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B.A. (Hons) Economics-Ist Year Paper – III STATISTICAL METHOD FOR ECONOMICS QUESTION BANK 2010-11 SHIVAJI COLLEGE HOUSE EXAMINATION – 2011 1. (a) If the first three moments about the origin for a distribution are 10, 225 and 0 respectively. Calculate the first three moments about the value 5 for the distribution. (b) Karl Pearson’s measure of skewness of a distribution is 0.5. The median and mode of the distribution are respectively, 42 and 32. Find (i) Mean, (ii) standard deviation, (iii) coefficient of variation. (c) Prove that A.M. G.M. H.M. OR (d) (i) The first four moments of a distribution about a value 3 of the variable are 2, 10 and 30 respectively. Do you think that the distribution is leptokurtic? (ii) A student received a mark of 84 on a final examination in economics for which the mean marks was 76 and standard deviation was 10. On the final examination in physics for which the mean marks was 82 and standard deviation was 16, he received a mark of 90. In which subject was his relative standing higher? (e) Calculate the Quartile Deviation and Coefficient of skewness from the following data Q 2 = 188.8”. Q1 = 14.6” and Q3 = 25.2”. Also indicate the nature of the distribution. (f) The mean and standard deviation of 100 observations were found to be 60 and 10 respectively. At the time of calculations, two items were wrongly taken as 5 and 45 instead of 30 and 20. Calculate the correct mean and correct coefficient of variation. 2. (a) (i) Given P (A) = 0.59., P (B) = 0.30, and P A B = 0.21., Find (i) P A B' (ii) P A' B' and (iii) P A' B' (ii) If a random variable X is normally distributed with mean μ (b) and variance σ 2 , show that the mean of the variable Z = X - μ / σ is always zero and variance is one. (i) A firm produces steel pipes in 3 plants with daily production volumes of 500, 1000 and 2000 units respectively. According to the past experience, it is known that fraction of defective output produced by the 3 plants are respectively .005, .008 and .10. If a pipe is selected from a day’s total production and found to be Bliss Point Studies 9891555578 1 Ravinder N. Jha 9811343411 B.A. (Hons) Economics-Ist Year (c) (d) (e) Paper – III defective, what is the probability that it came from the first plant? (ii) A box of fuses contains 20 fuses, of which five are defective. If three of the fuses are selected at random and removed from the box in succession without replacement, what is the probability that all the three fuses are defective? (i) The following table provides a probability distribution of the random variable X. X: 2 4 7 8 P(X) 0.2 0.3 0.4 0.10 Find (i) Var (X) (ii) If Y = 4X – 1 find Var (Y) (ii) For a normally distribution variable X, what proportion of the observations would be found between μ - σ and μ - 3σ ? OR (i) The time needed to complete a final examination in a particular examination in a particular course is normally distributed with a mean 80 minutes and a standard deviation of 10 minutes. (a) What is the probability of completing the exam in one hour or less? (b) What is the probability that a student will complete the exam in more than 60 minutes but less than 75 minutes? (c) Assume that class has 60 students and the examination period is 90 minutes in length. How many students do you expect will be unable to complete the exam in the allotted time? (ii) For a binomial distribution of defective bolts, the probability of defective bolts is given to be 0.2 and n is 500. Find the mean and standard deviation of the distribution. (i) An Institute surveyed 1000 students ages 16-22 about their personal finance. The survey found that 33% of the students have their own credit card. (a) In a sample of 6 students, what is the probability that at least two will have their own credit card? (b) In sample of 6 students majority of students will have their own credit card? (ii) The joint distribution of X and Y is as follows: X -2 -1 0 1 2 Y 10 .09 .15 .27 .25 0.04 20 .01 .05 .08 .05 0.01 (a) Find the marginal distributions of X & Y. (b) Find the conditional distribution of X given y = 20 Bliss Point Studies 9891555578 2 Ravinder N. Jha 9811343411 B.A. (Hons) Economics-Ist Year 3. (a) (b) (c) (a) Paper – III (c) Find the conditional distribution of Y given x = 2 You are given below the following information about