Question Paper Quantitative Methods II (132) : April 2006
| · Answer all questions. · Marks are indicated against each question. |

1. | In a partially destroyed laboratory record of an analysis of correlation data, the following results only are legible: Variance of X = 7 Regression equations: 6X – 2Y + 6 = 0 and 30X – 18Y + 174 = 0. One of these two regression equations is for Y on X (with X as the independent variable) and the other is for X on Y (with Y as the independent variable). Find the mean value of X. (a) 10 (b) 11 (c) 18 (d) 5 (e) 7. (1 mark) | < Answer >
2. | If the standard error of mean for a sample of size 16 is 3, then the population standard deviation will be equal to (a) 2 (b) 3 (c) 6 (d) 12 (e) 18. (1 mark) | < Answer >
3. | Which of the following statements is not true with regard to correlation analysis? (a) It derives the mathematical relationship between two or more series (b) It compares the variation in two or more series (c) It proves the presence of a cause and effect relationship between two or more series (d) It helps in determining the degree of association between two or more series (e) It helps in estimating the value of one variable on the basis of the other variable. (1 mark) | < Answer >
4. | Which of the following statements is not true? (a) An unweighted aggregates price index does not attach any importance to the consumption or usage levels of the items covered (b) Laspeyres price index is an unweighted aggregates price index (c) Paasches price index is a weighted aggregates price index (d) A price index reflects the overall change in price for the basket of goods considered (e) Relatives are the ratios of the values for the current period over that of the base period. (1 mark) | < Answer >
5. | In a correlation study on variables X and Y, the following results are obtained:
| X | Y
Mean | 45 | 47
Standard deviation | 3.5 | 5.5
Coefficient of correlation | 0.70
Find the regression equation of Y on X with X as the independent variable. (a) = 1.52 - 5.8X (b) = 2.65 + 12.9X (c)= - 2.5 + 1.1X (d) = – 13.9 + 0.65X (e)= – 3.5 - 2.12X. (1 mark) | < Answer >
6. | The area under the standard normal curve between the points 0 and 2.9 is 0.4981, then the area between the points – 2.9 and 2.9 is (a) 0.9962 (b) 0.9999 (c) 0.9972 (d) 0.9872 (e) 0.9852. (1 mark) | < Answer >
7. | There are 25 questions each with 4 multiple choices. What is the probability of answering exactly 5 questions correctly is (a) 0.3381 (b) 0.1645 (c) 0.2067 (d) 0.1550 (e) 0.2453. (1 mark) | < Answer >
8. | Formula-I race consists of 110 cars. On observing the cars, one person found that 20 cars are colored red, 13 cars are colored yellow and 3 cars are found to have both the colors. The probability of detecting a car of either colors among the total cars is (a) 4/110 (b) 3/110 (c) 2/11 (d) 3/11 (e) 30/11. (1 mark) | < Answer >
9. | If the variable X resembles same as variable Y, then the covariance between the two variables reduces to (a) The covariance of either variables (b) The correlation of either variables (c) The standard deviation of either variables (d) The variance of either variables (e) The expected value of either variables. (1 mark) | < Answer >
10. | A decision maker is using the Hurwicz criterion to take a decision under conditions of uncertainty. In which of the following cases can he be said to have a tendency towards pessimism? I. Index of pessimism lies between 0.1 and 0.3. II. Index of pessimism lies between 0.6 and 0.7. III. Index of pessimism lies between 0.7 and 0.9. (a) Only (I) above (b) Only (II) above (c) Only (III) above (d) Both (I) and (II) above (e) Both (II) and (III) above. (1 mark) | < Answer >
11. | The binomial distribution may be approximated to a Poisson distribution, if the probability of success (p) is ________ and the number of trials (n) is _______. (a) Large, small (b) Small, large (c) 1, finite (d) Finite, infinite (e) 1, infinite. (1 mark) | < Answer >
12. | In the game of cricket, umpire tosses a coin at captains of team A and team B. Then the probability of winning the toss by team B captain is 1/3. The standard deviation of the number of successes in a binomial distribution is given by. Find the probability of obtaining exactly 3 successes? (a) 0.0003 (b) 0.0175 (c) 0.0018 (d) 0.1079 (e) 0.0076. (1 mark) | < Answer >
13. | The time taken to find a taxi when leaving a restaurant follows a left skewed distribution with a mean of 20 minutes and a variance of 100 minutes. If 64 restaurant patrons are randomly sampled and the average time that it takes for them to find a taxi is calculated, then what is the probability that the sample mean will be between 18 and 23 minutes? (a) 0.4918 (b) 0.4452 (c) 0.9918 (d) 0.9452 (e) 0.9370. (2 marks) | < Answer >
14. | The probability of an event is ‘p’. If the experiment is repeated ‘n’ times then the probability of the event occurring every time is (a) p/n (b) (c) pn (d) np (e) n(1– p). (1 mark) | < Answer >
15. | Which of the following statements 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 and lower control limits are based on the overall sample mean (d) The lower control limit is equal to the overall sample proportion (e) The upper control limit is equal to the overall sample proportion. (1 mark) | < Answer >
16. | In the test involving ANOVA the F-statistic is calculated on the basis of (a) The variance of the largest sample (b) The mean of the largest sample (c) The standard deviation of the smallest sample (d) The estimated population mean based on the mean among the sample variances and the estimated population variance based on the variance between the samples (e) The estimated population variance based on the variance among the sample means and the estimated population variance based on the variance within the samples. (1 mark) | < Answer >
17. | The demand 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:
Days | Mon | Tue | Wed | Thurs | Fri | Sat
No. of spare parts demanded | 1032 | 1026 | 1020 | 1019 | 1023 | 1012
We need to test the hypothesis that the number of spare parts demanded does not depend on the day of the week. What is the value of the test statistic for the above test? (a) 0.3159 (b) 0.2226 (c) 0.2079 (d) 0.2159 (e) 0.3079. (2 marks) | < Answer >
18. | The following table gives the probability distribution of a daily supply of a particular item.
