Gandhinagar Institute of Technology Question Bank PDF 2024-2025

Gandhinagar Institute of Technology DOC NO: 4026 Question Bank Program (UG): Bachelor of Technology Name of Department: Mathematics...

Gandhinagar Institute of Technology DOC NO: 4026 Question Bank Program (UG): Bachelor of Technology Name of Department: Mathematics Academic Year: 2024-25 Semester: 3 Subject Code:10000301 Subject Name: Probability & Statistics Name of the Institute: Gandhinagar Institute of Technology Department of Mathematics 10000301 PS  Gandhinagar Institute of Technology Basic Statistics 1. Find the arithmetic mean from the following frequency distribution: Ans. 8.83 𝑥 5 6 7 8 9 10 11 12 13 14 𝑓 25 45 90 165 112 96 81 26 18 12 2. The daily earnings (in rupees) of employees working on a daily basis in a firm are Daily earnings (in 100 120 140 160 180 200 220 rupees) Number of 3 6 10 15 24 42 75 Employees Calculate the mean of daily earnings. Ans. 194.51 3. Calculate the arithmetic mean of the following distribution: Ans. 42.5 Class 0-10 10-20 20-30 30-40 40-50 50-60 60-70 70-80 Interval Frequency 3 8 12 15 18 16 11 5 4. Calculate the average overtime work done per employee for the following distribution which gives the [pattern of overtime work done by 100 employees of a company. Overtime Hours 10-15 15-20 20-25 25-30 30-35 35-40 Number of 11 20 35 20 8 6 Employees Ans.23.1 hours 5. The following table gives the weekly expenditures of 100 workers. Find the median weekly expenditure. Ans.24.815 Weekly 0-10 10-20 20-30 30-40 40-50 Expenditure (in ₹) Number of 14 23 27 21 15 Workers 6. The following table gives the marks obtained by 50 students in Mathematics. Find the median. Ans. 29.5 Marks 10-14 15-19 20-24 25-29 30-34 35-39 40-44 45-49 No. of 4 6 10 5 7 3 9 6 Students 7. Find the mode for the following data: Ans. 256.25 Profit per 0-100 100-200 200-300 300-400 400-500 500-600 Shop No. of 12 18 27 20 17 6 Shops 8. Find the mode for the following distribution: Ans. 1.47 Class Intervals 0-10 10-20 20-30 30-40 40-50 Frequency 45 20 14 7 3 9. Find the standard deviation for the following distribution: Ans.14.285 Marks 10-20 20-30 30-40 40-50 50-60 60-70 70-80 Number of 5 12 15 20 10 4 2 Students Department of Mathematics 10000301 PS  Gandhinagar Institute of Technology 10. Find the standard deviation for the following data: Ans. 11.285 Age (in 10-19 20-29 30-39 40-49 50-59 60-69 70-79 80-89 Years) Number of 1 0 1 10 17 38 9 3 Cases 11. The runs scored by two batsmen A and B in 9 consecutive matches are given below: A 85 20 62 28 74 5 69 4 13 B 72 4 15 30 59 15 49 27 26 Which of the batsmen is more consistent? Ans. Batsman B CV=64.18% 12. The number of matches played and goals scored by two teams A and B in world cup Football 2002 were as follows: Matches played by Team 27 9 8 5 4 A Matches played by Team 17 9 6 5 3 B No. of Goals scored in a 0 1 2 3 4 match Find which team may be considered more consistent. Ans. Team B CV=109.17% 13. Calculate Karl Pearson’s coefficient of skewness for the following data: Ans. Skewness = 0.08 𝑥 0 1 2 3 4 5 6 7 𝑦 12 17 29 19 8 4 1 0 Curve Fitting 1. A simply supported beam carries a concentrated load P(lb) at its mid-point. Corresponding to various values of P, the maximum deflection Y(in) is measured. The data are given below P 100 120 140 160 180 200 Y 0.45 0.55 0.60 0.70 0.80 0.85 Find a law of the form Y = a + bP using least square method. Ans. 