too much electricity for the current transmission lines and must be shut down. What percentage of the time will the station be shut down? (a) 0.21% (b) 2.1% (c) 9.79% (d) 28.6% (e) 50%. (1 mark) | < Answer >
4. | The chance of an individual being vegetarian is 1/2. Assuming that each of 100 investigators takes a sample of 10 individuals to see whether they are vegetarian, how many investigators would you expect to report that 3 people or less were vegetarians? (Round of your answer) (a) 5 (b) 12 (c) 15 (d) 17 (e) 21. (1 mark) | < Answer >
5. | In a normal distribution, 30.85% of the items are under 45 and 8.08% are over 64. Find the mean of the distribution. (a) 47 (b) 48 (c) 49 (d) 50 (e) 51. (1 mark) | < Answer >
6. | 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. Determine the standard deviation of the distribution of marks, assuming this to be normal. (a) 17.2 (b) 18.3 (c) 19.4 (d) 20.5 (e) 21.6. (1 mark) | < Answer >
7. | A baker has studied his record and notices that for the past 310 working days in the year, the demand for his product (bread) has varied as follows:
Demand (‘000 units) | 5 | 6 | 7 | 8 | 9 | 10
Number of days | 20 | 60 | 80 | 120 | 20 | 10
What is the expected demand for his product? (a) 5270.08 units (b) 6280.16 units (c) 7290.32 units (d) 8200.64 units (e) 9210.72 units. (1 mark) | < Answer >
8. | There are 25 students in a class, which consists of 14 boys and 11 girls. 5 students of the class were absent on a particular day. What is the probability that at least 4 boys were absent? (a) 0.20725 (b) 0.03768 (c) 0.24493 (d) 0.75507 (e) 0.79275. (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 pessimistic about the future. Which asset should he invest in? (a) Asset A (b) Asset B (c) Asset C (d) Either asset B or C (e) Either asset A or B. (1 mark) | < Answer >
10. | In a decision making problem the expected profit of the optimal decision without perfect information is Rs.40,000 and the expected profit with perfect information is Rs.45,000. What is the expected value of perfect information? (a) – Rs.5,000 (b) Rs.5,000 (c) Rs.85,000 (d) Rs.42,500 (e) Rs.2,500. (1 mark) | < Answer >
11. | Assuming the applicability of the Central Limit Theorem, if the 95% confidence limits for the population mean are 73 and 80, which of the following could be the 99% confidence limits? (a) 72 and 81 (b) 73 and 82 (c) 72 and 82 (d) 71 and 83 (e) 70 and 84. (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 value is 2.330 and the null hypothesis is accepted (d) The critical value is 2.069 and the null hypothesis is rejected (e) The critical value is 1.714 and the null hypothesis is accepted. (2 marks) | < Answer >
13. | The following details are available with regard to a hypothesis test on population mean: H0: H1: Significance level = 0.05 n = 64 It is later known that the true population mean is 12. Which of the following can be said with regard to the test? (a) There is insufficient information for doing the test (b) The t distribution should be used (c) The test does not lead to either type I or type II error (d) The test leads to a type I error (e) The test leads to a type II error. (2 marks) | < Answer >
14. | The deviations of the observations in a simple random sample, from zero are given below:
6 | 11 | 9 | 11 | 8 | 15
What is the point estimate of the population mean? (a) 6 (b) 9 (c) 10 (d) 11 (e) 60. (1 mark) | < Answer >
15. | A population has a mean of 75 and a standard deviation of 10. Samples of size 20 are chosen and the sample means recorded. What is the expected value of the sampling distribution of mean? (a) 2.236 (b) 10 (c) 75 (d) 10 (e) 7.5. (1 mark) | < Answer >
16. | 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 = 0.37 Significance level = 0.05 It is later known that the true population proportion is 0.35. 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 >
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) –3.578 (b) 3.578 (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 70. The population variance is not known. The sample size is 16 and the lower limit of a 95% confidence interval for population mean is 48.69. What is the sample variance? (a) 1,400 (b) 1,500 (c) 1,600 (d) 3,200 (e) 1,000. (1 mark) | < Answer >
19. | A sample of size 96 has been taken from a population and the sample proportion has been found to be 0.40. The lower limit of a 95% confidence interval for population proportion is 0.302. What is the upper limit of the confidence interval? (a) 0.196 (b) 0.204 (c) 0.098 (d) 0.40 (e) 0.498. (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 commiting a type I error? (a) 0.25 (b) 0.025 (c) 0.0025 (d) 0.05 (e) 0.005. (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 sample proportion? (a) 0.12 (b) 0.06 (c) 0.64 (d) 0.36 (e) 0.18. (1 mark) | < Answer >
22. | A manufacturer of light bulbs claims that the distribution of the light bulb life span has a mean of 600 hours and a standard deviation of 60 hours. The competitors decide to check this claim by purchasing 36 light bulbs and testing them to determine the life span of the light bulbs. If the manufacturer's claim is true, how will the sampling distribution of the mean of 36 light bulbs compare to the reported population distribution? (a) The sampling distribution of mean of 36 light bulbs will have a mean of less than 600 hours (b) The sampling distribution will have a variance of 3600 hours (c) The sampling distribution will have a variance of more than 3600 hours (d) The sampling distribution will be approximately normal with a mean of 600 hours (e) The sampling distribution will be approximately the same as the population distribution which is unknown. (1 mark) | < Answer >