Supply (’00 units) | 4 2 4
Probability | 20% 70% 10%
The opening inventory was 900 units and the daily supply is to be estimated. If in a run the random number generated was 55, then what is the closing inventory? (a) 845 (b) 700 (c) 900 (d) 745 (e) 55. (1 mark) | < Answer >
19. | From the given options which is applicable by Bienayme - Chebyshev’s theorem? (a) Normal Distribution (b) Rectangular Distribution (c) Symmetrical Distributions (d) Non-symmetrical distributions (e) Distributions of any kind. (1 mark) | < Answer >
20. | Which of the following statements is correct about, the value for “Rapid development decision”? (a) It will be the value of the outcome multiplied with the probabilities of outcomes (b) It will be the value of the outcome added to the probabilities of outcomes (c) It will be the value of the outcome multiplied with the total probability (d) It will be value of the outcome added to the possible outcomes (e) It will be the value of the outcome multiplied with the probability of decision variables. (1 mark) | < Answer >
21. | Which of the following is the appropriate test statistic to use, when comparing two population means by using large independent random samples selected from populations with equal and known variances? (a) t-statistic (b) z-statistic (c) F-statistic (d) Chi-square statistic (e) Mean of the two sample means. (1 mark) | < Answer >
22. | The following details are available with regard to a hypothesis test on means of two populations: H0: H1: n1 = 36 n2 = 18 The samples collected from the two populations are independent. What is the value of the test statistic? (a) 0.5346 (b) 0.4329 (c) 0.3354 (d) 0.2125 (e) 0.5091. (1 mark) | < Answer >
23. | The ‘a’ and ‘b’ coefficients of a simple regression line are 5 and 0.5 respectively. What will be the estimated value of the dependent variable if the independent variable is equal to 10? (a) 0.5 (b) 5 (c) 10 (d) 15 (e) 20. (1 mark) | < Answer >
24. | Which of the following statements is false? (a) Interval estimates are used for small populations only (b) An interval estimate is a range of values within which the actual value of the population parameter may fall (c) An interval estimate indicates the probability of the true population parameter falling within the range of values given (d) A point estimate is a single number that is used to estimate an unknown population parameter (e) The sample variance is a point estimator of the population variance. (1 mark) | < Answer >
25. | For a basket of commodities the price of each commodity in a year X is thrice the price in the base year. What is the Marshall Edgeworth price index for year X? (a) 50 (b) 100 (c) 300 (d) 250 (e) 150. (1 mark) | < Answer >
26. | The following data pertains to the consumption of needs by a bar:
Inputs | Units | Prices (in Rs. per unit) | Quantities Used (Units)
2001 | 2004 | 2001 | 2004
Meat | Kilogram | 13 | 39 | 459 | 600
Eggs | Dozen | 9 | 11 | 99 | 50
Water | Litres | 7 | 15 | 201 | 180
Oil | Kilogram | 45 | 16 | 70 | 60
Laspeyres quantity Index for the year 2004 with the year 2001 as base year, is (a) 109.41 (b) 106.96 (c) 198.23 (d) 206.04 (e) 110.19. (1 mark) | < Answer >
27. | The central limit theorem claims that the sampling distribution of the mean (a) Is always normal (b) Is always normal for large sample sizes (c) Approaches normality as sample size increases (d) Is a t-distribution if N is less than 30 (e) Approaches normality if N is greater than or equal to 30. (1 mark) | < Answer >
28. | In testing of hypothesis, the decision to use a single-tailed or two-tailed test depends upon (a) The size of the sample (b) The null hypothesis (c) The values of the sample statistic (d) The alternative hypothesis (e) Shape of the distribution. (1 mark) | < Answer >
29. | Which of the following statements is true when the sample size is large? (a) will be larger than (b) will be normally distributed (c) will be normally distributed (d) will have standard deviation equal to (e) will be equal to . (1 mark) | < Answer >
30. | If the sample of size 52 is collected from a population of size 795, then the finite population correction factor to be used for calculation of standard error is equal to (a) 0.9673 (b) 0.6498 (c) 0.7189 (d) 0.9363 (e) 0.9981. (1 mark) | < Answer >
31. | Consider the following probability distribution of a variable X
X | 5.0 | 10.0 | 15.0 | 20.0 | 25.0 | 30.0 | 35.0
p | 0.1 | 0.05 | 0.05 | 0.1 | 0.3 | 0.2 | 0.2
In a Monte Carlo simulation process the number 70 was generated in a particular run from a uniform distribution of 0 – 99. The value of X in this run will be (a) 35 (b) 20 (c) 25 (d) 15 (e) 30. (1 mark) | < Answer >
32. | The placement division of NGR is interested in finding out the annual earnings of NGO’s numbering around 1500. How large a sample it should take in order to determine the mean annual earnings within Rs.2,000 and at 95% confidence level? You may assume that the earnings of NGO’s follow normal distribution with a standard deviation of Rs.5,000. (a) 35 (b) 20 (c) 45 (d) 24 (e) 40. (2 marks) | < Answer >
33. | Puja, the chief manager of a Telephone manufacturing company, feels that the longer a group of employees work together on an assembly line, the higher the daily output rate is. She has gathered the following data for a group of employees who worked together for 10 days:
Daily output (units) | 3 | 6 | 8 | 9 | 10 | 4 | 7 | 11 | 12 | 8
Days worked together | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10
Which of the following is the intercept of the estimated regression line with the daily output as the dependent variable? (a) 4.35 (b) 4.80 (c) 4.45 (d) 5.50 (e) 6.56. (1 mark) | < Answer >
34. | A sports magazine conducted a survey among the spectators of major cricket tournament and found that for any randomly selected person the probability that he supports Team A is 0.50, the probability that he supports Team B is 0.30 and the probability that he supports Team C is 0.20. It is assumed that these probabilities are accurate. What is the probability that out of a randomly chosen group of eight persons at least six do not support Team B? (a) 0.238 (b) 0.325 (c) 0.448 (d) 0.552 (e) 0.905. (2 marks) | < Answer >
35. | A survey firm conducts door-to-door surveys on a variety of issues. Some individuals co-operate with the interviewer and complete the interview questionnaire and others do not. The following data is available.