advertisement and sales: Adv. Exp. (x) Sales (y) (in crores of Rs.) (in crores of Rs.) Mean 20 120 S.D. 5 25 Correlation coefficient (r) = 0.8 Find (i) Calculate two regression equations. (ii) Find the likely sales when advertisement expenditure is Rs. 25 crores. (iii) Standard error of estimate of regression Y on X. Prove that coefficient of correlation lies between – 1 and + 1. Given r 2 0.98, SX 26.67, SY 17.78, for 8 pairs of observation calculate (i) Regression coefficient b YX and b XY (ii) 95% confidence interval for population regression coefficient β (iii) Test the significance of β at 5% level of significance. OR For a given set of a bivariate data, the following results were obtained: X 190, Y 20, n = 20, x 2 20, y2 90, xy 100, where x and y are deviations from respective means. Find: (i) Two regression equations. (ii) Obtain the standard error of estimate Y on X. (iii) Test the significance of β at 5% level of significance. (b) (c) For a regression of Y on X, the total sum of squares was found to be 800 and the explained sum of squares was 392. If the regression was based on 20 pairs of values, find: (i) Coefficient of determination. (ii) Standard error of estimate for the regression. Data on advertising expenditures and revenue (in thousands of dollars) for the Delhi Seasons Restaurant follow. Advertising Expenditure (x): 1 2 4 6 10 14 20 Revenue (y): 19 32 44 40 52 53 54 (i) Estimate the regression line Y on X. (ii) Find the predicted revenue earnings if the advertisement expenditures is 30. (iii) Did the estimated regression equation provide a good fit? Explain. Bliss Point Studies 9891555578 3 Ravinder N. Jha 9811343411 B.A. (Hons) Economics-Ist Year Paper – III MH B.A (HONS) ECONOMICS IST YEAR 2010-2011 1. (a) Prove that the standard deviation is independent of change of origin but not of scale. (b) For a distribution, the first four moments about zero are 1,7,38 and 155 respectively. Compute the moment coefficient of skewness and kurtosis. Also, comment on the nature of the distribution. (c) If the price of a commodity doubles in a period of 4 years, What is the average annual percentage increase? Or (a) Find out the weighted harmonic mean of the first n natural numbers where the weights are equal to the corresponding numbers. (b) The first four central moments are given as 0, 4, 8 and 144. Comment on the nature of the distribution. (c) The arithmetic mean of two observations is 127.5 and their geometric mean is 60. Find their harmonic mean and the two observations. 2. (a) In 4 years, the prices of oil were $0.8, $0.9, $ 1.05 and $1.25 per gallon. What is the average price for this period is $1000 was spend every year? (b) You are given the following probability distribution of X X 1 2 3 4 5 6 P(X) 0.1 0.15 0.2 0.25 0.18 K (c) (a) (b) (i) What is the value of k (ii) What is E(X) and V(X)? (i) Find the probability of getting the sum 7 on atleast 1 of 3 tosses of a pair of fair dice. (ii) How many tosses are needed in order that probability in (i) is greater than 0.95? Or If you spend Rs. 100 per week on apples and the price of apples for 3 weeks is Rs.25, Rs.20 and Rs. 10 per kg respectively, what is the average price of apples for you? In the following probability distribution, X and Y can take the values 1, 2 and 3. X Y 1 2 3 1 2 3 5k K 3k 4k 3k 4k 2k 3k 2k Bliss Point Studies 9891555578 4 Ravinder N. Jha 9811343411 B.A. (Hons) Economics-Ist Year (c) 3. (a) (b) (c) (a) (b) (c) 4. (a) Paper – III (i) What is the value of k? (ii) Calculate E(X) and V(X) Suppose that a central university has to form a committee of 5 members from a list of 20 candidates out of whom 12 are teachers and 8 are students. If the members of the committee are selected at random, what is the probability the majority of the committee members are students? Small cars get better gas mileage but they are not as safe as bigger cars. Assume the probability of a small car being involved in an accident is 0.18. The probability of an accident involving a small car leading to a fatality is 0.128 and the probability of an accident not involving a small car leading to a fatality is 0.5. Suppose you learn of an accident involving a fatality, what is the possibility a small car was involved? A machine produces an average of 20% defective bolts. A batch is accepted if a sample of 5 bolts taken from that batch