23. | The observations from a simple random sample are given below:
15 | 9 | 11 | 8 | 11 | 6
The sample size is less than 5 percent of the population size. What is the estimated standard error of mean? (a) 3.098 (b) 1.265 (c) 3.919 (d) 0.516 (e) 1.6. (1 mark) | < Answer >
24. | Other things remaining the same, if the confidence level is decreased then the width of the confidence interval for the population mean will (a) Increase (b) Remain unchanged (c) Decrease by half the previous width (d) Increase by half the previous width (e) Decrease. (1 mark) | < Answer >
25. | If the sampled population has a normal distribution, then the sampling distribution of mean is a normal distribution (a) For only large sample sizes (b) For only small sample sizes (c) For any sample size (d) For only samples of size 100 or more (e) For only samples of size less than 10. (1 mark) | < Answer >
26. | For a given hypothesis test, if we do not reject the null hypothesis, and the null hypothesis is true, then which of the following can be stated? (a) Neither type I error nor type II error has been committed (b) Type I error has been committed (c) Type II error has been committed (d) Both type I and type II error have been committed (e) The test statistic is incorrect. (1 mark) | < Answer >
27. | In testing the equality of two population means, if the means of the two samples are found to be equal, the value of the test statistic will be (a) Equal to one (b) Greater than zero (c) Less than zero (d) Equal to zero (e) Greater than one. (1 mark) | < Answer >
28. | Which of the following is/are false? (a) 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 (b) The critical region is the region of rejection of null hypothesis (c) The calculation of the z-statistic in a hypothesis test of mean may be based on the standard deviation of the sample (d) The standard error of mean indicates how well a sample mean estimates the value of the population mean (e) The critical value demarcates between the acceptance region and rejection region of the sampling distribution, in a hypothesis test. (1 mark) | < Answer >
29. | In hypothesis testing, if the level of significance is made smaller, the rejection region (a) Becomes larger (b) Becomes smaller (c) Remains the same (d) Becomes equal to the acceptance region (e) Becomes larger than the acceptance region. (1 mark) | < Answer >
30. | The standard error of mean (a) Is the mean of the distribution of means of samples of a specific size taken from the same population (b) Is same for two different sampling distributions which have the same means (c) Is independent of the standard deviation of the population from which the samples are taken (d) Is independent of the size of the sample (e) Decreases as the sample size increases. (1 mark) | < Answer >
31. | The following details are available with regard to a basket of goods: Laspeyres price index = 125 Weighted average price of the goods in the current year using current year quantities as weights = Rs.56 Weighted average price of the goods in the base year using current year quantities as weights = Rs.50 What is Fisher’s ideal price index? (a) 89.3 (b) 112 (c) 118.32 (d) 125 (e) 84.52. (1 mark) | < Answer >
32. | The following details are available with regard to a basket of goods: SP1Q0 = 20000 SP1Q1 = 13000 Value index calculated on the basis of the same basket of goods = 81.25 What is Laspeyres price index? (a) 138.68 (b) 65 (c) 81.25 (d) 125 (e) 153.85. (1 mark) | < Answer >
33. | The following details are available with regard to a baskets of goods: Weighted average quantity consumed during the base year = 125 Weighted average quantity consumed during the current year = 81.25 Both the weighted averages are calculated on the basis of the prices during the current year as weights. Which of the following is correct? (a) Laspeyers price index = 153.85 (b) Laspeyres quantity index = 153.85 (c) Laspeyres quantity index = 65 (d) Paasches quantity index = 65 (e) Paasches price index = 65. (1 mark) | < Answer >
34. | The following details are available with regard to a group of commodities: Average quantity consumed during the base year = 127.5 units Average quantity consumed during the current year = 85 units Both the averages are simple averages. Which of the following is correct? (a) Paasches price index = 66.67 (b) Paasches quantity index = 66.67 (c) Laspeyres quantity index = 66.67 (d) Unweighted aggregates price index = 66.67 (e) Unweighted aggregates quantity index = 66.67. (1 mark) | < Answer >
35. | The following details are available with regard to a basket of commodities:
Commodity | P0Q0 (Rs.) |
A | 750 | 4/3
B | 2000 | 2/1
C | 2240 | 7/5
D | 1560 | 13/20
What is Laspeyres quantity index for the basket of commodities? (a) 69.64 (b) 37.5 (c) 84.92 (d) 117.76 (e) 266.67. (1 mark) | < Answer >
36. | The prices and quantity 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 weighted average of relatives price index for the year 2002 with the year 1992 as the base year and the year 2002 for weighting. (a) 136.19 (b) 137.29 (c) 138.31 (d) 139.42 (e) 140.53. (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 b in the estimating equation? (a) 4.7 (b) 5.8 (c) 6.9 (d) 7.0 (e) 8.1. (2 marks) | < Answer >
38. | Using the linear trend relationship between sales and year estimated by the method of least squares from the data given below, predict the sales for the year 2003.