𝒀 = 𝟎. 𝟎𝟒𝟕𝟔 + 𝟎. 𝟎𝟎𝟒𝟏𝑷 2. Fit the exponential curve 𝑦 = 𝑒 to the following data. x 0 2 4 6 8 y 150 63 28 12 5.6 Ans. y= 𝒆 𝟎.𝟒𝟏𝟏𝟕𝒙 3. Fit a second-degree parabola 𝑦 = 𝑎𝑥 + 𝑏𝑥 + 𝑐 in least square sense for the following data: x 1 2 3 4 5 y 10 12 13 16 19 4. By using Least Square Method for second degree polynomial using the following data. x -3 -2 -1 0 1 2 3 y 12 4 1 2 7 15 30 5. If P is the pull required to lift a load W by means of a pulley block. Find a linear approximation of the form P=mW+C connecting P and W using the following data. P 13 18 23 27 W 51 75 102 119 Department of Mathematics 10000301 PS  Gandhinagar Institute of Technology Where, P and W are taken in kg.wt. 6. Fit a straight line to the following data regarding x as the independent variable. x 1 2 3 4 5 6 y 1200 900 600 200 110 50 Ans. 𝒚 = 𝟏𝟑𝟔𝟏. 𝟗𝟕 − 𝟐𝟒𝟑. 𝟒𝟐𝒙 7. Fit as second-degree parabola to the following data. x 1 1.5 2 2.5 3 3.5 4 y 1.1 1.3 1.6 2.0 2.7 3.4 4.1 𝟐 Ans. 𝒚 = 𝟏. 𝟎𝟒 − 𝟎. 𝟏𝟗𝟖𝒙 + 𝟎. 𝟐𝟒𝟒𝒙 8. Fit a second-degree parabola to the following data x 0 1 2 3 4 y 1 1.8 1.3 2.5 6.3 Ans. 𝒚 = 𝟏. 𝟒𝟐 − 𝟏. 𝟎𝟕𝒙 + 𝟎. 𝟓𝟓𝒙𝟐 9. Find the curve of best fit of the type 𝑦 = 𝑎𝑒 to the following data by the method of least squares. x 1 5 7 9 12 y 10 15 12 15 21 𝟎.𝟎𝟓𝟗𝒙 Ans. 𝒚 = 𝟗. 𝟒𝟕𝟓𝟒𝒆 10. Using the method of least square fit the non-linear curve of the form 𝑦 = 𝑎𝑒 to the following data. x 0 2 4 y 5.012 10 31.62 Ans. 𝒚 = 𝟒. 𝟔𝟒𝟏𝟔𝒆𝟎.𝟒𝟔𝟎𝟓𝒙 11. Fit a curve 𝑦 = 𝑎𝑥 for the following data. x 61 26 7 2.6 y 350 400 500 600 Ans. 𝒚 = 𝟕𝟎𝟐. 𝟎𝟔𝟒𝟒𝒙 𝟎.𝟏𝟕𝟎𝟗 12. Fit a curve 𝑦 = 𝑎𝑏 for the following data. x 2 3 4 5 6 8 y 8.3 15.4 33.1 65.2 126.4 146 Ans. 𝒚 = 𝟑. 𝟖𝟏𝟖𝟏(𝟏. 𝟔𝟔𝟕𝟔)𝒙 Correlation and Regression 13. Ten competitors in a music competition are ranked by three judges in the following order 1st judge 1 6 5 10 3 2 4 9 7 8 2nd judge 3 5 8 4 7 10 2 1 6 9 3rd judge 6 4 9 8 1 2 3 10 5 7 Use the rank correlation to discuss which pair of judges has the nearest approach to common tastes in music. Ans. Pair of Judges 1 & 3 has the nearest common approach. 14. The following data give the experience of machine operators and their performance rating as given by the number of good parts turned out per 100 piece. Department of Mathematics 10000301 PS  Gandhinagar Institute of Technology Operators 1 2 3 4 5 6 Performance rating (x) 23 43 53 63 73 83 Experience(y) 5 6 7 8 9 10 Calculate the regression line of performance rating on experience and also estimate the performance if an operator has 11years experience. Ans. 𝒙 = 𝟏𝟏. 𝟒𝟐𝟗𝒚 − 𝟐𝟗. 𝟑𝟖𝟕𝟓. At 𝒚 = 𝟏𝟏, 𝒙 = 𝟗𝟔. 𝟑𝟑𝟏𝟓 15. Calculate the coefficient of correlation and obtain the lines of regression for the following: X 1 2 3 4 5 6 7 8 9 Y 9 8 10 12 11 13 14 16 15 16. A study of the amount of rainfall and the quantity of air pollution removed produced Daily rainfall x 4.3 4.5 4.9 5.6 6.1 5.2 3.8 2.1 7.5 (0.01cm) Particulate 126 121 116 118 114 118 132 141 108 removed, y(𝜇𝑔/𝑚 ) the following data: (i) Find the equation of the regression line to predict the particulate removed from the amount of daily rainfall. (ii) Find the amount of particulate removed when daily rainfall is x = 4.8 units. 