contains no defective and rejected if the sample contains 3 or more defective. In other cases, a second sample is take. What is the probability that the second sample is required? In a large institution, 2.28% of the employees receive income less than Rs.4500 and 15.8% of the employees receive income more than Rs.7500 per month. Assuming incomes to be normally distributed, calculate X and σ of the distribution Or Research shows that 30% of doctors in government hospitals leave their jobs to start their own practice. Among those who leave their jobs. 60% have a higher degree in specialization while 20% of these who do not leave have a higher degree. If a doctor has a specialization, what is the probability he will leave his job to start a private practice? Assuming that it is true that 2 in 10 industrial accidents are due to fatigue, find the probability that (i) Exactly 2 of 8 industrial accidents will be due to fatigue. (ii) atleast 2 of 8 industrial accidents will be due to fatigue The marks of a student in a certain examinations are normally distributed with mean marks as 40% and standard deviation as 20%. 60% of the students failed. The result was moderated and 70% students passed. Find the pass marks before and after the moderation. The S.D in the amount of time it takes to train a worker to perfume a tax is 40 mins. A random sample of 64 workers is taken. Bliss Point Studies 9891555578 5 Ravinder N. Jha 9811343411 B.A. (Hons) Economics-Ist Year (b) (a) (b) 5. (a) (i) What is the probability that the sample mean will exceed the population mean by more than 5 mins? (ii) What is the probability that the sample mean is more than 8 mins. Less than the population mean? A radio shop sells, in an average 200 radios per day with a S.D of 50 radios. After an extensive advertising campaign, the management will compute the average sales for next 25 days to see whether an improvement has occurred. Assume that the daily sale of radio is normally distributed. (i) Test the hypothesis at 5% level of significance if X = 216. (ii) How large must X be in order that the null hypothesis is rejected at 5% level of significance? Or A random sample of 81 purchased at a department store was take to estimate the mean of all purchases. The population S.D is $25. (i) What is the probability that the sample mean will not overstate the population mean by more than $4? (ii) What is the probability that the sample mean will understate the population mean by more than $1? Cars running on normal petrol have an average engine life of 1,20,000 kms with a S.D of 15,000 kms. A random sample of 100 cars using premium petrol reported mean engine life of 1,22,000 kms. (i) Test whether premium petrol increases car engine life using α = 0.05. (ii) What is the highest value of α that allows you to conclude that the engine life improves with the use of premium petrol? The following table shows how a sample of Maths and Statistics scores of 25 students are distributed. Maths (b) Paper – III 40–70 70–100 Statistic 40–70 5 2 70–100 7 11 Test at 5% level of significance whether scores in Maths and Statistics obtained by students are related. In a sample of 100 bulbs, the mean lifetime was found to be 100 hrs. with a S.D of 200 hrs. In another sample of 100 bulbs, the mean and S.D. were 1200 hrs & 100 hrs. respectively. Could they have come from the sample population? Use α = 0.05 Or Bliss Point Studies 9891555578 6 Ravinder N. Jha 9811343411 B.A. (Hons) Economics-Ist Year (a) (b) Paper – III The scientific experiment with peas revealed the following observations -315 round and yellow, 108 round and green, 101 wrinkled and yellow and 32 wrinkled and green. According to a previous theory, the numbers should be in proportion 9:3:3:1. Is there any evidence to doubt the theory at 1% and 5% level of significance? In a sample of 1500, the mean and S.D are found to be 18.5 and 4.5 respectively. In another sample of 1000, the mean is 20 and S.D is 4. Assuming that the sample s are independent, can they arise from the same population? Use α = 5% Bliss Point Studies 9891555578 7 Ravinder N. Jha 9811343411 B.A. (Hons) Economics-Ist Year Paper – III SATYAWATI COLLEGE (EVE.) MID-TERM EXAMINATION – 2010-11 1. (a) Cities A, B and C are equidistant from each other. A motorist travels from A to B at 30 miles/hr, from B to C at 40 miles/hr, and