Year | 1996 | 1997 | 1998 | 1999 | 2000
Sales of TV sets (in ‘000 units) | 12 | 18 | 20 | 23 | 27
(a) 34200 units (b) 35300 units (c) 36400 units (d) 37500 units (e) 38600 units. (2 marks) | < Answer >
39. | In a correlation study on variables X and Y, the following results are obtained:
| X | Y
Mean | 65 | 67
Standard Deviation | 2.5 | 3.5
Coefficient of correlation | 0.80
Find the regression equation of X on Y with Y as the independent variable. (a) = 24.56 + 0.3542 Y (b) = 25.46 + 0.4562 Y (c) = 26.7162 + 0.5714 Y (d) = 27.85 + 0.6785 Y (e) = 28.56 + 0.7584 Y. (2 marks) | < Answer >
40. | For a given data set consisting of 5 paired observations of X and Y values, we have åxy = - 26, åx2 = 40, åX = 30 and åY = 40, where x and y are the deviations of X and Y from their respective means. 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. | You are given the following information relating to a paired data set of variables X and Y comprising 10 observations:
= 5.5, | = 4.0, | åX2 = 385, | åY2 = 192, | å(X + Y)2 = 947
Find the value of the coefficient of correlation between X and Y. (a) 0.681 (b) 0.896 (c) -0.681 (d) -0.735 (e) -0.895. (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 |
Given: Variance among the sample means = 8.5 Variance within the samples = 3.5 We want to find out whether the mean number of personal loan applications processed per day is the same for the four employees. Use 0.05 level of significance. Find the appropriate test statistic for the above test. (a) 1.318 (b) 2.429 (c) 3.530 (d) 4.641 (e) 5.752. (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 Estimate the population variance using the variance within the samples. (a) 15.825 (b) 16.625 (c) 17.425 (d) 18.325 (e) 19.125. (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 lower limit of the confidence interval for the true population variance. (a) 5.13 (b) 6.36 (c) 7.69 (d) 8.31 (e) 9.63. (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 infer that the population standard deviation is not 50. What is the critical value in the right tail? (a) 42.557 (b) 43.773 (c) 45.772 (d) 46.979 (e) 49.588. (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 is the value of the test statistic for the above test? (a) 1.215 (b) 1.315 (c) 1.415 (d) 1.515 (e) 1.615. (2 marks) | < 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 value of the correct test statistic for the above test? (a) 0.154 (b) 0.167 (c) 0.179 (d) 0.182 (e) 0.193. (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 value of the test statistic for the appropriate statistical test? (a) 44.26 (b) 45.38 (c) 46.40 (d) 47.42 (e) 48.44. (1 mark) | < Answer >
49. | The manager of a warehouse wants to construct a control chart for the 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 upper control limit for the p chart. (a) 0.32 (b) 0.31 (c) 0.30 (d) 0.29 (e) 0.28. (1 mark) | < Answer >
50. | The following data provides the values of sample mean, , and range R, for the 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 lower control limit for the chart. (a) 7.4585 (b) 7.3474 (c) 7.2362 (d) 7.1251 (e) 7.0249. (1 mark) | < Answer >
51. | Calculate multiple correlation coefficient using following information with regard to the correlations among as variables X1, X2 and X3 : r12 (Between X1 and X2) = 0.5, r13 (Between X1 and X3) = 0.6 and r23 (Between X2 and X3) = 0.3. (a) 0.654 (b) 0.665 (c) 0.676 (d) 0.687 (e) 0.698. (1 mark) | < Answer >
52. | The following data provides the values of sample range, R, for samples of size 5 each.
Sample No. | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10
Range (R) | 7 | 4 | 8 | 5 | 7 | 4 | 8 | 4 | 7 | 9
Calculate the value of upper control limit for the R chart (Use the relevant D3 or D4 factor) (a) 13.1245 (b) 13.3182 (c) 13.4365 (d) 13.5648 (e) 13.6428. (1 mark) | < Answer >
53. | The coefficient of correlation between variables X and Y is 0.60. Cov (X, Y) = 300 sY = 20 What is the variance of 4X? (a) 5 (b) 25 (c) 100 (d) 5000 (e) 10000. (1 mark) | < Answer >
54. | The coefficient of correlation between 3X and 4Y is 0.75 and What is the covariance between 5X and 6Y? (a) 300 (b) 1200 (c) 3600 (d) 9000 (e) 1500. (1 mark) | < Answer >
55. | The manager of Pioneer Enterprises wants to formulate a relationship between total cost and sales revenue, which may be used to estimate the total cost for a given level of sales. He collects the following information from the past records.