17. Obtain the line of regression of monthly sales (Y) on advertisement expenditure (X) and estimate the monthly sales when the company will spend Rs.50,000 on advertisement, if the data on Y and X are as follows: Y(in Lac) 74 76 60 68 79 70 71 94 X(In Thousand) 43 44 36 38 47 40 41 54 18. Determine the coefficient of correlation if 𝑥̅ = 5.5, 𝑦 = 4, 𝑥 = 385, 𝑦 = 192, (𝑥 + 𝑦) = 947. 19. Find the coefficient of correlation between the weight of father and the son from the following data. Weight of father 55 56 57 58 59 60 61 Weight of son 57 56 59 62 60 60 59 Ans.0.579 20. The following table gives the distribution of the total students in first year as they obtained marks in Calculus and the number of students passed in Physics. Find out if there is any correlation between Calculus and Physics. Calculus marks 0-10 10-20 20-30 30-40 40-50 50-60 60-70 No. of students 75 20 25 100 30 35 15 No. of students 40 15 20 80 10 25 8 passed in Physics Ans. −𝟎. 𝟐𝟔𝟑 21. It is known that covariance between girls and boys is 20 and the two-standard deviation are 5 and 8. Determine the coefficient of correlation. Ans. 0.5 Department of Mathematics 10000301 PS  Gandhinagar Institute of Technology 22. Find Karl Pearson’s coefficient of correlation, given that Σ𝑥 = 140, Σ𝑦 = 150, Σ(𝑥 − 10) = 180, Σ(𝑦 − 15) = 215, Σ(𝑥 − 10)(𝑦 − 15) = 60, 𝑛 = 10. Ans. 0.91 23. Determine the two regression coefficients. x =2,5,6,3,4 and y = 9,6,8,5,4. Ans. −𝟎. 𝟎𝟓𝟖, −𝟎. 𝟏 24. From the following data obtain the two regression lines and the correlation coefficient. Sales 100 98 78 85 110 93 80 Purchase 85 90 70 72 95 81 74 Ans. (𝐢) 𝒚 = 𝟎. 𝟖𝟓𝟖𝒙 + 𝟐. 𝟎𝟔𝟒 (𝒊𝒊) 𝒙 = 𝟏. 𝟎𝟖𝟓𝒚 + 𝟒. 𝟏𝟏𝟓 25. The following data represents the rank of 10 students in two subjects Calculus and Physics. Find the rank correlation. Calculus 4 5 7 8 10 1 3 6 2 9 Physics 3 4 7 9 10 8 6 5 2 1 Ans.0.24 26. Find the rank coefficient of correlation. 𝑥 12 10 17 14 13 18 20 𝑦 110 210 108 135 160 104 70 Ans. −𝟎. 𝟖𝟗𝟑 27. Obtain the rank correlation coefficient for the data. 𝑥 20 25 33 17 25 17 70 60 17 𝑦 35 30 45 30 20 10 30 50 40 Ans. 0.333 Basic Probability 28. A company has two plants to manufacture hydraulic machine. Plat I manufactures 70% of the hydraulic machines and plant II manufactures 30%. At plant I, 80% of hydraulic machines are rated standard quality and at plant II, 90% of hydraulic machines are rated standard quality. A machine is picked up at random and is found to be of standard quality. What is the chance that it has come from plant I? Ans. 0.6747 29. In a bolt factory, three machines A, B and C manufacture 25%, 35% and 40% of the total product respectively. Of these outputs 5%, 4% and 2% respectively, are defective bolts. A bolt is picked up at random and found to be defective. What are the probabilities that it was manufactured by machines A, B and C? Ans.(1) 0.3623 (2) 0.4058 (3) 0.2319. 30. In a certain assembly plant, three machines, B1, B2, and B3, make 30%, 45%, and 25%, respectively, of the products. It is known from past experience that 2%, 3%, and 2% of the products made by each machine, respectively, are defective. Now, suppose that a finished product is randomly selected. What is the probability that it is defective? 31. Four cards are drawn from a pack of cards. Find the probability that (i) all are diamonds (ii) there is one card of each suit (iii) there are two spades and two hearts. 