from C to A at 50 miles/hr. Determine his average speed for the entire trip. (b) The first four moments of a distribution about 3 are 1, 3, 6, and 8. Is the data consistent? Explain. (c) Prove that variance of first “n” natural numbers is n 2 1 /12. Or (a) The first four moments of a distribution about value 4 are -1.5, 17, -30 and 108 respectively. Calculate four moments about mean, β1 and β 2 Comment on nature of the distribution. (b) (c) 2. (a) (b) (c) (a) (b) Prove that A.M. G.M. H.M. taking positive observations a & b. The mean and variance of 10 observations are known to be 17 and 33 respectively Later it is found that one observation (i.e. 26) is inaccurate and is removed. What is the man and standard deviation of the remaining observations? Suppose that a Central University has to form a committee of 5 members from a list of 20 candidates out of whom 12 are teachers and 8 are students. If the members of the committee are selected at random, what is the probability that the majority of the committee members are students? “A frequency distribution can be described almost completely by the first four moments and the two measures based on moments.” Examine the statement What is the chance that a leap year selected at random will contain 53 Sundays? Or A factory has two machines, X and Y. Empirical evidence has established that machines X and Y produce 30% and 70% of the output respectively. It has also been established that 5% and 1% of the output produced by these machines respectively was defective. A defective item is drawn at random. What is the probability that the defective item was produced by machine X? The IQ’s of army volunteers in a given year are normally distributed with μ = 110 and σ = 10. The army wants to give advance training to the 20% of those recruits with the highest scores. What is the lowest IQ score acceptable for advanced training? Bliss Point Studies 9891555578 8 Ravinder N. Jha 9811343411 B.A. (Hons) Economics-Ist Year 3. (a) (b) (c) (a) Past experience indicates that 60% of the students entering college get their degrees. Using the normal approximation to binomial, find the probability that out of 30 students picked at random from entering class, more than 20 will receive their degrees Assume that the family incomes are normally distributed with μ = Rs 16000, and σ = Rs 2000. What is the probability that a family picked at random will have an income i) between Rs. 15000 and Rs. 18000, ii) Below Rs 15000, iii) Above Rs 18000 and iv) Above Rs 20000? If the random variable X is normally distributed with mean μ and variance σ 2 derive the mean and variance of the variable z = x μ /σ . Define with suitable examples. (i)A random variable, (ii) A discrete random variable, (iii) A continuous random variable. Or What is a Normal distribution and what is its usefulness. How is it different from a Standard Normal distribution? Explain with suitable diagrammatic illustrations. Prove the following: (i) E (aX + b) = aE (X) + b (ii) E (X + c) E (X) + c (iii) (b) (c) (a) (b) (c) Paper – III Var (X) = E X 2 - E X 2 Where X is a random variable and a, b, & c are constants The completion times for a job task range from 10.2 minutes to 18.3 minutes and are thought to be uniformly distributed. What is the probability that it will require between 12.7 and 14.5 minutes to perform this task. Distinguish between independent and mutually exclusive events. When will the events be both independent are mutually exclusive? Find the probability of getting the sum 7 on at least 1 out of 3 tosses of a fair dice. How many tosses are needed in order that such probability is greater than 0.95? Or The lifetime of light bulbs is known to be normally distributed with μ = 100 hrs and σ = 8 hrs. What is the probability that a bulb picked at random will have a lifetime between 110 and 120 burning hours. What is the probability of: (a) two 6s on 2 rolls of a die, (b) a 6 on each die on rolling two dice once, (c) 3 girls in a family with 3 children. Discuss the axioms of probability with suitable examples. Bliss Point Studies 9891555578 9 Ravinder N. Jha 9811343411 B.A. (Hons) Economics-Ist Year 4. (a) (b) (c) (a) (b) (c) Paper – III Show that Laspeyre’s Price Index can be written as a weighted average of price relatives. What are the weights? An index number is at 100 in year 2001. It rises