Respondents | Sample Size | Non Co-operating
Men | 200 | 110
Women | 300 | 210
It is to be tested at a significance level of 0.05, whether the proportions of men and women who co-operate with the interviewer are equal. Which of the following can be inferred from the test? (a) Proportions of men and women who co-operate with the interviewer are equal (b) Proportions of men and women who co-operate with the interviewer are not equal (c) Proportion of men who co-operate with the interviewer is less than the proportions of women who co-operate (d) All men co-operate with the interviewer (e) All women co-operate with the interviewer. (2 marks) | < Answer >
36. | Which of the following statements is not true? (a) The purpose of hypothesis testing is to make a judgment about the difference between the sample statistic and the hypothesized population parameter (b) When the null hypothesis is rejected the alternative hypothesis is accepted (c) The purpose of hypothesis testing is to test the correctness of the computed value of the sample statistic (d) The acceptance of the null hypothesis does not prove that the null hypothesis is true (e) The acceptance of the alternative hypothesis does not prove that the null hypothesis is false. (1 mark) | < Answer >
37. | A survey was conducted on the retail prices of a specific type of pulses in a city. At a confidence level of 95 percent it was estimated that the mean retail price of the commodity fell between Rs.25.50 per kg and Rs.27.46 per kg. The sample size was 36 and less than 5 per cent of the population of the grain retailers in the city were sampled. Which of the following is the correct combination of the mean and standard deviation of the prices in the sample? (a) Mean price = Rs.52.96 Standard deviation of the prices = Rs.6.00 per kg (b) Mean price = Rs.26.48 per kg Standard deviation of the prices = Rs.3.00 per kg (c) Mean price = Rs.52.96 Standard deviation of the prices = Rs.3.00 per kg (d) Mean price = Rs.26.48 per kg Standard deviation of the prices = Rs.6.00 per kg (e) Mean price = Rs.6.00 per kg Standard deviation of prices = Rs.26.48 per kg. (2 marks) | < Answer >
38. | A t-statistic calculated on the basis of a sample of a specific size, has a t distribution with number of degrees of freedom equal to (a) The sample size (b) One less than the sample size (c) One more than the sample size (d) Twice the sample size (e) Half the sample size. (1 mark) | < Answer >
39. | Voyage Industries Ltd. manufactures heavy-duty voltage stabilizers. The general manager of the company wants to compare the performance of the stabilizers manufactured by the company with the industry standards. He knows that 18% of all the stabilizers sold in the industry requires repairs during the first year of sale. The company sampled 144 customers and found that in case of 36 customers the stabilizers required repairs in the first year of sale. The manager of the company wants to determine if the performance of the stabilizers manufactured by the company is different from the industry standards. What is the value of the appropriate test statistic for the above test? Use normal distribution and a significance level of 5%. (a) 1.98 (b) 2.07 (c) 2.19 (d) 2.38 (e) 2.57. (2 marks) | < Answer >
40. | Bhanu Chits is a firm involved in the business of supplying processed food products as per the tender specifications of various purchasing parties. On the basis of its past experience it has estimated that there is a 65% probability that it can secure the supply contract with the party inviting the tender. The firm has presently submitted quotations in response to tenders invited by 4 parties. Find out the probability that it will secure supply contracts with at least 3 of the parties inviting the tender. (a) 0.4305 (b) 0.2415 (c) 0.3525 (d) 0.4535 (e) 0.5629. (2 marks) | < Answer >
| 41. | The following data provides the values of sample range R, for samples of size 7 each.
Sample No. | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10
Range (R) | 7 | 4 | 8 | 5 | 17 | 4 | 8 | 10 | 7 | 9
Calculate the value of upper control limit for the R chart (Use the relevant D3 or D4 factor) (a) 16.2481 (b) 16.7006 (c) 16.4365 (d) 15.5648 (e) 17.6428. (1 mark) | < Answer >
| 42. | Each lot in 20 lots of rubber belts contains 1500 rubber belts. The number of defective rubber belts in these lots are as follows:
410 | 420 | 324 | 332 | 292 | 310 | 282 | 300 | 320 | 296
392 | 432 | 294 | 324 | 220 | 400 | 258 | 226 | 460 | 280
Calculate the value of central line for the p chart for the fraction of defectives. (a) 0.0921 (b) 0.0632 (c) 0.2190 (d) 0.2652 (e) 0.3662. (1 mark) | < Answer >
| 43. | The simple correlation coefficients between temperature (X1), corn yield (X2) and rainfall (X3) are r12 (Between X1 and X2) = 0.63, r13 (Between X1 and X3) = 0.21 and r23 (Between X2 and X3) = 0.87. Calculate the multiple correlation coefficient R1.23 for the three variables. (a) 0.4562 (b) 0.5678 (c) 0.9283 (d) 0.9311 (e) 0.6024. (1 mark) | < Answer >
| 44. | From a dataset containing paired values of variables X and Y we have the following:
åX = 45, | åY= 108, | åX2 = 285, | åY2 = 1356, | åXY = 597 and | n = 10.
Find the coefficient of correlation between the variables X and Y. (a) 0.91 (b) 0.92 (c) 0.93 (d) 0.94 (e) 0.95. (1 mark) | < Answer >
| 45. | Consider the following information:
Spices |  | Pn Qn
A | 120.58 | 11,265
B | 152.30 | 4,125
C | 150 | 1,854
D | 133 | 7,600
E | 142.63 | 12,250
The weighted average of relative price index is (a) 128.42 (b) 30.77 (c) 135.18 (d) 137.78 (e) 129.20. (2 marks) | < Answer >
| 46. | Which of the following statements 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) Paasches 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) Laspeyres price index has a downward bias (e) The weighted average of relatives price index uses base year prices as weights. (1 mark) | < Answer >
| 47. | A composite price index where the prices of the items in the composite are weighted by their relative importance is known as the (a) Price relative (b) Unweighted aggregates price index (c) Weighted aggregates price index (d) Value index (e) Quantity index. (1 mark) | < Answer >
| 48. | The price per unit of goods X, Y and Z was Rs.70, Rs.50 and Rs.40 respectively in the year 2004. The unweighted aggregates price index for the year 2004 was 128 and the base year is 2000. The prices per unit of goods X and Y were Rs.45 and Rs.25 respectively in the year 2000. What was the price per unit of good Z in the year 2000? (a) Rs.50 (b) Rs.45 (c) Rs.40 (d) Rs.55 (e) Rs.60. (1 mark) | < Answer >
| 49. | For basket of goods the Laspeyres price index for the year 2002 was 175. The aggregate value of the same basket of goods using base year quantities in the year 2002 was Rs.6,800. The aggregate value of the same basket of goods using base year quantities in the year 2004 was Rs.7,800 and the base year is 1999. What was the Laspeyres price index for the year 2004? (a) 125 (b) 220 (c) 200 (d) 176 (e) 180. (1 mark) | < Answer >
