Question Paper Quantitative Methods – II (132) : October 2003
| Section A : Basic Concepts (40 Points) |
| · This section consists of questions with serial number 1 - 40. · Answer all questions. · Each question carries one point. |
1. | If each of the values of a random variable is transformed into a Z-value, the distribution of the resulting values will have a standard deviation equal to a. Zero b. One c. The mean of the original distribution d. The standard deviation of the original distribution e. A variable, depending upon the shape and spread of the original distribution. | < Answer >
2. | Which of the following is false with regard to standard error of mean? a. It is less than the standard deviation of the population b. It decreases as the sample size increases c. It measures the variability of the mean from sample to sample d. It is the standard deviation of the sampling distribution of mean e. It is the standard deviation of the sample. | < Answer >
3. | Which of the following is true with regard to covariance? a. It is the average of all the values that may be assumed by two random variables, considered together b. It is the variance of all the values that may be assumed by two random variables, considered together c. It is the standard deviation of all the values that may be assumed by two random variables, considered together d. It is the range of all the values that may be assumed by two random variables, considered together e. It is a single number that captures the extent to which two random variables move together. | < Answer >
4. | If two random variables are independent of each other then their covariance is equal to a. 0 b. 1 c. A positive value d. A negative value e. An infinite value. | < Answer >
5. | Which of the following is not true with regard to the binomial distribution? a. The outcomes are independent of each other b. Each outcome can be classified as a success or failure c. The probability of success is constant from trial to trial d. The random variable of interest is continuous e. The probability of failure on any trial can be determined on the basis of the probability of success. | < Answer >
6. | Which of the following best describes the expected value of a discrete random variable? a. It is the geometric average of all possible outcomes of the variable b. It is the simple average of all possible outcomes of the variable c. It is the weighted average of all possible outcomes of the variable d. It is the outcome of the variable, which has the highest probability of occurrence e. It is the highest probability of occurrence in probability distribution of the random variable. | < Answer >
7. | When sampling without replacement from a finite population such that the probability of success is not constant from trial to trial, the data follow a a. Binomial distribution b. Uniform distribution c. Normal distribution d. Continuous distribution e. Hypergeometric distribution. | < Answer >
8. | A graphical method of representing states of nature and courses of action may be referred to as a a. Decision tree b. Histogram c. Scatter diagram d. Frequency distribution e. Probability distribution. | < Answer >
9. | 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. None of the above. | < Answer >
10. | The sampling distribution of the mean is a distribution of a. Means of individual populations b. Observations within a population c. Observations within a sample d. Means of all possible samples of a specific size taken from a population e. Means of samples of a specific size taken from different populations. | < Answer >
11. | According to the Central Limit Theorem, the sampling distribution of mean can be approximated by the normal distribution a. As the number of samples decreases b. As the sample size increases c. As the population standard deviation increases d. As the sample standard deviation decreases e. Only if the population distribution is normal. | < Answer >
12. | As the sample size increases a. The variation of the sample mean from the population mean becomes larger b. The variation of the sample mean from the population mean becomes smaller c. The variance of the sample becomes less than the variance of the population d. The standard deviation of the sample becomes less than the standard deviation of the population e. The standard deviation of the sample comes close to zero. | < Answer >
13. | When estimating the population mean from a sample taken from a normal population with unknown variance, the t distribution may be used with number of degrees of freedom equal to a. Sample size b. Sample size + 1 c. Sample size – 1 d. 0 e. 1. | < Answer >
14. | The sample mean is an unbiased point estimator of a. The population variance b. The population mean c. The population proportion d. The standard error of mean e. The standard error of proportion. | < Answer >
15. | Which type of sampling is appropriate when the population consists of well-defined groups such that the elements within each group are heterogeneous and each group is a representative of the population as a whole? a. Simple random sampling b. Cluster sampling c. Stratified sampling d. Systematic sampling e. Judgmental sampling. | < Answer >
16. | In which of the following conditions a Type I error is said to have occurred, in testing a hypothesis? a. The sample statistic is incorrectly calculated b. The sample variance is incorrectly calculated c. Null hypothesis is rejected though it is true d. Null hypothesis is accepted though it is false e. The sample size is small. | < Answer >
17. | If a hypothesis is tested at a significance level of 10% then it means that a. There is a 10% probability that the null hypothesis will be rejected though it is true b. There is a 10% probability that the null hypothesis will be false c. There is a 10% probability that the null hypothesis will be true d. There is a 90% probability that the alternative hypothesis will be false e. There is a 90% probability that the alternative hypothesis will be true. | < Answer >
18. | Which of the following distributions should be used in hypothesis testing when the population standard deviation is not known and the sample size is less than 30? a. Chi-square distribution b. F-distribution c. Normal distribution d. Binomial distribution e. t-distribution. | < Answer >
19. | The property of efficiency with regard to an estimator refers to the a. Sample size b. Size of the standard error of the sample statistic c. Size of the standard deviation of the sample d. Size of the sample variance e. Size of the sample mean. | < Answer >
20. | Which of the following criteria is applied for making decisions under conditions of uncertainty if the decision maker is perfectly optimistic? a. Maximax criterion b. Maximin criterion c. Hurwicz criterion d. Regret criterion e. Expected value. | < Answer >
21. | The Y intercept in the simple linear regression equation (X is the independent variable and Y is the dependent variable) represents the a. True value of Y when X = 0 b. Change in average value of Y per unit change in X c. Expected value of Y when X = 0 d. Standard deviation of the values of X e. Mean of the values of X. | < Answer >