by 4% in year 2002, falls by 6% in 2003, falls 4% in 2004 and rises 3% in 2005. Calculate index number for five years with 2003 as base. The Consumer Price Index over a certain period increased from 110 to 200 and the salary of the workers increased from Rs 3,500 to Rs 5000. What is the gain or loss to the workers? Or Explain time reversal, factor reversal and circular tests. Examine whether Laspeyre’s and Paasche’s index numbers satisfy these tests. Given the two price index series, splice them on the base 2004 = 100. By what percent did price of steel rise between 2000 and 2005 YEAR Old Price Index New Price Index Base 1995 = 100 Base 2004 = 100 2000 141.5 2001 163.7 2002 158.2 2003 156.8 2004 157.1 100.0 2005 102.3 The consumer price index for a group of workers was 250 in 2004 with 1990 as the base: (i) Compute the purchasing power of a rupee in 2004 compared to 1990. (ii) At what value of Consumer Price Index would the purchasing power of a Rupee be 25 paisa. Bliss Point Studies 9891555578 10 Ravinder N. Jha 9811343411 B.A. (Hons) Economics-Ist Year Paper – III KALINI COLLEGE HALF – YEARLY EXAMINATION 2010-2011 1. (a) Given the first four moments of a distribution μ 1 = 0, μ 2 = 40, μ 3 = -100, μ 4 = 200, test the skewness and kurtosis of the distribution. (b) Find mean and S.D. of the following observations: X: 1, 2, 4, 8, 9. Transform the above observations such that the mean of the transformed observation becomes double of the mean of X, std. Remaining unchanged. OR 1. (a) an income tax assesse depreciated the machinery of his factory by 20% in each of the first 2 years and 40% in the third year. How much average depreciation relief should he claim from the taxation department? (b) The first four moment of a distribution about the value 5 of the variable are 2, 20, 40 and 50 show that the mean is 7, variance 16 and μ 3 = -64. 2. (a) Does a zero value of Karl Pearsons co-efficient of correlation between two variable X&Y imply that X&Y are not related explain. (b) What is the co-efficient of determination. How is it useful in interpreting the value of an observed correlation. Given by.x = 1.4 & bxy = -0.5 calculate the coefficient of correlation & interpret the result. (c) The co-efficient of correlation between the two variable X&Y is 0.48. The covariance Is 36. The variance of X is 16. Find the Std. Dev of Y. OR 2. (a) The correlation co-efficient between eh two variables X & Y is found to be 0.4. What is the correlation between 2X & (-Y). (b) If the correlation co-efficient between the annual value of exports & the annual No. of children born during the last ten years is + 0.9, what inference if any, would you draw. (c) Family income and its percentage spent on food in the case of hundred families gave the following bivariate frequency distribution. Calculate the coefficient of correlation and interpret its value. Food Family income (Rs.) Expenditure (in %) 200-300 300-400 400-500 500-600 600-700 10-15 – – – 3 7 15–20 – 4 9 4 3 20–25 7 6 12 5 – 25–30 3 10 19 8 – Bliss Point Studies 9891555578 11 Ravinder N. Jha 9811343411 B.A. (Hons) Economics-Ist Year 3. (a) (b) (c) (d) (e) (f) 3. (a) (b) (c) Paper – III ‘Mutual independence and mutual exclusiveness are not synonymous’. Prove. It is given that PA B = 5/6, PA B = 1/3 and P(B) = ½ show that A & B are independence events. What do you understand by a random variable and its probability distribution? Define mathematical expectation of a random variable. Time taken by the crew of a company to construct a small bridge is a normal variate with mean 400 labor hours and standard deviation of 100 labor hours. (i) What is the probability that the bridge gets constructed between 300 to 450 labour hours? (ii) If the company promises to construct the bridge in 450 labour hours or less and agrees to pay a penalty of Rs. 100 for each labour hour spent in excess of 450. What is the probability that the company pays a penalty of at least Rs. 2000. If the probability of a defective bolt is 0.1, find (i) the mean and (ii) the standard deviation for the distribution of defective bolt in a total of 400. Two sets of candidates compete for positions of Board of Directors of a company. The probabilities for winning are 0.7 & 0.3 for the two sets. If the first set wins they will introduce a new product with a probability 0.4. Similarly value for