Sales (Rs.in lakhs) | 20 | 23 | 38 | 28 | 45 | 40
Total cost (Rs.in lakhs) | 17 | 22 | 35 | 22 | 39 | 36
Formulate a relationship between sales and total cost with sales as the independent variable using simple regression analysis. What is the estimate of total cost when sales revenues are Rs.35 lakh, according to the regression relationship? (a) Rs.28.908 lakhs (b) Rs.30.897 lakhs (c) Rs.32.786 lakhs (d) Rs.34.675 lakhs (e) Rs.36.564 lakhs. (2 marks) | < Answer >
56. | The coefficient of correlation between variables X and Y is 0.75. The coefficient of variation of X is 20% and the coefficient of variation of Y is 10%. The mean of X is 30 and the mean of Y is 40. What is the covariance between X and Y? (a) 4 (b) 6 (c) 18 (d) 24 (e) 48. (1 mark) | < Answer >
57. | The coefficient of correlation between variables X and Y is 0.80. What is the coefficient of correlation between X – 10 and Y – 20? (a) 0.08 (b) 0.04 (c) 0.80 (d) 0.40 (e) 0.72. (1 mark) | < Answer >
58. | The estimated regression relationship between variables X (independent variable) and Y (dependent variable) is = 10 + 5X Let What will be the estimated regression relationship between Y and Q (Q is the independent variable)? (a) = 10 + 5Q (b) = 6 + 5Q (c) = 30 + 5Q (d) = 30 – 5Q (e) = 30 –Q. (1 mark) | < Answer >
59. | The estimated regression relationship between variables Y and X (X is the independent variable) is: a and b are unknown. Let Q = X + 10 The regression relationship between Y and Q (Q is the independent variable) is known and the slope of the same is 2. What is the slope of the regression relationship between Y and X (X is the independent variable)? (a) 2 (b) 12 (c) 20 (d) -8 (e) 0.20. (1 mark) | < Answer >
60. | A test of ANOVA is applied to data obtained from five samples, where each sample contains six observations. What is the number of degrees of freedom for the denominator of the F statistic? (a) 6 (b) 4 (c) 5 (d) 25 (e) 30. (1 mark) | < Answer >
61. | The linear trend estimating equation for a time series is given below: = 139 + 7.5x where x = Year – 1998 is the estimated value of the dependent variable Y. The observed value of Y for the year 2000 is 160. What is the relative cyclical residual for the year 2000? (a) –3.9 (b) 10.8 (c) 3.9 (d) 96.25 (e) 103.9. (1 mark) | < Answer >
62. | Which of the following serves as the centre line in an chart, when the population mean is known? (a) The smallest sample mean (b) The largest sample mean (c) The grand mean (d) The population mean (e) The population median. (1 mark) | < Answer >
63. | Which of the following is used to find out the upper and lower control limits of an chart when the population standard deviation is known? (a) Sample standard deviation (b) Sample range (c) Sample coefficient of variation (d) Actual standard error of mean for the specific sample size (e) Sample interquartile range. (1 mark) | < Answer >
64. | In a control chart there are some observations that lie outside the upper and lower control limits. These observations are referred to as the (a) Outliers (b) Attributes (c) Parameters (d) Statistics (e) Constants. (1 mark) | < Answer >
65. | Which of the following is false regarding a chi-square distribution? (a) Chi-square distribution has only one parameter (b) The distribution becomes sharply positively skewed when the number of degrees of freedom is large (c) For large number of degrees of freedom the distribution approaches symmetry (d) The distribution is continuous (e) The distribution is unimodal. (1 mark) | < Answer >
66. | In performing a chi-square test of independence, as the difference between the respective observed and expected frequencies decreases, the likelihood of accepting the null hypothesis (a) Decreases (b) Increases (c) May decrease or increase depending on the number of rows and columns in the contigency table (d) Remains same (e) Tends to become equal to the likelihood of rejecting it. (1 mark) | < Answer >
67. | Which of the following distributions have to be used for testing the equality of means of more than two populations? (a) Chi-square distribution (b) F distribution (c) Normal distribution (d) t distribution (e) Binomial distribution. (1 mark) | < Answer >
68. | If all the values considered in calculating an index are of equal importance, the index is called (a) Weighted index (b) Unweighted index (c) Average of relatives index (d) Value index (e) Quantity index. (1 mark) | < Answer >
69. | Which of the following is true with regard to Fisher’s ideal price index? (a) For a given set of commodities it is less than both Laspeyres and Paasches price indices (b) For a given set of commodities it is more than both Laspeyres and Paasches price indices (c) For a given set of commodities the value of Fisher’s ideal price index falls in between the values of Laspeyres and Paasches price indices (d) It is the ratio of Laspeyres price index to Paasches price index (e) It is the ratio of Paasches price index to Laspeyres price index. (1 mark) | < Answer >
70. | To measure changes in total monetary worth, one should calculate (a) Price index (b) Value index (c) Quantity index (d) Price relative (e) Quantity relative. (1 mark) | < Answer >
71. | In which of the following methods of calculating index numbers is it possible to change the base year without changing the weights used in the calculation? (a) The Paasches index (b) The Laspeyres index (c) The unweighted average of relative price index (d) The value index (e) The unweighted aggregates quantity index. (1 mark) | < Answer >
72. | Which of the following is an example of the simplest form of an index number? (a) Unweighted aggregates index (b) Weighted aggregates index (c) Laspeyres index (d) Paasche’s index (e) Fisher’s ideal index. (1 mark) | < Answer >
73. | Which of the following is true with regard to Fisher’s ideal price index? (a) It is free from the effect of the base year prices (b) It is free from the effect of the base year quantities (c) It is free from the effect of the current year prices (d) It is free from the effect of 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. | The movement of the dependent variable above and below the secular trend line over periods longer than one year is known as (a) Cyclical variation (b) Secular trend (c) Seasonal variation (d) Irregular variation (e) Short term variation. (1 mark) | < Answer >