22. | The slope of the simple linear regression equation (X is the independent variable and Y is the dependent variable) represents the a. Mean value of Y when X = 0 b. Change in mean value of Y per unit change in X c. True value of Y for a fixed value of X d. Variance of the values of X e. Variance of the values of Y for a fixed value of X. | < Answer >
23. | Which of the following is true with regard to simple regression and correlation analyses? a. Correlation analysis provides a measure, which indicates how well an equation for estimating one variable from another, explains the changes in the estimated variable b. If the coefficient of correlation between two variables is close to 1, then there is no correlation between the two variables c. If the slope of a regression equation is positive, then the coefficient of correlation between the variables involved is negative d. If the slope of a regression equation is negative, then the dependent variable decreases as the independent variable decreases e. Correlation analysis necessarily indicates the presence of cause-effect relationship between the variables involved. | < Answer >
24. | Which of the following is indicated if the standard error of estimate for a regression equation is zero? a. The standard deviation of the observed values of the dependent variable is zero b. The standard deviation of the independent variable is zero c. The observed value of the dependent variable will always be equal to its estimated value, for a given value of the independent variable d. There is no correlation between the variables involved e. The correlation coefficient between the variables is greater than zero but less than 1. | < Answer >
25. | Which of the following is true when the slope of a regression line is negative? a. The correlation coefficient between the dependent and independent variables is 1 b. The correlation coefficient between the dependent and independent variables, lies between 0 and 1 c. There is a negative correlation between the dependent and independent variables d. The regression line is parallel to the horizontal axis e. The regression line passes through the intersection of the horizontal and vertical axes. | < Answer >
26. | 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. | < Answer >
27. | Which of the following is the graphical plot of the values of the dependent and independent variables, in the context of regression analysis ? a. Scatter diagram b. Frequency polygon c. Histogram d. p chart e. Ogive. | < Answer >
28. | A multiple regression relationship is given by : Y = a + b1X1 + b2X2 Which of the following conditions indicates the presence of multicollinearity ? a. The regression coefficients ‘a’, ‘b1’ and ‘b2’ are all negative b. Only the regression coefficients ‘b1’ and ‘b2’ are negative c. There is a strong correlation between the variables X1 and Y d. There is a strong correlation between the variables X2 and Y e. There is a strong correlation between the variables X1 and X2. | < Answer >
29. | Which of the following is false with regard to the coefficient of multiple correlation between the dependent variable, Y, and the independent variables, X1 and X2? a. It can take values in the range of 0 to 1 b. It is the positive square root of coefficient of multiple determination c. It does not depend upon the coefficient of correlation between X1 and X2 d. It depends upon the coefficient of correlation between Y and X1 e. It depends upon the coefficient of correlation between Y and X2. | < Answer >
30. | Which of the following is true with regard to a multiple regression equation ? a. It has only one dependent variable b. It has multiple dependent variables c. It has only one independent variable d. All the regression coefficients are equal to 1 e. For given values of the independent variables the estimated value of the dependent variable is always equal to its observed value. | < Answer >
31. | Which of the following price indices uses only current year quantities as weights? a. Fisher’s ideal price index b. Laspeyre’s price index c. Paasche’s price index d. Marshall-Edgeworth price index e. Fixed weight aggregates price index. | < Answer >
32. | Which of the following is true with regard to index numbers? a. The fixed weight aggregates price index does not allow the flexibility to select the base period for comparison of prices b. Paasche’s price index tends to underestimate the rise in prices c. Unweighted aggregates price index numbers link the price changes to the usage or consumption levels d. Laspeyre’s price index has a downward bias e. The weighted average of relatives price index uses base year prices as weights. | < Answer >
33. | 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. | < Answer >
34. | Which of the following components of time series can be separated using the percent of trend measure? a. Irregular variation b. Secular trend c. Seasonal variation d. Cyclical variation e. None of the above. | < Answer >
35. | Which of the following components of time series shows the long-term behavior of the variable? a. Seasonal variation b. Cyclical variation c. Secular trend d. Irregular variation e. None of the above. | < Answer >
36. | 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. | < Answer >
37. | 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. | < Answer >
38. | Which of the following techniques is suitable for testing whether there is a significant difference between more than two sample means? a. Chi-square test b. Analysis of variance c. Simple regression analysis d. Multiple regression analysis e. Correlation analysis. | < Answer >
39. | The value of chi-square statistic can not 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 c. 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. | < Answer >
40. | 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. | < Answer >
| END OF PART A |

| Section B : Problems (60 Points) |
| · This section consists of questions with serial number 41 - 70. · Answer all questions. · Points are indicated against each question. |
41. | The probability distributions of independent random variables, X and Y, are given below:
X | 10 | 15 | 20 | 25
Probability | 0.20 | 0.40 | 0.30 | 0.10
Y | 8 | 12 | 16
Probability | 0.25 | 0.55 | 0.20
If a random variable, Z, is defined as: Z = 2X + 5Y What is the expected value of the random variable Z? a. 6.6 b. 11.8 c. 33 d. 59 e. 92. (2 points) | < Answer >
42. | A random sample of 15 people is taken from a group in which 40% favor a particular political stand. What is the probability that at least 4 individuals in the sample favor this political stand? a. 0.0047 b. 0.0219 c. 0.0634 d. 0.0905 e. 0.9095. (2 points) | < Answer >
43. | The standard deviation of the number of successes in a binomial distribution is . The probability of success in a single trial is 1/4. What is the probability of obtaining exactly eight successes? a. 0.0197 b. 0.1001 c. 0.25 d. 0.75 e. 0.80. (1 point) | < Answer >