| 50. | The following details are available with regard to a basket of commodities: What is the Marshall-Edgeworth price index? (a) 155.3 (b) 126.2 (c) 135.6 (d) 123.0 (e) 194.1. (1 mark) | < Answer >
| 51. | For a regression relationship, the ratio of regression sum of squares to error sum of squares is 4:3. The total sum of squares is 850. What is the regression sum of squares? (a) 484.5 (b) 450.4 (c) 360.2 (d) 480.7 (e) 370.5. (1 mark) | < Answer >
| 52. | In the year 2000 the Eastern Publishing Ltd. employed 45 sales representatives. In 2004 they employed 55 sales representatives. Using 2000 as the base period, the index for the number of sales representatives in 2004 approximately is (a) 60 (b) 82 (c) 70 (d) 100 (e) 120. (1 mark) | < Answer >
| 53. | In Laspeyres price index the prices in the base year and current year are weighted by (a) Base year total price (b) Current year total price (c) Base year quantities (d) Current year quantities (e) Arithmetic mean of base year quantities and current year quantities. (1 mark) | < Answer >
| 54. | The following information was obtained from the manager of a city water department for predicting the consumption of water (in gallons) from the size of household: Household size = X Water consumption = Y SX = 57 S Y = 10,038 n = 10 SX2 = 433 SY2 = 11,641,474 SXY = 67,669. The equation of the least squares regression for water consumption on household size is given by: (a) = 97053.7 + 96.692X (b) = 999.220 + 0.803X (c) = –1.0028 + 0.0067X (d) = 452.656 + 96.692X (e) = 1003.8 – 96.692. (1 mark) | < Answer >
| 55. | A sample randomly collected from a population has the following details Sum of the squares of the observations = 2285 Sample mean = 15.53 Number of observations = 8 What is the estimated standard error of mean? (a) 3.246 (b) 2.517 (c) 5.452 (d) 4.783 (e) 3.658. (1 mark) | < Answer >
| 56. | In testing the significance of individual regression coefficients, which of the following statement(s) is/are true? I. The test statistic is the t-statistic. II. The independent variables are tested individually. III. The independent variables for which the null hypothesis is not rejected do not have a significant relationship with the dependent variable. (a) Only (I) above (b) Only (II) above (c) Both (I) and (II) above (d) Both (II) and (III) above (e) All (I), (II) and (III) above. (1 mark) | < Answer >
| 57. | The variable Y increases as variable X increases and vice versa. Variable X cannot explain 51 percent of the variations in Y. What is the coefficient of correlation between X and Y? (a) 0.70 (b) 0.20 (c) 0.64 (d) – 0.64 (e) – 0.80. (1 mark) | < Answer >
| 58. | If deseasonalized value in time series is 2.00 and actual data is 1.58, the seasonal index is (a) 79 (b) 21 (c) 179 (d) 121 (e) 100. (1 mark) | < Answer >
| 59. | The following information is regarding the time series analysis of profits of an automobile company. Relative cyclical residual measure = - 2.16% Estimated profits for the year 2003 = Rs.478.56 lakh Based on the above information, which of the following statement(s) is/are false? I. Actual profit is Rs.468.22 lakh. II. Percent of trend is 102.16. III. Percent of trend is 97.84. IV. Actual profit is Rs.298.90 lakh. (a) Only (I) above (b) Only (IV) above (c) Both (I) and (II) above (d) Both (I) and (III) above (e) Both (II) and (IV) above. (1 mark) | < Answer >
| 60. | If the percent of trend for a year in a time series is greater than 100%, it indicates that (a) The actual time series value lies below the trend line and the relative cyclical residual is positive (b) The actual time series value lies below the trend line and the relative cyclical residual is negative (c) The actual time series value lies above the trend line and the relative cyclical residual is negative (d) The actual time series value lies above the trend line and the relative cyclical residual is positive (e) The actual time series value is less than the estimated value, for the year. (1 mark) | < Answer >
| 61. | A sample of 20 observations from a normal distribution has a mean of 42 and a variance of 11.8. 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) 7.43 (d) 8.31 (e) 9.63. (1 mark) | < Answer >
| 62. | For samples of size greater than 1, the finite population correction factor (a) Is always equal to 1 (b) Is always more than 1 (c) Is always constant irrespective of the sample size (d) Is always equal to 0 (e) Is always less than 1. (1 mark) | < Answer >
| 63. | The sampling distribution of mean is (a) A distribution of the observations included in a large sample (b) A distribution of the observations included in a small sample (c) A distribution of sample means of all possible samples of a specific size, that can be taken from a population (d) A distribution of sample means of all possible samples of a specific size that can be taken from different populations (e) A distribution of sample means of samples of different sizes taken from different populations. (1 mark) | < Answer >
| 64. | The latest nationwide political poll indicates that for persons, who are randomly selected, the probability that they support Party A is 0.55, the probability that they support Party B is 0.30, and the probability that they support Party C is 0.15. It is assumed that these probabilities are accurate and that the randomly selected persons have independent opinions towards the different parties. What is the probability that out of a randomly chosen group of ten persons, less than eight persons do not support Party A? (a) 0.02739 (b) 0.02289 (c) 0.0045 (d) 0.99618 (e) 0.97261. (2 marks) | < Answer >
| 65. | Mohanty Bio-Chem. is a renowned company in providing bio-chemical and bio-physical products. It provides clones of popular genes such as AB-1, AB-2 and AB-3. The number of pairs sold for each brand on several different days is as follows:
AB-1 | AB-2 | AB-3
7 | 11 | 9
5 | 10 | 7
3 | 4 | 5
9 | 10 | 9
For a test of ANOVA, the estimated common population variance using the variance within the samples is (a) 3.0833 (b) 5.4853 (c) 7.6453 (d) 6.8609 (e) 8.4533. (2 marks) | < Answer >
| 66. | When we carry out a chi-square test of independence, which of the following can be stated in the null hypothesis? (a) The population mean is equal to a specified value (b) The population proportion is equal to a specified value (c) The population distribution follows a specified probability distribution (d) The population can be classified on the basis of two criteria, which are not associated (e) The population can be classified on the basis of two criteria, which are associated. (1 mark) | < Answer >