76. | 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 >
77. | According to the ‘method of least squares’ criterion, the regression line should be drawn on the scatter diagram in such a way that (a) The sum of the squared values of the vertical distances from each plotted point to the line is maximum (b) The sum of the squared values of the vertical distances from each plotted point to the line is minimum (c) The sum of the squared values of the horizontal distances from each plotted point to the line is maximum (d) The sum of the squared values of the horizontal distances from each plotted point to the line is minimum (e) The sum of the squared values of the vertical distances from each plotted point to the line is equal to zero. (1 mark) | < Answer >
78. | The standard error of estimate of a simple regression line is (a) Equal to the slope of the regression line (b) Equal to the square root of the coefficient of determination (c) Equal to the Y intercept of the regression line (d) The measure of the variability of the observed values around the regression line (e) Equal to the coefficient of correlation between two variables. (1 mark) | < Answer >
79. | In which method of sampling each possible sample has an equal probability of being chosen and each item in the entire population has an equal probability of being included in the sample. (a) Simple Random sampling (b) Systematic sampling (c) Stratified sampling (d) Cluster sampling (e) Judgemental sampling. (1 mark) | < Answer >
80. | Which of the following criteria cannot be used for decision making if the decision maker does not know the probabilities associated with the various states of nature? (a) Maximax criterion (b) Maximin criterion (c) Regret criterion (d) Expected value (e) Hurwicz criterion. (1 mark) | < Answer >
81. | Which of the following is false with regard to a Bernoulli process? (a) Each trial of the experiment has only two possible outcomes (b) The probability of the outcome of any trial remains fixed over time (c) The outcome of any trial affects the outcome of the following trial (d) The possible outcomes of any trial may or may not be equally likely (e) The process consists of a sequence of a specific number of identical trials. (1 mark) | < Answer >
82. | Which of the following is not a characteristic of normal probability distribution? (a) The mean of the normally distributed population lies at the center of the distribution (b) A normal distribution always has a variance of 1 (c) The mean, median and mode are equal (d) Both the tails extend indefinitely and never meet the horizontal axis (e) Half of the area under the normal distribution curve lies below the mean. (1 mark) | < Answer >
83 | Which criterion of decision-making is based upon the opportunity cost of making specific decisions? (a) Maximax criterion (b) Maximin criterion (c) Hurwicz criterion (d) Regret criterion (e) Expected value. (1 mark) | < Answer >
84. | The expected value and variance of a variable which follows the standard normal probability distribution are (a) 1 and 0 respectively (b) 1 and 1 respectively (c) 0 and 1 respectively (d) 0 and 0 respectively (e) –1 and 0 respectively. (1 mark) | < Answer >
85. | Which of the following elements is not commonly found in problems of decision making under uncertain conditions? (a) The decision maker has an objective to be achieved (b) There are several courses of action (c) There is a calculable measure of benefit or worth of the various courses of action (d) There are events which are not controllable by the decision maker (e) The decision maker has knowledge of the probabilities of occurrence of the events which are not under his control. (1 mark) | < Answer >
86. | 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 >
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. | 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 >
89. | 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 >

Suggested Answers
Quantitative Methods-II (132): July 2005

1. | Answer : (a) Reason : Let X be the number of questions that are correctly solved. Then X follows a binomial distribution with n = 10 and p = ¼ = 0.25 | < TOP >
2. | Answer : (c) Reason : Variance of a binomial distribution = npq Mean of a binomial distribution = np \ Variance = Mean . q = \ standard deviation = | < TOP >
3. | Answer : (a) Reason : Percentage of the time will the station be shut down = P(x > 45) = = P(z > 2.86) = 1 - (0.5 + 0.4979) = 0.0021 i.e. 0.21% | < TOP >
4. | Answer : (d) Reason : Given that p = 1/2 and q = 1/2 Probability of getting 3 people or less vegetarian in a sample of 10 = = \ The number of investigators (out of 100) expected to report that 3 or less people are vegetarian = 100 ´ 0.172 = 17.2 » 17 people. | < TOP >
5. | Answer : (d) 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 – 0.0808) = 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 : (c) 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 (using interpolation) Hence Or 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 Or 75 = m + 1.282s ……….(ii) Solving the two equations (i) and (ii), we get s = 19.4. | < TOP >
7. | Answer : (c) Reason : Let the demand (‘000 units) be denoted by X. Then E(X) = å[X.P (X)] = 7.29032 Therefore the expected demand = 7.29032 ´ 1000 = 7290.32 units. | < TOP >
8. | Answer : (c) Reason : The number of absentees who are boys follows a hypergeometric distribution. The following details are available: N = No. of elements in the population = 25 r = No. of elements in the population labelled success = No. of boys = 14 n = No. of trials = No. of absent students = 5 x = Desired number of successes In a hypergeometric distribution P(x) = | < TOP >
9. | Answer : (a) Reason : Since the businessman is perfectly pessimistic the appropriate decision criterion is Maximin criterion. The decision will be made as follows: Net gains (Rs.)