44. | The test scores of 600 students are normally distributed with a mean of 76 and standard deviation of 8. Approximately what is the number of students scoring between 70 and 82? a. 164 b. 150 c. 328 d. 272 e. 450. (2 points) | < Answer >
45. | For a sample randomly collected from a population, the following details are available: Sum of the squares of the observations = 1840 Sum of the observations = 160 Number of observations = 16 What is the estimated standard error of mean? a. 1 b. 2 c. 4 d. 10 e. 16. (1 point) | < Answer >
46. | Two variables, X and Y, are related by a regression equation such that X is the independent variable and Y is the dependent variable. The following details are available: Coefficient of correlation between X and Y = 0.80 Standard deviation of X = 5 Standard deviation of Y = 8 The slope of the regression equation is a. 0.80 b. 1.28 c. 6.4 d. 32 e. 64. (1 point) | < Answer >
47. | The following details are available with regard to a simple regression relationship: Total sum of squares = 15730 Error sum of squares = 1530 What percentage of the variations in the dependent variable is explained by the regression relationship? a. 0 b. 9.73% c. 0.8% d. 90.3% e. 100%. (1 point) | < Answer >
48. | The following details are available with regard to a regression relationship: Sum of the observations of the independent variable = 120 Sum of the observations of the dependent variable = 425 Number of data points = 10 Slope of the regression line = 2.5 What is the estimated value of the dependent variable if the independent variable is equal to 10? a. 2.5 b. 12.5 c. 25 d. 37.5 e. 42.5. (1 point) | < Answer >
49. | The following details are available with regard to a basket of commodities in a year: Laspeyre’s price index = 140.5 Fisher’s ideal price index = 143.03 What is the Paasche’s price index for the basket of commodities in that year ? a. 98.23 b. 101.8 c. 103.6 d. 135.6 e. 145.6. (1 point) | < Answer >
50. | The following data relate to the age of a sample of ten employees in an organization and the number of days they reported sick during a period of six months:
Age (in years) | 21 | 31 | 33 | 36 | 41 | 47 | 53 | 56 | 59 | 63
Number of sick days | 2 | 3 | 1 | 4 | 5 | 7 | 6 | 5 | 9 | 8
What is the coefficient of correlation between the age of the employees and the number of sick days? a. 0.163 b. 0.683 c. 0.871 d. 0.187 e. 0.759. (3 points) | < Answer >
51. | A simple regression equation is developed which relates the variables X and Y ; X is the independent variable and Y is the dependent variable. The following details are available:
Y | 58 | 10 | 88 | 11 | 11 | 13 | 15 | 16 | 14 | 20
= 21040 = 140 = 2528 Number of data points = 10 What is the standard error of estimate of the regression equation? a. 2.88 b. 12.37 c. 13.83 d. 98.74 e. 151.96. (3 points) | < Answer >
52. | A simple regression equation is developed which relates the variables X and Y ; X is the independent variable and Y is the dependent variable. The following details are available:
Y | 31 | 40 | 30 | 34 | 25 | 20
= 1000 = 30 = 200 Number of data points = 6 What percentage of the variations in Y does the regression line, not explain? a. 3.2% b. 17.4% c. 20% d. 82.6% e. 90%. (3 points) | < Answer >
53. | Ten salesmen in a firm were put to a competency test. The test scores obtained by the salesmen and the monthly sales made by them are given below:
Test score | 30 | 60 | 40 | 50 | 70 | 40 | 80 | 30 | 50 | 50
Monthly sales (Rs. '000) | 25 | 50 | 35 | 45 | 55 | 30 | 60 | 30 | 40 | 30
A regression equation has to be developed for estimating the amount of monthly sales from the test scores. On the basis of this regression equation what is the estimated monthly sales for a salesman who scores 90 in the competency test? a. Rs. 687.5 b. Rs. 5,625 c. Rs. 50,625 d. Rs. 61,875 e. Rs. 67,500. (3 points) | < Answer >
54. | A multiple regression equation has to be developed between the variables Y, X1 and X2. X1 and X2 are the independent variables, and Y is the dependent variable. The following details are available: Number of data points = 10 = 272 = 12005 = 4013 = 6485 = 441 = 147 = 19461 = 2173 On the basis of the multiple regression relationship what is the value of Y, if X1 = 40 and X2 = 10? a. –546.38 b. 35.772 c. –93.756 d. 19.722 e. 541.566. (3 points) | < Answer >
55. | From an association consisting of 540 individuals, a sample of 60 individuals is taken. From this sample, the average age of the individuals is found to be 31 years and the standard deviation is found to be 6.84 years. A 95 percent confidence interval for the mean age of the individuals in the association has to be constructed. The lower and upper confidence limits of the confidence interval are a. 24.16 years and 37.84 years respectively b. 28 years and 33 years respectively c. 29.37 years and 32.63 years respectively d. 30.17 years and 31.83 years respectively e. 17.6 years and 44.41 years respectively. (2 points) | < Answer >
56. | A computer store purchases untested microchips for computers, from a wholesaler. It has to estimate the proportion of faulty microchips in the consignments supplied by the wholesaler. From a large consignment sent by the wholesaler a sample of 200 microchips were tested and 20 microchips were found to be faulty. A 98 percent confidence interval has to be constructed for the true population proportion of faulty microchips. The lower and upper confidence limits of the confidence interval are a. 1 percent and 19 percent respectively b. 5.08 percent and 14.92 percent respectively c. 7.88 percent and 12.12 percent respectively d. 87.88 percent and 92.12 percent respectively e. 85.08 percent and 94.92 percent respectively. (2 points) | < Answer >
57. | The prices and quantities of some commodities consumed by a family in the years 1998 and 2002 are given below:
Commodity | Year
1998 | 2002
Price (Rs./kg) | Quantity (kgs) | Price (Rs./kg) | Quantity (kgs)
Tea | 600 | 10 | 800 | 8
Sugar | 14.50 | 150 | 15.00 | 120
Milk Powder | 90 | 15 | 120 | 12
Rice | 13.50 | 500 | 15.50 | 480
Flour | 12.00 | 300 | 13.50 | 250
What is the price index by Laspeyre’s method for the basket of commodities consumed by the family, for the year 2002, considering 1998 as the base year? a. 83.33 b. 118.50 c. 102.92 d. 112.07 e. 120.00. (2 points) | < Answer >
58. | The following data are collected from a wholesaler of commodities:
Commodity | Year
1997 | 2002 | 1999
Price (Rs./kg.) | Price (Rs./kg) | Price (Rs./kg) | Quantity (kg.)
Rice | 11.00 | 14.00 | 13.00 | 2500
Wheat | 8.50 | 11.50 | 9.75 | 1000
Salt | 3.50 | 5.00 | 4.25 | 500
Sugar | 12.50 | 14.00 | 13.00 | 750
Pulses | 24.50 | 29.50 | 27.00 | 1200
What is the weighted average of relative price index for the year 2002, using the year 1999 for weighting and the year 1997 for the base year? a. 123.33 b. 124.27 c. 127.57 d. 90.54 e. 111.67. (2 points) | < Answer >
59. | The following data are collected from a fruit merchant:
Commodity | Year
1996 | 2002
Price | Quantity | Price | Quantity
Mangoes | Rs. 10 per kg. | 500 kgs. | Rs. 15 per kg. | 400 kgs.
Bananas | Rs. 8 per doz. | 600 doz. | Rs. 12 per doz. | 500 doz.