the second set is 0.8. If the new product was introduced, what is the chance that the first set was as directors. Or A piece of electronic equipment has tow essential parts A&B. In the past, part A failed 30% of the time, part B failed 20% of the time and both parts failed simultaneously 5% of the times. Assuming that both part must operate to enable the equipment of function. What is the probability that the equipment will function. A state commission has been found to reduce response time of local fire departments. A group of experts is attempting to identify these city fire department whose response time is either in the lowest 10 percent or who take longer than 90 percent of all fire department in the study. Those in the first group are to serve as models for the less efficient fire units in the second group. Data show that the mean response time for a certain class of fire department is 12.8 minutes, with a standard deviation of 3.7 minutes. Obtain the variance of the binomial variate X show that V (cx)= C 2 V(x) where C is constant. Bliss Point Studies 9891555578 12 Ravinder N. Jha 9811343411 B.A. (Hons) Economics-Ist Year (d) (e) 4. (a) (b) (c) (d) 4. (a) (b) (c) (d) (e) Paper – III Assuming that each child has probability 0.4 of being a boy, find the probability distribution of number of boys in a family of 4 children. In a sample of 500 couples how many do vou expect to have (i) at least one boy (ii) exactly one girl. The median of a frequency distribution if 89.0 and the first quartile is 75.5. Find the Std. Dev. Assuming that the distribution is normal. Distinguish between mull and alternate hypotheses. State the null and alternate hypotheses regarding population mean that lead to (i) left tailed test and (ii) two tailed test. Explain clearly why hypotheses of Ho : μ 10 requires a right tailed and μ 10 requires a left tailed test. Explain the concept of confidence level and confidence coefficient. Give the critical value of z corresponding to 95% and 99% confidence level for one tailed test. Define simple random sampling with replace3ment (srswr). What do you know about the Sampling Distribution of the sample mean when sampling is srswr. What do you know about its standard error A Stenographer claims that she can take dictation at the rate of 120 words per minute. Can we reject her claim on the basis of 100 trials in which she demonstrates a mean of 116 words with a std.dev. Of 15 worlds? Use 5% level of significance & 1% level of significance. OR With the help of the diagrams, Explain Type 1 and Type 11 errors. A population consists of three numbers 2, 4, 6. Consider all possible sample of size two without replacement. Show that the variance of sampling distribution of means is less than the variance of the population. A random sample of 400 items is found to have a mean of 82 and S.D. of 18. Find 95% confidence limits for the mean of the population from which the sample is drawn. A man buys 50 electric bulbs of ‘Philips’ and 50 electric bulb of ‘HMT’. He finds that Philips bulbs give an average life of 1500 hrs. with S.D. of 60 hrs. and HMT bulbs give an average life of 1512 hrs. with a S.D. of 80 hrs. Is there a significant difference in the mean life of the two makes of bulb? Distinguish between (a) point estimation and interval estimation or a parameter and a statistic. Bliss Point Studies 9891555578 13 Ravinder N. Jha 9811343411 B.A. (Hons) Economics-Ist Year Paper – III DAULAT RAM COLLEGE HOUSE EXAMINATIONS (JAN’,2011) 1. (a) Employee dissatisfaction at ‘Bates Electronics’ is reflected in the number of official complaints over the past four months: 23, 41, 37 and 49. Based on this data, what is the average monthly increase in complaints? (b) The shareholders research centre of India has conducted a research study on price behavior of three leading industrial shares ‘P’, ‘Q’ and ‘R’ for the period 2000-2005, the results of which are published in its quarterly journal: Share Average Price S.D. ( ) (Rs.) Current (Rs.) Selling Price (Rs.) P 18.0 5.4 36.00 Q 22.5 4.5 34.75 R 24.0 6.0 39.00 Which share appears to be more stable in value. Which one would you like do dispose of at present and why? (Given that you are a holder of all three shares.) (c) The first four moments of a distribution about 3 are 1,3,6 and 8. Is the data consistent? Explain. OR (c) Of a total of ‘N’ numbers, the fraction ‘P’ are ones and the fraction q = 1-p are zeros. Find the moment coefficient of skewness and kurtosis. OR 1. (a) A large firm selling sports equipment is testing the effect of two advertising plans on sales over the last four months. Given the sales seen here, which advertising program seems to be producing the highest mean growth in monthly sales? Month Plan-1 Plan-2 January Rs. 1657.00 Rs. 4735.00 February Rs. 1998.00 Rs. 5012.00 March Rs. 2267.00 Rs. 5479.00 April Rs. 3432.00 Rs. 5589.00 (b) The following table gives the length of life of 400 radio tubes: Length of life (hour) No. of radio tubes 1000–1199 12 1200–1399 30 1400–1599 65 1600–1799 78 1800–1999 90 2000–2199 55 2200–2399 36 2400–2599 25 Bliss Point Studies 9891555578 14 Ravinder N. Jha 9811343411 B.A. (Hons) Economics-Ist Year 2 2. Paper – III 2600–2799 9 Calculate the percentage number of tubes where length of life of tube falls within X 2σ Compare with the theoretical result. (c) Given that the mean of a distribution is 5, variance is 9 and moment coefficient of skewness is – 1. Find the first three moments about the origin. Or (c) Given that the mean of the distribution is 50 and mode is 58 :(i) Calculate the median (ii) What can you day about the shape of the distribution? Explain. (a) Distinguish between independent and mutually exclusive events. When will the events ‘A’ and ‘B’ be both independent and mutually exclusive? (b) The employees of a firm are classified according to length of service with the firm (X) and salary grade (Y). There are three grades 1, 2 and 3, grade ‘1’ being the lowest and grade ‘3’ the highest. The results for the firm’s 200 employees are summarized by the following joint frequency table. Length of Solary Grade (Dollars Total Service Y (Years) X 1 2 3 1 20 0 0 20 2 40 10 0 50 3 24 24 12 60 4 16 26 8 50 5 0 0 20 20 From the data contained in this table: (i) Construct a bivariate probability table, showing the marginal distribution of ‘X’ and ‘Y’. (ii) Find the conditional probability distributions of ‘Y’ for the given values of ‘X’. Are ‘X’ and ‘Y’ statistically independent? (c) If the random variable ‘X’ is normally distributed with mean ‘ μ ’ x - μ is and variance σ 2 , Show that the mean of the variable Z σ always zero? Or (a) Each of three identical jewelry boxes has tow drawers. In each drawer of the first box, three is a gold watch. In each drawer of second box, there is a silver watch. In one drawer of the third box, there is a gold watch while in the other, there is a silver watch. If we select a box at random. Open one of the drawers and find it to contain a silver watch, what is the probability that the other drawer has the gold watch.? (b) The mean weight of 500 male students at a certain college is 151 1b and the standard deviation is 151 1b. Assuming that Bliss Point Studies 9891555578 15 Ravinder N. Jha 9811343411 B.A. (Hons) Economics-Ist Year 3. 3. Paper – III weights are normally distributed, find how many students weight (i) between 120 and 155 1b (ii) more than 185 1b. Or (b) What do you understand by a random variable? A random variable has the following probability distribution: Value of X: 0 1 2 3 Probability: 0.1 0.3 0.4 0.2 Find: (i) E (X) (ii) Var (X) (c) Cans of Happy – Tale dog Food average 16 ounces, with a range of 4.2 ounces. (i) What is the smallest can in ounces that one can buy for weiner a toy poodle? What is the largest can one can buy for wolfhound a killer? (ii) If one picks a can at random, what is the probability it will weigh between 15.8 and 16.5 ounces? (a) During a certain period the consumer price index increased from 110 to 200 and salary of a worker also increased from Rs. 3500 to Rs. 5000. What is the real if any to the worker? (b) If Laspeyre’s price index is equal to paasche’s index show that paasche’s index number will satisfy the time reversal test? Or (a) Show that in general, the laspeyre’s price index is greater than the passche’s index. When would this not be so? (b) The following data relate to the average weekly income of workers and the price index: Year Weekly Income (Rs) Price Index (Rs.) (1995=100) 1995 800 100 1996 819 105 1997 825 110 1998 876 120 1999 920 125 2000 924 135 Calculate the real income of the workers during the years 1995-2000. ***** Bliss Point Studies 9891555578 16 Ravinder N. Jha 9811343411