Asset | State of nature | Minimum gains
S1 | S2 | S3
A | 30,000 | 12,000 | 10,000 | 10,000
B | 40,000 | 37,000 | –6,000 | –6,000
C | 50,000 | 25,000 | –30,000 | –30,000
The maximum of the minimum gains is offered by asset A. Hence he should invest in asset A. | < TOP >
10. | Answer : (b) Reason : EVPI = Expected profit with perfect information – Expected profit without perfect information = 45,000 – 40,000 = Rs. 5,000. | < TOP >
11. | Answer : (a) Reason : 95% confidence limits for the population mean: 73 = (A) 80 = (B) Adding the two equations and solving : Subtracting (A) from (B) and solving : \99% confidence limits are: Lower limit : = 72 (approx.) Upper limit : = 81 (approx.) | < TOP >
12. | Answer : (a) 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 in the rejection region. Hence we reject the null hypothesis. | < TOP >
13. | Answer : (d) Reason : H0 : m = 12 H1 : m ¹ 12 The sample is large. So we shall use the normal distribution. = = = 1.125 z = = = –2.67 At a = 0.05, Critical values are ± 1.96. The test statistic is less than the left tail critical value. So it falls in the rejection region. \ H0 is rejected. But H0 is true. So the test leads to a Type I error. | < TOP >
14. | Answer : (c) Reason : The point estimate of population mean = sample mean = | < TOP >
15. | Answer : (c) Reason : The expected value of the distribution of sample means = population mean = 75 | < TOP >
16. | Answer : (e) 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 = = –1.162 At a = 0.05, the critical values are ±1.96. The test statistic is more than the left tail critical value. So it falls in the acceptance region. \ We accept H0. The true proportion is 0.35. So H0 is false. Hence the test leads to a type II error. | < TOP >
17. | Answer : (b) Reason : Estimated standard error of difference between means: Value of the test statistic = | < TOP >
18. | Answer : (c) Reason : Since the sample is small and population variance is unknown the distribution to be used is tn–1 (or t15) distribution. LCL = or = = 70 – 48.69 = 21.31 Or = = 10 Or = 10 or s2 = 100n = 100 ´ 16 = 1600. \ Sample variance = 1600. | < TOP >
19. | Answer : (e) Reason : LCL = or 0.302 = 0.40 – or z = 0.40 – 0.302 = 0.098 \ UCL = = 0.40 + 0.098 = 0.498. | < TOP >
20. | Answer : (d) 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 : (d) Reason : UCL = = 0.4776 LCL = = 0.2424 \ = = = 0.36. | < 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 = 600 = = = = 100 Hence statement (d) is correct. | < TOP >
23. | Answer : (b) Reason : When the sample size is less than 5 percent of the population size the estimated standard error of mean = Estimated standard error of mean = | < TOP >
24. | Answer : (e) Reason : Other things remaining the same, if the confidence level is decreased then the width of the confidence interval for the population mean will decrease. | < TOP >
25. | Answer : (c) Reason : Whenever the sampled population has a normal distribution, the sampling distribution of mean is a normal distribution for any sample size. | < TOP >
26. | Answer : (a) Reason : Since the null hypothesis is true and it has been accepted there is no instance of either type I or type II error. | < TOP >
27. | Answer : (d) Reason : The test statistic = Under the null hypothesis that , if then The test statistic will be zero. | < TOP >
28. | Answer : (a) Reason : The probability of making a type I error is same in both one tailed and two tailed tests for a given significance level. | < TOP >
29. | Answer : (b) Reason : In hypothesis testing, if the level of significance is made smaller, the rejection region becomes smaller. | < TOP >
30. | Answer : (e) Reason : The standard error of mean is defined as (where n is the sample size) and hence it decreases as n increases. (a), (b), (c) and (d) are incorrect with regard to standard error of mean. | < TOP >
31. | Answer : (c) Reason : Fisher’s ideal price index = = = | < TOP >
32. | Answer : (d) Reason : Laspeyres price index = = Value index = | < TOP >
33. | Answer : (d) Reason : Paasches quantity index = | < TOP >
34. | Answer : (e) Reason : Unweighted aggregates quantity index = = = (Where ‘n’ represents the no. of commodities in the basket or group and it is the same for both the base and current years) | < TOP >
35. | Answer : (c) Reason : Laspeyres quantity index = = = ´ 100 = | < TOP >
36. | Answer : (a) Reason : For weighted average of relatives price index we do the following calculations: = 7442.639 ´ 100 = 5465 Weighted average of relative price index = = » 136.19 Therefore the value of weighted average of relatives price index is 136.19. | < TOP >
37. | Answer : (c) 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 =6.9 ………………… (iii) | < TOP >
38. | Answer : (d) Reason : The estimating equation can be calculated as below:
Year (X) | Sales (Y) | x = X- | xY | x2
1996 | 12 | -2 | -24 | 4
1997 | 18 | -1 | -18 | 1
1998 | 20 | 0 | 0 | 0
1999 | 23 | 1 | 23 | 1
2000 | 27 | 2 | 54 | 4
Total | 100 | 0 | 35 | 10
= 1998 The equation of the straight line trend is Y = a + bx Since Sx = 0, Thus the equation of the straight line trend is : = 20 + 3.5x For year 2003, the value of x = 2003 – 1998 = 5, Hence Y2003 = 20 + 3.5 (5) = 20 +17.5 = 37.5 (in thousands) Thus the estimated sales of television sets for the year 2003 is 37,500. | < TOP >
39. | Answer : (c) Reason : = a + bY b = Equation of regression line of X on Y : X – 65 = 0.80 (2.5 / 3.5) (Y – 67) or, = 26.7162 + 0.5714 Y. | < TOP >
40. | Answer : (a) Reason : The regression equation of Y on X is : Yc = a + bX. b = a = = (40/5) – (– 0.65)(30/5) = 11.9 Therefore the regression equation is, = 11.9 – 0.65X. | < TOP >