Oranges | Rs. 30 per kg. | 350 kgs. | Rs. 50 per kg. | 300 kgs.
Pineapple | Rs. 8 per piece | 450 pieces | Rs. 12 per piece | 400 pieces
Apple | Rs. 40 per kg. | 300 kgs. | Rs. 70 per kg. | 200 kgs.
What is the price index by Marshall-Edgeworth method for the fruits traded by the merchant, for the year 2002, considering 1996 as the base year? a. 60.38 b. 81.82 c. 162.87 d. 165.63 e. 122.22. (2 points) | < Answer >
60. | According to a recently completed survey in a city, the mean number of hours of television viewing per household is 7.25 hours per day with a standard deviation of 2.4 hours per day. The survey involved a sample of 200 households spread in different parts of the city. One of the private television channels assumes that the mean number of hours of television viewing per household in the city, is 6.70 hours per day. It is to be tested whether the mean number of hours of television viewing per household in the city is more than 6.70 hours per day. At a significance level of 5 percent, what is the conclusion? a. The sample mean is incorrect b. The sample standard deviation is incorrect c. The mean number of hours of television viewing per household in the city is equal to 6.70 hours per day d. The mean number of hours of television viewing per household in the city is less than 6.70 hours per day e. The mean number of hours of television viewing per household in the city is more than 6.70 hours per day. (3 points) | < Answer >
61. | A random sample of 300 loans was collected from the loans made during the recent five year period, by a financial institution which finances small scale enterprises. This sample showed that 120 of the loans were made to women entrepreneurs. A complete census of all the loans made by the financial institution five years ago showed that 42 percent of the loans were made to women entrepreneurs. It is to be tested whether the proportion of loans made to the women entrepreneurs by the financial institution has reduced in the past five years. At a significance level of 2 percent, what is the conclusion? a. The sample information is incorrect b. The population proportion is incorrect c. The proportion of loans made to the women entrepreneurs has reduced in the past five years d. The proportion of loans made to the women entrepreneurs has increased in the past five years e. The proportion of loans made to the women entrepreneurs has not reduced in the past five years. (3 points) | < Answer >
62. | A commodity merchant knows that the mean retail price of a specific variety of rice three months ago was Rs. 14.50 per kg. In the current month the merchant has collected the information on the price charged for the same variety of rice by 16 randomly selected merchants in the same city. It was found from the sample that the mean retail price was Rs. 15.00 per kg and the standard deviation was Rs. 1.25 per kg. It is to be tested whether the mean retail price of the rice in the current month is more than Rs. 14.50 per kg. At a significance level of 5 percent, what is the conclusion? a. The sample standard deviation is incorrect b. The sample mean is incorrect c. The mean retail price of the rice in the current month is more than Rs. 14.50 per kg d. The mean retail price of the rice in the current month is not more than Rs. 14.50 per kg e. The mean retail price of the rice in the current month is less than Rs. 14.50 per kg. (3 points) | < Answer >
63. | For a simple regression equation the following results were obtained: If X = 5 , If X = 10 , where X is the independent variable and is the estimated value of the dependent variable Y. Number of data points = 6 What is the mean of the observed values of the dependent variable, Y ? a. 15 b. 32 c. 47 d. 18.2 e. 25. (3 points) | < Answer >
64. | The production (in thousand quintals) of a sugar factory during seven years is given below:
Year | 1996 | 1997 | 1998 | 1999 | 2000 | 2001 | 2002
Production ( '000 quintals) | 75 | 85 | 87 | 80 | 89 | 95 | 90
What is the estimated production of sugar for the year 2003 on the basis of a linear estimating equation that describes the trend in the sugar production by the factory ? a. 85857 quintals b. 95429 quintals c. 88250 quintals d. 89857 quintals e. 487904 quintals. (3 points) | < Answer >
65. | 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 point) | < Answer >
66. | An automobile manufacturer is planning to launch a new model of scooter. In order to find out the opinion of the prospective customers about the proposed model, the firm has taken a random sample from the audience which attended a preview of the proposed model. The results obtained are given below:
Opinion of the persons | Age group (in years)
Below 20 | 20 - 30 | 30 - 40 | 40 - 50 | Total
Persons who liked the model | 140 | 80 | 50 | 30 | 300
Persons who disliked the model | 60 | 50 | 30 | 60 | 200
Total | 200 | 130 | 80 | 90 | 500
A chi-square test has to be performed to find out whether the opinion of the persons and the age groups are independent. At a 5 percent significance level what is the conclusion? a. The sample data are correct b. The sample data are incorrect c. The opinion of the persons and the age groups are independent d. The opinion of the persons and the age groups are dependent e. The data are insufficient to perform a chi-square test. (3 points) | < Answer >
67. | In a test involving ANOVA the following details are obtained: Estimated population variance based on the variance among the sample means = 20 Estimated population variance based on the variances within the samples = 14.77 What is the F statistic for the data? a. 0.739 b. 1.354 c. 5.23 d. 34.77 e. 295.4. (1 point) | < Answer >
68. | The following details are available for a test involving ANOVA: Number of samples = 4 Size of the first sample = 5 Size of the second sample = 5 Size of the third sample = 6 Size of the fourth sample = 4 The number of degrees of freedom in the numerator and denominator of the F ratio are a. 4 and 6 respectively b. 4 and 4 respectively c. 5 and 4 respectively d. 3 and 16 respectively e. 4 and 20 respectively. (1 point) | < Answer >
69. | The following details are available with regard to a product: Cost per unit of the product = Rs. 120 Selling price per unit of the product = Rs. 210 Salvage value of each unsold unit of the product = Rs. 30 What is the minimum required probability of selling an additional unit of the product which justifies stocking that unit? a. 0.25 b. 0.33 c. 0.50 d. 0.67 e. 1.00. (1 point) | < Answer >
70. | A multiple regression relationship contains two independent variables. The standard error of estimate is 4.8. Error sum of squares = 576. What is the number of data points? a. 24 b. 25 c. 26 d. 27 e. 28. (1 point) | < Answer >
| END OF PART A |

Suggested Answers
Quantitative Methods – II (132) : October 2003