| 67. | The following details are available with regard to a hypothesis test on means of two populations: n1 = 64 n2 = 36 The standard error of difference between means is . The samples collected from the two populations are independent. What is the standard deviation of the first population? (a) 4 (b) 16 (c) 256 (d) 2 (e) 48. (1 mark) | < Answer >
| 68. | The following details are available with regard to the sample data collected for a hypothesis test on means of two populations: n1 = 16 n2 = 9 s2 = 12 The samples are independently collected and it is assumed that the two populations are normally distributed and have the same variance. The pooled estimate of the underlying common population variance is 217.0435. What is the variance of the first sample? (Round off your answer to the nearest integer). (a) 16 (b) 256 (c) 64 (d) 128 (e) 240. (1 mark) | < Answer >
| 69. | 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. (1 mark) | < Answer >
| 70. | The property of efficiency with regard to an estimator relates 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. (1 mark) | < Answer >
| 71. | The weekly number of automobile accidents in 12 weeks in a certain city were as follows:
8 | 3 | 15 | 7 | 9 | 21 | 19 | 5 | 9 | 11 | 6 | 16
It is believed that the traffic conditions were same during the 12 weeks. We want to test whether the numbers obtained are in agreement with the belief. What is the critical value for the appropriate statistical test at a significance level of 5%? (a) 13.362 (b) 19.675 (c) 15.987 (d) 16.919 (e) 18.307. (1 mark) | < Answer >
| 72. | A senior researcher in a marketing research firm observes from his past experiences that potential respondents are more likely to reply to questionnaire printed on light colored paper. Questionnaires were sent on a random basis with the following results:
Color used | Response received | No response | Total
Light | 120 | 80 | 200
Dark | 100 | 100 | 200
Total | 220 | 180 | 400
He wants to test his observation from the past experience on the basis of the above data. What is the value of the appropriate test statistic for the above test? (a) 2.06 (b) 3.05 (c) 4.04 (d) 5.03 (e) 6.02. (1 mark) | < Answer >
| 73. | Five coins are tossed 3,200 times and the number of heads appearing each time was noted. At the end, the following results were obtained:
No. of heads | 0 | 1 | 2 | 3 | 4 | 5
Frequency | 80 | 570 | 1100 | 900 | 500 | 50
With the help of the above data, we want to test whether the coins are unbiased. Which of the following is an appropriate null hypothesis for this test? (a) The coins are biased (b) The coins are unbiased (c) Some of the coins are biased (d) Exactly two coins are biased (e) Exactly three coins are biased. (1 mark) | < Answer >
| 74. | If the Laspeyres price index and Paasches price index are 111 and 123 respectively, then the Fisher’s ideal price index will be approximately equal to (a) 106.36 (b) 107.54 (c) 116.85 (d) 110.91 (e) 111.16. (1 mark) | < Answer >
| 75. | The ratio of the weighted average price of the goods in a basket in the current year to their weighted average price in the base year is 7:5. Both the weighted averages have been calculated using the consumption quantities in the current year as weights. Which of the following is correct? (a) Laspeyres price index = 1.40 (b) Laspeyres price index = 140 (c) Laspeyres price index = 71.43 (d) Paasches price index = 1.40 (e) Paasches price index = 140. (1 mark) | < Answer >
| 76. | 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) Both seasonal and irregular variations. (1 mark) | < Answer >
| 77. | The linear trend estimating equation for a time series is given below: = 139 + 7.5X Where X = Year – 2002 is the estimated value of the dependent variable Y. The observed value of Y for the year 2004 is 160. What is the relative cyclical residual for the year 2004? (a) – 3.9 (b) 10.8 (c) 3.9 (d) 9.25 (e) 10.9. (1 mark) | < Answer >
| 78. | A two-wheeler dealer has sold 35,000 two-wheelers during the year 2003-04. According to the secular trend equation, the estimated number for the year is 37,000. The cyclical variation for the year measured by the relative cyclical residual technique is (a) – 5.40 (b) 0.94 (c) 6.25 (d) – 6.67 (e) 6.67 (1 mark) | < Answer >
| 79. | Which of the following represents the movements of a variable during the same months over successive years? (a) Secular trend (b) Cyclical fluctuation (c) Seasonal variation (d) Irregular variation (e) Standard deviation. (1 mark) | < Answer >
| 80. | The quality manager of the bottling plant of a cola major wants to ensure that the quantity of soft drinks in the bottles is as printed on the label of the bottle. He takes the samples of ten bottles every day and plots a control chart for the mean quantity of beverage in the bottle. Towards the end of the week he observes that the observations are within the control limits but there are jumps in the level around which the observations vary. Which of the following inferences is most appropriate? (a) The variations have been reduced to a great extent (b) The means of two different populations are being observed (c) The process mean has been shifting (d) The process is out of control (e) The process mean is constant. (1 mark) | < Answer >
| 81. | Olympus Sporting Goods has implemented a special trade promotion for its tennis balls and feels that the promotion should result in a price benefit for the consumer. Olympus knows that before the promotion began, the average retail price of the ball was Rs. 44.95, with a standard deviation of Rs.5.75. Olympus selects 25 samples of its retailers after the promotion was launched and finds the mean retail price for the ball is now Rs.42.95. Olympus wants to examine whether the price of balls has significantly reduced after the trade promotion. Which of the following conclusions can be drawn on the basis of the above information at a significance level of 2%? (a) The average retail price to the consumers has increased insignificantly (b) The average retail price to the consumers has not decreased significantly (c) The average retail price to the consumer has decreased significantly (d) We cannot draw any conclusion, as the sample size is too small (e) We cannot draw any conclusion, as information is insufficient. (2 marks) | < Answer >
| 82. | A Steno has completed a draft of a 700 page document required by a statistician. The statistician has randomly selected some pages and estimated that there was 89.9 percent probability that the average number of typographical errors per page in the entire document will lie between 4.2317 and 4.7683. The average number of typographical errors per page for the pages selected by the statistician was 4.5 and the standard deviation was 1.2. The statistician selected a large sample of the pages in the document and the number of pages selected by the statistician was more than 5 percent of the number pages in the entire document. How many pages are selected by the statistician for estimating the average number of typographical errors per page in the entire document? (Round off your answer to the nearest integer). (a) 5 (b) 25 (c) 35 (d) 50 (e) 70. (3 marks) | < Answer >