41. | Answer : (c) Reason : Given that, å(X + Y)2 = 947 or, å X2 + å Y2 + 2 å XY = 947 or, 385 +192 + 2 å XY = 947 or, å XY = 185. Now, å X = n ´ = 10 ´ 5.5 = 55 and å Y = n ´ = 10 ´ 4.0 = 40. \ Coefficient of correlation is given by ; This can be rewritten after simplification, as, | < TOP >
42. | Answer : (b) Reason : This problem requires the application of ANOVA. Therefore we use F ratio as the test statistic for the given test. Finding out the F ratio: F = = = 2.429. | < TOP >
43. | Answer : (e) Reason : We find the variance within samples with the help of following table:
| x1i | (x1i-)2 | x2i | (x2i-)2 | x3 | (x3i-)2 | x4 | (x4i-)2
| 78 | 36 | 94 | 16 | 74 | 4 | 79 | 1
| 87 | 9 | 91 | 1 | 78 | 4 | 83 | 25
| 81 | 9 | 87 | 9 | 70 | 36 | 78 | 0
| 89 | 25 | 90 | 0 | 82 | 36 | 69 | 81
| 85 | 1 | 88 | 4 | 76 | 0 | 81 | 9
Sum | 420 | 80 | 450 | 30 | 380 | 80 | 390 | 116
Mean | 84 |  | 90 |  | 76 |  | 78 |
Therefore variance within the samples = | < TOP >
44. | Answer : (c) Reason : The lower 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 right tail and (20 – 1) = 19 degrees of freedom is 30.144. \ The lower confidence limit for the estimated population variance is . | < TOP >
45. | Answer : (c) 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 right tail and (30 – 1) = 29 degrees of freedom is 45.722. | < TOP >
46. | Answer : (c) Reason : We calculate the variances of the two samples as follows:
Number | X1 | (X1-) | (X1-)2 | X2 | (X2-) | (X2-)2
1 | 66 | -14 | 196 | 64 | -19 | 361
2 | 67 | -13 | 169 | 66 | -17 | 289
3 | 75 | -5 | 25 | 74 | -9 | 81
4 | 76 | -4 | 16 | 78 | -5 | 25
5 | 82 | 2 | 4 | 82 | -1 | 1
6 | 84 | 4 | 16 | 85 | 2 | 4
7 | 88 | 8 | 64 | 87 | 4 | 16
8 | 90 | 10 | 100 | 92 | 9 | 81
9 | 92 | 12 | 144 | 93 | 10 | 100
10 |  |  |  | 95 | 12 | 144
11 |  |  |  | 97 | 14 | 196
Sum | 720 |  | 734 | 913 |  | 1298
Means | 80 |  |  | 83 |  |
Sample Variance |  |  | 91.75 |  |  | 129.8
The test statistic F . | < TOP >
47. | Answer : (c) Reason : This is a test of goodness of fit (chi-square test). Let us take the null hypothesis that ‘the number of parts demanded is uniformly distributed over the week. The total number of spare parts demanded in a week is 6720 and if it is uniform over the week we should expect a demand of 6720/6 i.e. 1120 spare parts on each day of the week.
Day | O | E | (O – E)2 / E
Monday | 1124 | 1120 | 0.014
Tuesday | 1125 | 1120 | 0.022
Wednesday | 1110 | 1120 | 0.089
Thursday | 1120 | 1120 | 0
Friday | 1126 | 1120 | 0.032
Saturday | 1115 | 1120 | 0.022
|  | Total | 0.179
The value of Chi square statistic, c2 = = 0.179 | < TOP >
48. | Answer : (b) Reason : This is a chi-square test of independence The table of expected frequencies can be found out as follows:
Persons who | Under 20 | 20 - 40 | 40 - 50 | 50 and over | Total
Liked the car | 120 | 78 | 48 | 54 | 300
Disliked the car | 80 | 52 | 32 | 36 | 200
Total | 200 | 130 | 80 | 90 | 500
The chi square value can be obtained as follows:
Observed Frequency (O) | Expected Frequency (E) | (O-E)2 / E
146 | 120 | 5.633
54 | 80 | 8.450
78 | 78 | -
52 | 52 | -
48 | 48 | -
32 | 32 | -
28 | 54 | 12.519
62 | 36 | 18.778
The value of the test statistic, Chi square statistic, c2 = = 45.38 | < TOP >
49. | Answer : (a) Reason : Upper Control Limit for p chart = | < TOP >
50. | Answer : (e) 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 units. Mean of sample range = = åR / n = 63 /10 = 6.3 Lower Control Limit of chart = - A2 = 10.66 - 0.577 ´ 6.3 = 7.0249 (The value of A2 for sample of size 5 is 0.577 from control chart factors table) | < TOP >
51. | Answer : (d) Reason : We have to calculate the multiple correlation coefficient treating variable X1 as dependent and X2 and X3 as independent variables, i.e., we have to find Substituting the given values . | < TOP >
52. | Answer : (b) Reason : The central line of for the R chart = Mean of the sample range () = åR / n = 63 /10 = 6.3. The value of the upper control limit for the R chart = D4 = 6.3 ´ 2.114 = 13.3182. | < TOP >
53. | Answer : (e) Reason : Given that, | < TOP >
54. | Answer : (d) Reason : Given that, | < TOP >
55. | Answer : (b) Reason : The equation of the regression line is = a + bX where, X = Sales (Rs. in lakhs) Y = Total costs (Rs. in lakhs) b = ; a = . = 194; = 171; n = 6; = 5987 ; = 6782; = 28.5. b = == 0.899. a = = - 0.568 = -0.568 + 0.899 x Sales (x) = Rs.35 lakhs (given) = -0.568 + (0.899) (35) = 30.897 Total costs = Rs.30.897 lakhs. | < TOP >
56. | Answer : (c) Reason : Coefficient of variation = Or = 4 \ Cov (X, Y) = = 0.75 ´ 6 ´ 4 = 18. | < TOP >
57. | Answer : (c) Reason : Cov(X – 10, Y – 20) = Cov(X, Y) = = \ = = rx,y = 0.80. | < TOP >
58. | Answer : (c) Reason : = a + bX = 10 + 5X = Q = X – 4 = = = = = = = = a + 4b = 10 + 4 ´ 5 = 30 = 30 + 5Q. | < TOP >
59. | Answer : (a) Reason : Where b is the slope of the regression between Y and X \ b = 2. | < TOP >
60. | Answer : (d) Reason : No. of degrees of freedom for the denominator of the F statistic = nT – k = (5 ´ 6 ) – 5 = 25. | < TOP >
61. | Answer : (c) Reason : Year = 2000 \ x = 2000 – 1998 = 2 Relative cyclical residual = Y = 160 \ Relative cyclical residual for the year 2000 = = 3.9. | < TOP >
62. | Answer : (d) Reason : When the population mean is known the population mean serves as the center line. | < TOP >
63. | Answer : (d) Reason : When the population standard deviation is known the upper and lower control limits of the chart, are found out on the basis of the actual standard error of mean for the specific sample size. | < TOP >