1. | Answer : (b) Reason : Z = (x – μ)/σ When each value is reduced by a constant the mean of the resulting values = previous mean - constant. When the mean itself is subtracted from each value the mean of the resulting values = mean – mean = 0. When each value is reduced by a constant the standard deviation remains unchanged. However when each value is divided by a constant the standard deviation of the resulting values = previous standard deviation/constant. When each value is divided by the standard deviation the standard deviation of the resulting values = standard deviation/standard deviation = 1 | < TOP >
2. | Answer : (e) Reason : a. The standard error of mean is less than the population standard deviation(σ) because it is equal to σ/√n. b. From above we can see that it will decrease as the sample size increases. c. The standard error is a measure of the variability of the mean across various samples of the same size taken from the population. d. It is the standard deviation of the distribution of means of all possible samples of a specific size that can be taken from the population. e. From above we can see that the standard error of mean is not the standard deviation of the sample. | < TOP >
3. | Answer : (e) Reason : a. The covariance is not the average of all the values that may be assumed by both the random variables, considered together. b. The covariance is not the variance of all the values that may be assumed by both the random variables, considered together. c. The covariance is not the standard deviation of all the values that may be assumed by both the random variables, considered together. d. The covariance is not the range of all the values that may be assumed by both the random variables, considered together. e. The covariance is a single number which measures the extent to which two random variables move together. | < TOP >
4. | Answer : (a) Reason : a. If two random variables are independent of each other then their covariance is equal to 0. b, c, d & e. If two random variables are independent of each other then their covariance cannot be equal to any value different from 0. | < TOP >
5. | Answer : (d) Reason : a. In a binomial distribution the outcomes are independent of each other. b. In a binomial distribution each outcome can be classified as a success or failure. c. In a binomial distribution the probability of success is constant from trial to trial. d. In a binomial distribution the random variable of interest is continuous e. In a binomial distribution the probability of failure can be determined from the probability of success. | < TOP >
6. | Answer : (c) Reason : a. The expected value of a discrete random variable is not a geometric average of the outcomes of the variable. b. The expected value of a discrete random variable is not a simple average of the outcomes of the variable. c. The expected value of a discrete random variable is a weighted average of the outcomes of the variable. d. The expected value of a discrete random variable is not the outcome, which has the highest frequency. e. The expected value of a discrete random variable is not the highest probability of occurrence in the distribution of the random variable | < TOP >
7. | Answer : (e) Reason : a. The data may follow a binomial distribution if the probability of success is constant from trial to trial. b. The data may follow a uniform distribution if the probabilities of the outcomes of the trial are equal. c. The data may follow a normal distribution if the probability of success is constant from trial to trial. d. The data may follow a continuous distribution if the outcome is any value within a given range of values. e. When sampling is done without replacement from an finite population such that the probability of success is not constant from trial to trial, the data follow a hypergeometric distribution. | < TOP >
8. | Answer : (a) Reason : a. A decision tree is a graphical method which represents states of nature and courses of action. b. A histogram is a graphical representation of a frequency distribution. c. A scatter diagram shows the distribution of data points in regression analysis. d. A frequency distribution is a distribution of data along with their frequencies. e. A probability distribution is a distribution of values of a random variable along with their respective probabilities | < TOP >
9. | 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. | < TOP >
10. | Answer : (d) Reason : a. The sampling distribution of mean is not a distribution of means of individual populations. b. The sampling distribution of mean is not a distribution of observations within a population c. The sampling distribution of mean is not a distribution of observations within a sample. d. The sampling distribution of mean is a distribution of means of all possible samples of a specific size taken from a population. e. The sampling distribution of mean is not a distribution of means of samples of a specific size taken from different population | < TOP >
11. | Answer : (b) Reason : b. According to the Central Limit Theorem the sampling distribution of the mean can be approximated by the normal distribution as the sample size increases. a, c, d & e. These alternatives are not the correct interpretations of the Central Limit Theorem | < TOP >
12. | Answer : (b) Reason : a. As the sample size increases the variation of the sample mean from the population mean becomes smaller. b. From above we can see that (b) is incorrect. c. It can not be said with certainty that as the sample size increases the variance of the sample becomes less than the variance of the population. d. It can not be said with certainty that as the sample size increases the standard deviation of the sample becomes less than the standard deviation of the population. e. It can not be said with certainty that as the sample size increases the standard deviation of the sample comes close to zero. | < TOP >
13. | Answer : (c) Reason : c. When estimating population mean from a sample taken from a normal population with unknown variance the t distribution may be used with number of degrees of freedom equal to sample size – 1. a, b, d & e. represent incorrect degrees of freedom for using the t distribution | < TOP >
14. | Answer : (b) Reason : a. The sample mean is not an estimator of the population variance. b. The sample mean is an unbiased estimator of the population mean. c. The sample mean is not an estimator of the population proportion. d. The sample mean is not an estimator of the standard error of mean. e. The sample mean is not an estimator of the standard error of proportion | < TOP >