| 83 | From a group of 600 children belonging to the same age group, a sample of 25 children is taken. From this sample, the average weight of the children is found to be 24 kg and the variance is found to be 10.24 (kg2). A 95 percent confidence interval for the mean weight of the children in the group has to be constructed. The lower and upper confidence limits of the confidence interval are (a) 24.16 kg and 37.84 kg respectively (b) 20.25 kg and 27.50 kg respectively (c) 22.68 kg and 25.32 kg respectively (d) 25.5 kg and 27.5 kg respectively (e) 17.6 kg and 24 kg respectively. (2 marks) | < Answer >
| 84. | From a population with a known variance two samples are independently collected. The first sample has a size a and has a standard error of mean as 2.4. The second sample has a size b and has a standard error of mean as 2.0. If a sample of size a + b is taken from this population then what will be the standard error of mean? (a) 1.536 (b) 2.437 (c) 1.440 (d) 1.207 (e) 0.834. (2 marks) | < Answer >

Suggested Answers
Quantitative Methods II (132) : April 2006

1. | Answer : (d) Reason : The mean value of the X series and Y series is the point of intersection of the two regression lines of Y on X and X on Y. Therefore the points of intersection could be found by solving the two equations: 6X – 2Y + 6 = 0 and 30X – 18Y + 174 = 0. We get the values of = 5 and = 18. | < TOP >
2. | Answer : (d) Reason : Standard error of mean = 3 n =16 i.e. =34 =12. | < TOP >
3. | Answer : (d) Reason : (a), (b), (c) and (e) are true; (d) is false about correlation analysis. | < TOP >
4. | Answer : (b) Reason : Laspeyres price index is a weighted aggregates price index. Alternatives (a), (c), (d) and (e) are true. | < TOP >
5. | Answer : (c) Reason : = a + bX b = Equation of regression line of Y on X : Y – 47 = 0.70 (5.5 / 3.5) (X – 45) or, Y = - 2.5 + 1.1 X. | < TOP >
6. | Answer : (a) Reason : The area under the standard normal curve between the points 0 and 2.9 is 0.4981 and thus the area under the standard normal curve between the points –2.9 and 0 is also 0.4981,by symmetry. Hence the area under the standard normal curve between the points –2.9 to 2.9 is .9962. | < TOP >
7. | Answer : (b) Reason : Let r=5 and given n=25. The probability of answering a question correctly is taken as a success, then we have p=0.25, q=0.75, (i.e. each multiple choice has 0.25 probability of success). Then applying the binomial distribution for r number of success out of n trials is: P(r) = . | < TOP >
8. | Answer : (d) Reason : P(R or Y) = P(R) + P(Y) – P(R and Y) = 20/110 + 13/110 – 3/110 = 33/110 – 3/110. = 30/110 = 3/11. | < TOP >
9. | Answer : (d) Reason : When the variables X and Y are same that implies the covariance between the variables reduces to the variance of either variables. | < TOP >
10. | Answer : (a) Reason : The index of pessimism is (1-) and it lies between 0.1 and 0.3. | < TOP >
11. | Answer : (b) Reason : This situation occurs when the Poisson distribution is limiting case of binomial distribution. Where the probability of success (p) is infinitesimally small and the number of trials (n) is so large that the product ‘np’ equals, a finite constant. | < TOP >
12. | Answer : (a) Reason : Probability of success p = 1/3. Then q = 1-p = 2/3. To find the value of n, put the values of p and q in variance. i.e. variance = npq 8.2 = n = 37 . | < TOP >
13. | Answer : (e) Reason : By Central Limit Theorem for large samples the sample mean is approximately normally distributed with mean = population mean and standard deviation \ = m = 20 = = = = = 1.25 \ = = P(–1.60 < z < 2.40) = P(z < 2.40) – P(z < –1.60) = P(z < 2.40) – P(z > 1.60) = P(z < 2.40) – [1 – P(z < 1.60)] = (0.50 + 0.4918) – 1 + (0.50 + 0.4452) =0.937. | < TOP >
14. | Answer : (c) Reason : If the experiment is repeated ‘n’ times the probability of the event occurring every time is = p ´ p ´ p……..n terms = pn | < TOP >
15. | 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) and (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 >
16. | Answer : (e) Reason : 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. | < TOP >
17. | Answer : (b) Reason : Here we set up the null hypothesis, H0 that the number of parts demanded does not depend on the day of week. Under the null hypothesis, the expected frequencies of the spare part demanded on each of the six days would be: This is a chi-square test of independence and the contingency table is given as:
Days | Frequency Observed (fi) Expected (ei) | (fi-ei)2 |
Mon. Tues. Wed. Thurs. Fri. Sat. | 1032 1026 1020 1019 1023 1012 | 1022 1022 1022 1022 1022 1022 | 100 16 4 9 1 100 | 0.097 0.015 0.0039 0.0088 0.0009 0.097
Total | 6132 | 6132 |  | 0.2226
Then the value of chi-square statistic is given by: i.e. | < TOP >
18. | Answer : (b) Reason :
Daily supply (’00) | Probability | Cumulative probability | Random numbers allocated
4 | 0.2 | 0.2 | 00-19
2 | 0.7 | 0.9 | 20-89
4 | 0.1 | 1.00 | 90-99
Since the random number generated was 55 that fall in between 20-89, the daily supply 200. So, the closing inventory = opening inventory – sales = 900-200 = 700. | < TOP >
19. | Answer : (e) Reason : Bienayme-Chebyshev theorem is applicable to any kind of distribution regardless of its shape. | < TOP >
20. | Answer : (a) Reason : The value for Rapid development decision will be multiplying the value of the outcome with the probabilities of outcomes. | < TOP >
21. | Answer : (b) Reason : When comparing two population means by using large independent random samples selected from populations with equal and known variances the appropriate test statistic to use is the z statistic. | < TOP >
22. | Answer : (a) Reason : Value of the test statistic = | < TOP >
23. | Answer : (c) Reason : = a + bx Given: a = 5 and b = 0.5 \ = 5 + 0.5x For x = 10, = 5 + 0.5 (10) = 10. | < TOP >
24. | Answer : (a) Reason : Interval estimates can be used for both small and large populations. | < TOP >
25. | Answer : (c) Reason : P1 = 3P0 Marshall Edgeworth price index = = = = 300. | < TOP >
26. | Answer : (b) Reason : Laspeyres Quantity Index for the given data is x 100 i.e., x 100 = = 106.96. | < TOP >
27. | Answer : (c) Reason : The central limit theorem claims that the sampling distribution of the mean approaches normality as sample size increases. | < TOP >
28. | Answer : (d) Reason : In testing of hypothesis, the decision to use a single-tailed or two-tailed test depends upon the alternative hypothesis. | < TOP >
29. | Answer : (b) Reason : When the sample size is large then, we know that the sample mean follows normal distribution. | < TOP >
30. | Answer : (a) Reason : When the sample size and population size are given the standard error can be found by the formula = = 0.9673. | < TOP >