32. A microchip company has two machines that produce the chips. Machine I produces 65% of the chips, but 5% of its chips are defective. Machine II produces 35% of the chips and 15% of its chips are defective. A chip is selected at random and found to be defective. What is the probability that it came from Machine I? 33. If 𝐴 and 𝐵 are two events with 𝑃(𝐴) = , 𝑃(𝐵) = , 𝑃(𝐴 ∩ 𝐵) = , find (𝑖)𝑃(𝐴/𝐵) (𝑖𝑖)𝑃(𝐵/𝐴)(𝑖𝑖𝑖)𝑃(𝐵/𝐴̅)(𝑖𝑣)𝑃(𝐴 ∩ 𝐵 ). Ans.(i) 1/3 (ii) ¼ (iii) ¼ (iv) ¼ Department of Mathematics 10000301 PS  Gandhinagar Institute of Technology 34. An equipment consists of two parts A and B. In the process of manufacturing of part A, 9 out of 100 are likely to be defective and that of B 5 out of 100 are likely to be defective. Find the probability that the assembled article will not be defective. Ans.0.8645 35. Three bags contain 10%, 20% and 30% defective items. An item is selected at random which is defective. Determine the probability that it come from 3rd bag. Ans 0.5 36. It came to know that the marks given by a certain examiner is correct in 90% of the cases. Suppose that 40% of the answer books are given to the examiner which can be given full marks in actual. What is the probability of the actual answer book given to the examiner have been corrected actually with fall marks? Ans. 0.857 Random Variables 37. Find the value of k from the following data. 𝑥 0 10 15 𝑃(𝑋 = 𝑥) 𝑘 − 6 2 14 5 𝑘 5𝑘 Also, find the distribution function and expectation of 𝑥. 38. For the following distribution 𝑋 -3 -2 -1 0 1 2 𝑃(𝑋 = 𝑥) 0.01 0.1 0.2 0.3 0.2 0.15 Find (𝑎)𝑃(𝑋 ≥ 1) (𝑏)𝑃(𝑋 < 0) (𝑐)𝐸(𝑋) (𝑑)𝑉𝑎𝑟(𝑋) 39. A random variable 𝑋 has the following probability function: 𝑋 0 1 2 3 4 5 6 7 8 𝑃(𝑋 = 𝑥) 𝑘 𝑘 𝑘 𝑘 2𝑘 6𝑘 7𝑘 8𝑘 4𝑘 45 15 9 5 45 45 45 45 45 Determine (𝑎)𝑘 (𝑏)𝑀𝑒𝑎𝑛 (𝑐)𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒 (𝑑)𝑆𝐷 40. A player tosses two fair coins. He wins Rs.1 or Rs. 2 as 1 tail or 1 head appears. On the other hand, he loses 5 if no head appears. Find the expected gain or loss of the player. 41. A man runs an ice cream parlour at a holiday resort. If the summer is mild, he can sell 2500 cups of ice cream; if it is hot, he can sell 4000 cups: if it is very hot, he can sell 5000 cups. It is known that for any year, the probability of summer to be mild is 1/7 and to be hot is 4/7. A cup of ice cream costs Rs. 2 and is sold for Rs. 3.50. What is his expected profit? 42. Calculate the first four moments about the mean and also the value of 𝛽 from the following table. 𝑥 0 1 2 3 4 5 6 7 8 𝑓 1 8 28 156 170 56 28 8 1 43. The first four moments about the working mean 28.5 of a distribution are 0.294, 7.144, 14.409 and 454.98. Calculate the moments about the mean. Also , evaluate 𝛽 and 𝛽. 44. Karl Pearson’s coefficient of skewness of a distribution is 0.32. Its standard deviation is 6.5 and the mean is 29.6. Find the mode and median for the distribution. 45. If the probability density function of a random variable is given by 𝑘𝑥 0≤𝑥≤2 𝑓(𝑥) = 2𝑘 2≤𝑥≤4 6𝑘 − 𝑘𝑥 4 ≤ 𝑥 ≤ 6 Find (𝑎)𝑘 (𝑏)𝑃(1 ≤ 𝑥 ≤ 3) (𝑐)𝑋 46. A continuous random variable has the probability density function Department of Mathematics 10000301 PS  Gandhinagar Institute of Technology 𝑓(𝑥) = 2𝑒 𝑥>0 0 𝑥