15. | Answer : (b) Reason : a. Simple random sampling may not be appropriate when the population is known to consist of well defined groups such that the elements within each group are heterogeneous and each group is a representative of the population as a whole, because even if the sample is random it may not reflect the nature of the population. b. When the population is known to consist of well-defined groups such that the elements within each group are heterogeneous and each group is a representative of the population as a whole, the cluster sampling is appropriate. c. When the population is known to consist of well-defined groups such that the elements within each group are homogeneous and the groups vary from each other significantly, the stratified sampling is appropriate. d. When the population is known to consist of well defined groups such that the elements within each group are heterogeneous and each group is a representative of the population as a whole, because even if the sample is random it may not reflect the true nature of the population. e. When the population is known to consist of well defined groups such that the elements within each group are heterogeneous and each group is a representative of the population as a whole, judgmental sampling may not be appropriate because the representativeness of the sample depends upon the knowledge and judgment of the decision maker | < TOP >
16. | Answer : (c) Reason : c. A type I error occurs if the null hypothesis is rejected though it is true. d. A type II error occurs if the null hypothesis is accepted though it is false. a, b & e are incorrect interpretations of type I error | < TOP >
17. | Answer : (a) Reason : a. If a hypothesis is tested at a 10% significance level then, it means that there is a 10% probability that the null hypothesis will be rejected though it is true. b, c, d & e are incorrect interpretations of the significance level | < TOP >
18. | Answer : (e) Reason : a. The chi-square distribution is applied for comparing more than two sample proportions. b. The F-distribution is applied for comparing more than two sample means. c. The normal distribution is applied if the sample is large (or the population standard deviation is known if the sample is small). d. In the given situation the binomial distribution is not appropriate. e. The t-distribution is appropriate for hypothesis testing when the population standard deviation is not known and the sample size is less than 30. | < TOP >
19. | Answer : (b) Reason : a. The property of efficiency with regard to an estimator does not refer to the sample size. b. The property of efficiency with regard to an estimator refers to the size of the standard error. c. The property of efficiency with regard to an estimator does not refer to the size of the standard deviation of the sample. d. The property of efficiency with regard to an estimator does not refer to the size of the sample variance. e. The property of efficiency with regard to an estimator does not refer to the size of the sample mean | < TOP >
20. | Answer : (a) Reason : a. For decision making under conditions of uncertainty the maximax criterion is applied if the decision maker is perfectly optimistic. b. For decision making under conditions of uncertainty the maximin criterion is applied if the decision maker is perfectly pessimistic. c. For decision making under conditions of uncertainty the Hurwicz criterion is applied if the decision maker cannot be classified as perfectly optimistic or perfectly pessimistic. d. For decision making under conditions of uncertainty the regret criterion is applied if the decision maker wants to take into account the opportunity cost his decisions. e. The expected value criterion is applied under conditions of risk. | < TOP >
21. | Answer : (c) Reason : a. The Y intercept of the regression line does not represent the true value of Y when X= 0. b. The Y intercept of the regression line does not represent the change in average value of Y per unit change in X. c. The Y intercept of the regression line represents the mean value of Y when X = 0. d. The Y intercept of the regression line does not represent the standard deviation of the values of X. e. The Y intercept of the regression line does not represent mean of the values of X | < TOP >
22. | Answer : (b) Reason : a. The slope of the simple regression equation does not represent the mean value of Y when X = 0. b. The slope of the simple regression equation represents the change in average value of Y per unit change in X. c. The slope of the simple regression equation does not represent the true value of Y for a fixed value of X. d. The slope of the simple regression equation does not represent the variance of the values of X. e. The slope of the simple regression equation does not represent variance of the values of Y for a fixed value of X. | < TOP >
23. | Answer : (a) Reason : a. Correlation analysis provides a measure (coefficient of determination) which indicates, how well an estimating equation explains the changes in the estimated (i.e. dependent variable). b. If the coefficient of correlation between two variables is close to 1 then there is a very high correlation between the two variables. c. If the slope of a regression line is positive then, the coefficient of correlation between the variables involved is positive. d. If the slope of a regression line is negative then the dependent variable decreases as the independent variable increases. e. Correlation analysis may indicate a possibility of a cause-effect relationship. It does not necessarily indicate that there is a cause-effect relationship between the variables involved. | < TOP >
24. | Answer : (c) Reason : a. If the standard error of estimate for a regression equation is zero, then it does not indicate that the standard deviation of the observed values of the dependent variable is zero. b. If the standard error of estimate for a regression equation is zero, then it does not indicate that the standard deviation of the independent variable is zero. c. If the standard error of estimate for a regression equation is zero, then it indicates that the observed value of the dependent variable will always be equal to its estimated value, for a given value of the independent variable. d. If the standard error of estimate for a regression equation is zero, then it indicates that there is a perfect correlation between the variables. e. If the standard error of estimate for a regression equation is zero, then it indicates that there is a perfect correlation between the variables; so the correlation coefficient will be either –1 or 1. | < TOP >
25. | Answer : (c) Reason : a. When the slope of a regression line is negative the correlation coefficient need not be 1. b. When the slope of a regression line is negative, there is a negative correlation between the variables; hence the correlation coefficient lies between –1 and 0. c. When the slope of a regression line is negative, there is a negative correlation between the variables. d. When the slope of a regression line is zero, the regression line will be parallel to the horizontal axis. e. When the y-intercept of a regression line is zero, the regression line passes through the intersection of horizontal and vertical axes | < TOP >