31. | Answer : (e) Reason :
X | Probability | Cumulative Probability | Random numbers allotted
5 | 0.10 | 0.10 | 00 - 09
10 | 0.05 | 0.15 | 10 - 14
15 | 0.05 | 0.20 | 15 - 19
20 | 0.10 | 0.30 | 20 - 29
25 | 0.30 | 0.60 | 30 - 59
30 | 0.20 | 0.80 | 60 - 79
35 | 0.20 | 1.00 | 80 - 99
Since the random number generated was 70 (which fall in between 60 – 79), the value of X in this run is 30. | < TOP >
32. | Answer : (d) Reason : We are required to find the sample size so that the mean annual earnings of the NGO’s should fall within Rs.2000 at a confidence level of 95%. The Z-value for 95% level of confidence is 1.96. 1.96 = Rs.2000 = Rs.2000/1.96 = Rs.1020.408 the standard error is also given by the following formula = 1020.408 = = 5000/1020.408 = n = 24.01 or 24 Thus sample may contain 24 NGO’s at random. | < TOP >
33. | Answer : (b) Reason : To get the regression constants we tabulate the given data as below:
Days worked Together (X) | Daily Output(Y) | X2 | XY
1 | 3 | 1 | 3
2 | 6 | 4 | 12
3 | 8 | 9 | 24
4 | 9 | 16 | 36
5 | 10 | 25 | 50
6 | 4 | 36 | 24
7 | 7 | 49 | 49
8 | 11 | 64 | 88
9 | 12 | 81 | 108
10 | 8 | 100 | 80
Total = 55 | 78 | 385 | 474
The coefficient a = | < TOP >
34. | Answer : (d) Reason : The probability that a randomly selected person does not support Team B = 0.50 + 0.20 = 0.70 = p. Then the probability that he/she supports Team B is 1– 0.70 = 0.30 = q Therefore we can assume that it follows a binomial distribution with probability of success (does not support Team B), p= 0.70 and probability of failure (supports Team B), q = 0.30. The probability that at least six out of eight persons chosen does not support Team B is P(6) + P(7) + P(8) = = 0.2965 + 0.19765 + 0.057648 = 0.55179 = 0.552 (approx.) | < TOP >
35. | Answer : (b) Reason : p1 : proportion of men cooperating p2 : proportion of women cooperating Null hypothesis : p1 = p2 Alternative hypothesis : p1 ¹ p2 Estimated value of the population proportion , p = Test statistic : Critical values:± 1.96 The test statistic goes beyond the critical value in the right tail. Hence we reject null hypothesis. We can conclude that the proportions of men and women who cooperate with the interviewer are not equal. | < TOP >
36. | Answer : (c) Reason : The purpose of hypothesis testing is not to test the correctness of the computed value of the test statistic. So the option (c) is not true in the purpose of hypothesis testing. | < TOP >
37. | Answer : (b) Reason : The sample size is greater than 30 and the standard deviation of population is not known. Hence the appropriate distribution applicable is the normal distribution. A confidence interval of 95 per cent implies that 0.475 of the area of the normal curve is covered on either side of the sample mean. From the normal distribution table it is found that the appropriate z-value is 1.96. = ? s = ? = ? For the given confidence interval: Lower limit = Rs.25.50 per kg Upper limit = Rs.27.46 per kg. \ – 1.96 = 25.50. ………………(A). + 1.96 = 27.46 ………………………(B) Adding equations (A) and (B) we get: = 52.96 or = 26.48 (Rs. per kg.) Subtracting equation (A) from equation (B) we get: 3.92 = 1.96 \ = = 0.50 (Rs. per kg) Now, = \ 0.50 = or s = 0.50 = 3 (Rs. per kg) Mean price of the sample = Rs.26.48 per kg Standard deviation of the sample = Rs.3.00 per kg. | < TOP >
38. | Answer : (b) Reason : A t-statistic calculated on the basis of a sample of a specific size has a t distribution with degrees of freedom equal to one less than the sample size. | < TOP >
39. | Answer : (c) Reason : Sample size, n = 144 Sample proportion, = = 0.25 H0 : p = 0.18 ( = p0) H1 : p ¹ 0.18 a = 0.05 Standard error of proportion, = = = 0.032 Standardizing the sample proportion: Z = = » 2.19. | < TOP >
40. | Answer : (e) Reason : Given n = 4 p = 0.65 q = 0.35 P(r ³ 3) = P(r = 4) + P(r = 3) According to the binominal theorem, P(r) = P(r = 4) = = (0.65)4 x 1 = 1 x (0.65)4 = 0.1785 P(r = 3) = = (0.65) 3 (0.35) = 0.3844 \ P(r ³ 3) = 0.1785 + 0.3844 = 0.5629 The probability that Pragati Enterprises will secure supply contracts with at least 3 of the parties inviting the tender is 56.29%. | < TOP >
41. | Answer : (b) Reason : The central line of for the R chart = Mean of the sample range () = åR / n = 79 /10 = 7.9 The value of the upper control limit for the R chart = D4 = 7.9´ 2.114 = 16.7006. | < TOP >
42. | Answer : (c) Reason : Total number of defectives out of 30,000 items in 20 samples is: 410 + 420 + 324 + … + 460 + 280 = 6572 = Therefore the central line is set at 0.2190. | < TOP >
43. | Answer : (d) Reason : We have to calculate the multiple correlation coefficient treating X1 as dependent and X2 and X3 as independent variables, i.e., we have to find Substituting the given values | < TOP >
44. | Answer : (e) Reason : Coefficient of correlation is given by ; This can be rewritten after simplifying, as | < TOP >
45. | Answer : (c) Reason : Weighted average of relatives price index
Commodity |  | Pn Qn |
Rice | 120.58 | 11265 | 1358333.7
Wheat | 150.3 | 4125 | 619987.5
Salt | 150 | 1854 | 278100
Sugar | 133 | 7600 | 1010800
Pulses | 142.63 | 12250 | 1747217.5
|  | S = 37094 | S = 5014438.7
= 5014438.7 S PnQn = 37094. \ Weighted average of relatives price index for year is = | < TOP >
46. | Answer : (b) Reason : 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. Paasche’s price index tends to underestimate the rise in prices. Unweighted aggregates price index numbers do not link the price changes to the consumption levels. Laspeyres price index has an upward bias. The weighted average of relatives use the values (and not prices) as weights. | < TOP >
47. | Answer : (c) Reason : A composite price index where the prices of the items in the composite are weighted by their relative importance is known as the weighted aggregate price index. | < TOP >
48. | Answer : (d) Reason : Unweighted aggregates price index = Or 128 = Or 160 ´ 100 = 128 (45 + 25 + x) Or 16000 = 8960 + 128x Or X = | < TOP >
49. | Answer : (c) Reason : Laspeyres price index for 2001: = Or \ Laspeyres price index for 2003 = = = 200. | < TOP >
50. | Answer : (c) Reason : Marshall-Edgeworth price index = . = (Dividing both numerator and denominator by 2) = = = 135.6 (nearly) | < TOP >
51. | Answer : (a) Reason : Coefficient of determination = = \ or RSS = 0.57 ´ TSS = 0.57 ´ 850 = 484.5. | < TOP >
52. | Answer : (b) Reason : Index Number = 45/55 x100 = 81.8 = 82. | < TOP >
53. | Answer : (c) Reason : In Laspeyres price index the prices are weighted by base year quantities. | < TOP >
54. | Answer : (d) Reason : Slope, b = = = = 96.692 Intercept, a = = = 452.656 Equation = = 452.656 + 96.692X. | < TOP >
55. | Answer : (b) Reason : s = Estimated standard error of mean = . | < TOP >
56. | Answer : (e) Reason : In testing the significance of individual regression coefficients the test statistic is the t distribution, we test the independent variables individually and we usually delete the variables where the null hypothesis is not rejected. | < TOP >