64. | Answer : (a) Reason : In a control chart there are some observations that lie outside the upper and lower control limits. These points are referred to as the Outliers. | < TOP >
65. | Answer : (b) Reason : The distribution is positively skewed when the number of degrees of freedom is small. | < TOP >
66. | Answer : (b) Reason : In performing a chi-square test of independence, as the difference between the respective observed and expected frequencies decreases, the probability of accepting the null hypothesis increases. | < TOP >
67. | Answer : (b) Reason : F-distribution have to be used for testing the equality of more than two population means. | < TOP >
68. | Answer : (b) Reason : If all the values considered in calculating an index are of equal importance, the index is called unweighted index. | < TOP >
69. | Answer : (c) Reason : For a given set of commodities Fisher’s ideal price index falls in between Laspeyres price index and Paasches price index. | < TOP >
70. | Answer : (b) Reason : To estimate changes in total monetary worth, one should calculate value index. | < TOP >
71. | Answer : (a) Reason : It is possible to change the base year without changing the quantities used for weights when using the Paasche index. | < TOP >
72. | Answer : (a) Reason : Unweighted aggregates index is an example of the simplest form of Index Number. Hence from above discussion, we can infer that option (a) is correct. | < TOP >
73. | Answer : (e) Reason : Fisher’s ideal price index = = Fisher’s ideal price index is the geometric mean of the Laspeyres and Paasche’s price indices. So it is not free from the effects of base year and current year prices and quantities. | < TOP >
74. | Answer : (d) Reason : The technique used for studying seasonal variations is known as ratio to moving average method. | < TOP >
75. | Answer : (a) Reason : The movement of dependent variable above and below the secular trend line over periods longer than one year is known as cyclical variation. | < TOP >
76. | 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 >
77. | Answer : (b) Reason : (a) This is a wrong answer. The sum of the squared values of the vertical distances from each plotted point to the line should be minimum (not maximum). (b) This is the right answer. According to the ‘method of least squares’ criterion, the regression line should be drawn on the scatter diagram in such a way that the sum of the squared values of the vertical distances from each plotted point to the line are minimum. (c) This is the wrong answer. The sum of the squared values of the vertical (not the horizontal) distances from each plotted point to the line should be minimum (not maximum). (d) This is the wrong answer. The sum of the squared values of the vertical (not the horizontal) distances from each plotted point to the line should be minimum. (e) This is the wrong answer. The sum of the squared values of the vertical distances from each plotted point to the line should be minimum (not necessarily zero). | < TOP >
78. | Answer : (d) Reason : (a) This is a wrong answer. The standard error of the estimate of a regression line is not the slope of the regression line. (b) This is a wrong answer. The standard error of the estimate of a regression line is not The square root of the coefficient of determination. (c) This is a wrong answer. The standard error of the estimate of a regression line is not the Y intercept of the regression line. (d) This is the right answer. The standard error of the estimate of a regression line is the measure of the variability of the observed values around the regression line. (e) This is a wrong answer. The standard error of the estimate of a regression line is not the coefficient of correlation between two variables. | < TOP >
79. | Answer : (a) Reason : In simple random sampling each possible sample has an equal chance of being selected. | < TOP >
80. | Answer : (d) Reason : Under uncertain conditions the decision maker is unaware of the probabilities of the various states of nature. Hence the expected value criterion cannot be used. | < TOP >
81. | Answer : (c) Reason : Alternative (a), (b), (d) and (e) are the characteristics of a Bernoulli process. (c) is false because the trials are independent. | < TOP >
82. | Answer : (b) Reason : Normal distributions may have different variances and the variance need not be 1 always. | < TOP >
83. | Answer : (d) Reason : Regret criterion is based upon the opportunity cost of making specific decisions. | < TOP >
84. | Answer : (c) Reason : The expected value and variance of the standard normal probability distribution are 0 and 1 respectively. | < TOP >
85. | Answer : (e) Reason : All the elements stated in the alternatives (a) through (d) are commonly found in problems of decision making under uncertain conditions. Under uncertainty the decision maker has no knowledge of the probabilities of occurrence of the events not under his control. | < TOP >
86. | Answer : (c) Reason : A continuous uniform probability distribution is described by the upper and lower limits of the interval. | < 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: (b) Reason: If the coefficient of correlation between two variables lies between –1 and 0, then the covariance between them is negative. | < TOP >
89. | Answer: (a) Reason: The proportion of variations that is explained by the regression line is represented by the coefficient of determination. | < TOP >

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