26. | Answer : (a) Reason : a. The coefficient of determination is the square of coefficient of correlation (r). Hence it will always be 0 and 1. b. From above we can see that the coefficient of determination cannot be less than zero. c. The coefficient of determination is will be equal to 1, only if the coefficient of correlation is equal to –1 or 1; in other cases it will be > 0 and < 1. d. The coefficient of determination is always positive; The coefficient of correlation may be negative. e. Since -1 r 1, the coefficient of determination which is the square of coefficient of correlation (r), will always have magnitude less than or equal to r | < TOP >
27. | Answer : (a) Reason : a. The graphical plot of the values of the dependent and independent variables, in the context of regression analysis, is called scatter diagram. b. 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. c. A histogram is a graphical representation of a frequency distribution. d. A p chart is a quality control chart. e. An ogive is a graphical plot of a cumulative frequency distribution | < TOP >
28. | Answer : (e) Reason : a & b. The negative multiple regression coefficients does not necessarily indicate the presence of multicollinearity. c &d. A strong correlation between the dependent variable and any one of the independent variables does not necessarily indicate multicollinearity. e. Multicollinearity indicates the presence of a strong correlation between the independent variables in a multiple regression relationship | < TOP >
29. | Answer : (c) Reason : a. The coefficient of multiple correlation always takes values in the range of 0 and 1. b. It is the positive square root of the coefficient of multiple determination. c. It depends upon the coefficient of correlation between the independent variables. d & e. It depends upon the coefficient of correlation between the dependent variable and each of the independent variables | < TOP >
30. | Answer : (a) Reason : a. A multiple regression equation has only one dependent variable. b. From above we can see that b is false. c. A multiple regression equation has more than one independent variable. d. In a multiple regression equation the regression coefficients need not necessarily be 1. e. For given values of the independent variables the estimated values of the dependent variable need not necessarily be equal to the observed values | < TOP >
31. | Answer : (c) Reason : a. Fisher’s ideal price index uses both current year and base year quantities. b. Laspeyres price index uses only base year quantities. c. Paasche’s price index uses only current year quantities. d. Marshall Edgeworth price index uses both current year and base year quantities. e. Fixed weight aggregates price index uses the weights from a representative period | < TOP >
32. | Answer : (b) Reason : a. The fixed weight aggregates price index allows the flexibility to select the base period for comparison of prices, because it uses the weights from a representative period. b. Paasche’s price index tends to underestimate the rise in prices. c. Unweighted aggregates price index numbers do not link the price changes to the consumption levels. d. Laspeyres price index has an upward bias. e. The weighted average of relatives use values (and not prices) as weights | < TOP >
33. | 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 >
34. | Answer : (d) Reason : a. The irregular variations cannot be separated from a time series using the percent of trend measure. b. The secular trend is not separated using the percent of trend measure. c. The seasonal variations are not separated using the percent of trend measure. d. The cyclical variations can be separated using the percent of trend measure | < TOP >
35. | Answer : (c) Reason : a. The seasonal variations show the seasonal patterns in the variable. b. The cyclical variations show the cyclical patterns in the variable. c. Secular trend shows the long term behavior of the variable. d. Irregular variation shows the random variations in the variable | < TOP >
36. | 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 >
37. | 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 >
38. | Answer : (b) Reason : a. Chi-square test is suitable for testing whether there is a significant difference between more than two sample proportions. b. ANOVA is suitable for testing whether there is a significant difference between more than two sample means. c, d & e. Simple and multiple regression analysis, are techniques for establishing relationship between two or more variables. Correlation analysis is the technique of studying the nature and strength of relationship between two or more variables. | < TOP >
39. | 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 >
40. | 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 >
41. | Answer : (e) Reason : E(Z) = E(2X + 5Y) = 2E(X) + 5E(Y) = = 33 + 59 = 92 | < TOP >
42. | Answer : (e) Reason : n = 15 p = 0.40 q = 1– 0.40 = 0.60 P (r ³ 4) = 1– P (r £ 3) = 1– = = = 0.9095 | < TOP >
43. | Answer : (a) Reason : p = \ q = 1– p = 1 – Variance = npq \ = n or 3 = or n = 16 \ P(r = 8) = | < TOP >
44. | Answer : (c) Reason : x = 70 x = 82 from the tables = \ \ The number of students scoring between 70 and 82, is = 0.5468 ´ 600 = 328.08 @ 328 | < TOP >
45. | Answer : (a) Reason : Estimated standard error of mean = Sample standard deviation, \ s = \ Estimated standard error of mean = | < TOP >
46. | Answer : (b) Reason : Slope of a regression equation, b = = = | < TOP >
47. | Answer : (d) Reason : Proportion of variations in the dependent variable explained by the regression relationship = i.e. 90.3 percent | < TOP >
48. | Answer : (d) Reason : Regression equation : a = Given : b = 2.5 \ a = \ Regression equation is If X = 10, then | < TOP >
49. | Answer : (e) Reason : Fisher’s ideal price index (F) = Or 143.03 = \140.5P = 143.032 P | < TOP >
50. | Answer : (c) Reason : Let the following notations be used: X : Age (in years) Y : Number of sick days Coefficient of correlation
A | X | 21 | 31 | 33 | 36 | 41 | 47 | 53 | 56 | 59 | 63 |  | 440
B | Y | 2 | 3 | 1 | 4 | 5 | 7 | 6 | 5 | 9 | 8 | = | 50
C. |  | –23 | –13 | –11 | –8 | –3 | 3 | 9 | 12 | 15 | 19 |  |
D. |  | –3 | –2 | –4 | –1 | 0 | 2 | 1 | 0 | 4 | 3 |  |
E. |  | 529 | 169 | 121 | 64 | 9 | 9 | 81 | 144 | 225 | 361 |  | 1712
F |  | 9 | 4 | 16 | 1 | 0 | 4 | 1 | 0 | 16 | 9 |  | 60
G | C ´ D | 69 | 26 | 44 | 8 | 0 | 6 | 9 | 0 | 60 | 57 |  | 279
\ r = | < TOP >
51. | Answer : (c) Reason : Given: n = 10 \ a = Standard error of estimate, se = \ | < TOP >
52. | Answer : (b) Reason : Proportion of variation in Y, that is not explained by the regression line = 1– Coefficient of determination (r2) b = a = \ = \ Proportion of variation in Y that is not explained by the regression line = 1-r2 = 1- 0.826 = 0.174 Þ Percentage of variations in Y that is not explained by the regression line = 17.4% | < TOP >