57. | Answer : (a) Reason : (Positive square root because y increases as x increases and vice versa) | < TOP >
58. | Answer : (a) Reason : Deseasonalized value = ´ 100 \Seasonal Index = ´ 100 = . | < TOP >
59. | Answer : (d) Reason : Relative cyclical residual measure = -2.16% Estimated profits for the year 2003 = Rs.478.56 lakh Percent of trend measure = 100 + Relative cyclical residual = = 100 + (-2.16) = 97.84 \Actual profit = Estimated profit ´ = 478.56 ´ = Rs.468.22 lakh. | < TOP >
60. | Answer : (d) Reason : If the percent of trend for a year in a time series is greater than 100%, it indicates that the actual time-series value lies above the trend line and the relative cyclical residual is positive. | < TOP >
61. | 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 >
62. | Answer : (e) Reason : Finite population correction factor = (N – n) < (N – 1) \ Finite population correction factor is always less than 1 for n > 1. | < TOP >
63. | Answer : (c) Reason : The sampling distribution of mean is a distribution of sample means of all possible samples of a specific size, that can be taken from a population. | < TOP >
64. | Answer : (e) Reason : The probability that a randomly selected person supports party A is 0.55. Then the probability that he/she does not support party A is 1- 0.55 = 0.45 \The event that a randomly selected person does not support party A, follows a binomial distribution with probability of success = 0.45. The probability that less than 8 out of 10 person do not support party A =1- P(At least 8 persons do not support party A) =1– [P(8) + P(9) + P(10)] =1– =1– [0.02289 + 0.00416 + 0.00034] = 0.97261. | < TOP >
65. | Answer : (d) Reason : The Means of the samples are . The grand mean, = [(7 + 5 + 3 + 9) + (11 + 10 + 4 + 10) + (9 + 7 + 5 + 9)] / 12 = 7.42. We calculate Within column variance as below: . s12 = = 6.666. s22 = . = 10.25. s32 = = 3.6667. So, = x (6.666 + 10.25 + 3.6667) = 6.8609. Hence from above discussion, we can infer that option (d) is the answer. | < TOP >
66. | Answer : (d) Reason : When carrying out a chi-square test of independence it can be stated in the null hypothesis that the population can be classified on the basis of two criteria which are independent (i.e., not associated). | < TOP >
67. | Answer : (b) Reason : | < TOP >
68. | Answer : (b) Reason : Pooled estimate of the underlying common population variance = | < TOP >
69. | Answer : (a) Reason : 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 >
70. | Answer : (b) Reason : The property of efficiency with regard to an estimator refers to the size of the standard error. | < TOP >
71. | Answer : (b) Reason : The test is chi-square goodness of fit. Since the number of elements in the table is 12, the number of degrees of freedom for the chi square test is 12 – 1 =11. The value of chi-square for 11 degrees of freedom and 5% level of significance from the table is 19.675. | < TOP >
72. | Answer : (c) Reason : The test is chi-square test of independence The table of expected frequencies is: Expected frequencies of light are, The expected frequencies of dark are
Color used | Response received | No response | Total
Light | 0.5x220 =110 | 0.5x180 =90 | 200
Dark | 0.5x220 =110 | 0.5x180 =90 | 200
Total | 220 | 180 | 400
The test statistic (chi-square statistic) can be found as:
Observed frequency (O) | Expected frequency (E) | (O –E)2/E
120 | 110 | 0.909
100 | 110 | 0.909
80 | 90 | 1.111
100 | 90 | 1.111
The value of Chi square statistic, c2 = = 4.04. | < TOP >
73. | Answer : (b) Reason : The null hypothesis is that the coins are unbiased. The alternative hypothesis would be that the coins are biased. If the coins are unbiased, then the distribution of heads will follow binomial distribution | < TOP >
74. | Answer : (c) Reason : Fisher’s Ideal Index is the geometric mean of Laspeyres Index and Paasche’s Index. So, Fisher’s Ideal Index = = 116.85 (approx.) | < TOP >
75. | Answer : (e) Reason : Paasches price index = = = = | < TOP >
76. | Answer : (d) Reason : The cyclical variations can be separated using the percent of trend measure. | < TOP >
77. | Answer : (c) Reason : Year = 2004 \ x = 2004 – 2002 = 2 Relative cyclical residual = Y = 160 \ Relative cyclical residual for the year 2004 = = 3.9. | < TOP >
78. | Answer : (a) Reason : The required cyclical variation for the given year by relative cyclical residual technique will be = = – 5.40. | < TOP >
79. | Answer : (c) Reason : Seasonal variation represents movements of a variable during the same months over successive years. | < TOP >
80. | Answer : (c) Reason : If the observations can be described as jumps in the level around which the observations vary, then this pattern indicates that the process mean has been shifting. | < TOP >
81. | Answer : (b) Reason : We want to test the hypothesis H0 : m = 44.95 ( Null Hypothesis: Average retail price has not changed) H1: m < 44.95 (Alternative Hypothesis: Average retail price has reduced after sales promotion) The standard error of mean = = = 1.15 The standard normal value of the mean retail price of the sample = = = = -1.739 Looking at the table for the critical value corresponding the area under the normal curve as 2% in the left tail i.e. –2.05. We find that the sample statistic falls in the acceptance region therefore we accept null hypothesis and at significance level of 2 %. We can conclude that the price of the tennis balls has not reduced significantly after the trade promotions. | < TOP >
82. | Answer : (d) Reason : The sampling proportion is large as n/N > 0.05 (given) Therefore we will use finite population correction factor to find standard error of mean, = Since the statistician selected a large sample (this means n > 30), the appropriate distribution is normal distribution. Given, p(4.2317 < X < 4.7683) = 0.899 => Z = ± 1.64 \ 4.2317 = – 1.64 ------------ (A) 4.7683 = + 1.64 ------------ (B) Subtracting (A) from (B) 0.5366 = 3.28 or 3.28 = 0.5366 or = Since the sampling fractions exceeds 5%, = \ Squaring both sides or or (53.8033 x 700) – 53.8033n = 699n or n = 50.03 ~ 50 | < TOP >
83. | Answer : (c) Reason : s2 (Variance) = 10.24 n = 25 N = 600 Sampling fraction, \ Estimated standard error of mean, = Since the sample size is less than 30 and population variance is not known the t distribution will be used with 24 degrees of freedom. t-values for the upper and lower confidence limits are \ Upper confidence limit = Lower confidence limit = | < TOP >
84. | Answer : (a) Reason : Standard error of mean for sample size ‘a’ = = 2.4 = A Standard error of mean for sample size ‘b’ = = 2.0 = B = or = 1.2 or = 1.2 or = 1.2 or b = 1.44a Sample size = a + b = a + 1.44a = 2.44a \ Standard error of mean for sample size ‘a + b’ = = = = = 1.536 | < TOP >

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