53. | Answer : (e) Reason : Let the following notations be used: X : Test score Y : Monthly sales (Rs. ‘000) b = n = 10 \ a = \ For X = 90, \Estimated monthly sales = Rs.67500 | < TOP >
54. | Answer : (d) Reason : A multiple regression equation is obtained by solving the following equations: Putting in the relevant values in the equation : 272 = 10a + 441 b1 + 147b2………………….(A) 12005 = 441a + 19461 b1 + 6485 b2……………….(B) 4013 = 147a + 6485 b1 + 2173 b2 ……………...(C) Multiplying (A) by 441 and (B) by 10, and subtracting (B) form (A):
119952 = | 4410a + | 194481b1 + | 64827 b2
120050 = – – | 4410a + – | 194610b1 + – | 64850b2
– 98 = |  | – 129b1 | – 23b2
\ 98 = 129 b1 + 23b2………………………………(D) Multiplying (B) by 147 and (C) by 441, and substracting (C) from (B):
1764735 = | 64827a + | 2860767b1 + | 953295 b2
1769733 = – – | 64827a + – | 2859885b1 + – | 958293b2
– 4998 = |  | 882b1 – | 4998b2
\ 4998 = – 882b1 + 4998b2 ….(E) Multiplying (D) by 882 and (E) by 129, and adding the two equations:
86436 = | 113778b1 + | 20286 b2 |
644742 = | – 113778 b1 + | 644742b2 |
731178 = |  | 665028b2 |
\ b2 = Putting the values of b2 in (D) : 98 = 129 b1 + (23 ´ 1.099) or b1 = Putting the values of b1 and b2 (A): 272 = 10a + (441 ´ 0.564) + (147 ´ 1.099) or a = \ The multiple regression equation is : For | < TOP >
55. | Answer : (c) Reason : s = 6.84 n = 60 N = 540 Sampling fraction, \ Estimated standard error of mean, = = = 0.833 Since the sample size is greater than 30 the normal distribution will be used. Z-values for the upper and lower confidence limit are \ Upper confidence limit = Lower confidence limit = | < TOP >
56. | Answer : (b) Reason : Estimated standard error or proportion, = 0.0212 Z – Values for the upper and lower confidence limit are \ Upper confidence limit = Lower confidence limit = | < TOP >
57. | Answer : (e) Reason : Laspeyres price index = P0 = Base year (1998) price Q0 = Base year (1998) quantity P1 = Current year (2002) price Q1 = Current year (2002) quantity \ Laspeyres price index for the year 2002 = = | < TOP >
58. | Answer : (b) Reason : Weighted average of relatives price index = P0 = Base year (1997) price P1 = Current year (2002) price Pn = Price in the year 1999 to be used in weights Qn= Quantity in the year 1999 to be used in weights \ Weighted average of relatives price index for 2002 = = | < TOP >
59. | Answer : (c) Reason : Marshall-Edgeworth price index = P1 = Current year (2002) price P0 = Base year (1996) price Q1 = Current year (2002) quantity Q0= Base year (1996) quantity \ Marshall – Edgeworth price index for 2002 = = = = 162.87 | < TOP >
60. | Answer : (e) Reason : Estimated standard error of mean, Since the sample size is large (greater then 30), the appropriate distribution is the normal distribution. Standardized value of sample mean, = Critical Z – value = 1.64 Since the standardized sample statistic exceeds the critical value, the sample statistic falls in the rejection region. Hence the null hypothesis is rejected at a significance level of 5 percent and it can be concluded that the mean number of hours of television viewing per household in the city is more than 6.70 per day. | < TOP >
61. | Answer : (e) Reason : Sample proportion, Standardized value of sample proportion, = – 0.702 Since the sample size is large we shall use the normal distribution. Critical Z –value = – 2.05 Thus we can see that the sample statistic falls in the acceptance region. Hence we accept the null hypothesis at a significance level of 2 percent. It can be concluded that the proportion of loans made to the women entrepreneurs has not reduced in the past five years. | < TOP >
62. | Answer : (d) Reason : Estimated standard error of mean Since the population standard deviation is not known and the sample size is less than 30, the t-distribution will be used. Degrees of freedom = n – 1 = 16 – 1 = 15 Since this is a right tailed test with, the rejection area is 0.05 under the right tail. Hence we should look under the 0.10 column in the t-distribution (0.10 area in both tails combined). \Critical t-value = 1.753 Standardized sample statistic, t = We find that the sample statistic falls in the acceptance region. Hence we accept the null hypothesis. It is concluded that the mean price of the rice is not more than Rs.14.50 per kg. | < TOP >
63. | Answer : (c) Reason : A simple regression equation is of the form: Given: If X = 5, =31 \ 31 = a + 5b………(A) If X = 10, =47 \47 = a + 10b ……..(B) Subtracting (A) from (B) we get: 47 – 31 = (10 – 5)b 16 = 5b or b = Putting the values of b in (B): 47 = a + 10 ´ 3.2 = a + 32 or a = 47 – 32 = 15 \=15 + 3.2X Now, a = \ \Mean of the observed values of Y, i.e. = 47 | < TOP >
64. | Answer : (b) Reason : Let the following notations be used: X : Years x : Coded value of year Y : Production of sugar (thousand quintals)
X |  | Y | xY | x2
1996 | -3 | 75 | -225 | 9
1997 | -2 | 85 | -170 | 4
1998 | -1 | 87 | -87 | 1
1999 | 0 | 80 | 0 | 0
2000 | 1 | 89 | 89 | 1
2001 | 2 | 95 | 190 | 4
2002 | 3 | 90 | 270 | 9
\ \ For the year 2003 : x = 2003 – 1999 = 4 \ Estimated sugar production = 85.857 + (2.393 ´ 4 ) = 95.429 (thousand quintals) i.e. 95429 quintals | < TOP >
65. | Answer : (c) Reason : Year = 2000 \ x = 2000 – 1998 = 2 Relative cyclical residual = Y = 160 \ Relative cyclical residual for the year 2000 = = 3.9. | < TOP >
66. | Answer : (d) Reason : H0 : Opinion of the persons is independent of the age group H1 : Opinion of the persons is dependent on the age group. Expected frequency for any cell =
Row | Column | fo |  |  |  |
1 | 1 | 140 |  | 20 | 400 | 3.333
1 | 2 | 80 |  | 2 | 4 | 0.051
1 | 3 | 50 |  | 2 | 4 | 0.083
1 | 4 | 30 |  | –24 | 576 | 10.667
2 | 1 | 60 |  | –20 | 400 | 5.000
2 | 2 | 50 |  | –2 | 4 | 0.077
2 | 3 | 30 |  | –2 | 4 | 0.125
2 | 4 | 60 |  | 24 | 576 | 16
|  |  |  |  |  | 35.336
Number of degree of freedom = (2 – 1) (4 – 1) = 3 Critical value : The value for the sample data is much greater than the critical value. Hence we reject the null hypothesis and conclude that the opinion of the persons and the age groups are dependent. | < TOP >
67. | Answer : (b) Reason : F ratio = | < TOP >
68. | Answer : (d) Reason : In ANOVA: Number of degrees of freedom in the numerator of F ratio = Number of samples – 1 = 4 – 1 = 3 Number of degree of freedom in the denominator of F ratio = Total sample size – Number of samples = (5 + 5 + 6 + 4) – 4 = 16 | < TOP >
69. | Answer : (c) Reason : Minimum required probability of selling an additional unit of the product which justifies stocking that unit = Marginal profit, MP = Rs.(210 – 120) = Rs.90 Marginal loss, ML = Rs.(120 – 30) = Rs.90 \ | < TOP >
70. | Answer : (e) Reason : In multiple regression relationship: Standard error of estimate, Given : ESS = 576 n = ? k = 2 se = 4.8 \4.8 = \ or or n = | < TOP >

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