This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Step 1: Estimate the upper quartile from the cumulative frequency diagram. The total number of cars is 200. The upper quartile (Q3) corresponds to the 75^th percentile of the data. Upper quartile position = 0.75 × 200 = 150 From the cumulative frequency table (from the previous step): • At v 45, cumulative frequency is 112. • At v 50, cumulative frequency is 170. The 150^th car's speed lies between 45 km/h and 50 km/h. We estimate this value by interpolating between these points. The range of cumulative frequency is 170 - 112 = 58. The range of speed is 50 - 45 = 5. We need to find the speed corresponding to a cumulative frequency of 150. This is 150 - 112 = 38 units above the cumulative frequency at 45 km/h. Estimated upper quartile speed = 45 + ((38)/(58)) × (50 - 45) = 45 + ((38)/(58)) × 5 = 45 + (190)/(58) ≈ 45 + 3.276 ≈ 48.3 km/h Step 2: Estimate the number of cars with a speed greater than 35 km/h. First, find the cumulative frequency at 35 km/h from the diagram. From the cumulative frequency table: • At v 20, cumulative frequency is 16. • At v 40, cumulative frequency is 50. The speed 35 km/h lies between 20 km/h and 40 km/h. We estimate the cumulative frequency by interpolating. The range of speed is 40 - 20 = 20. The range of cumulative frequency is 50 - 16 = 34. We need to find the cumulative frequency at 35 km/h. This is 35 - 20 = 15 units above 20 km/h. Estimated cumulative frequency at 35 km/h = 16 + ((15)/(20)) × (50 - 16) = 16 + 0.75 × 34 = 16 + 25.5 = 41.5 This means approximately 41.5 cars have a speed less than or equal to 35 km/h. The number of cars with a speed greater than 35 km/h is the total number of cars minus the number of cars with speed less than or equal to 35 km/h. Number of cars with speed > 35 km/h = 200 - 41.5 = 158.5 Since the number of cars must be an integer, we round this to the nearest whole number. ≈ 159 cars Step 3: Find the probability that two randomly chosen cars both have a speed greater than 50 km/h. First, determine the number of cars with a speed greater than 50 km/h. From the cumulative frequency table, the cumulative frequency for v 50 is 170. So, the number of cars with speed > 50 km/h is 200 - 170 = 30. The probability that the first car chosen has a speed greater than 50 km/h is: P(1st car > 50) = Number of cars with speed > 50Total number of cars = (30)/(200) After choosing one such car, there are 29 cars left with a speed greater than 50 km/h, and 199 total cars remaining. The probability that the second car chosen also has a speed greater than 50 km/h (given the first did) is: P(2nd car > 50 1st car > 50) = (29)/(199) The probability that both cars have a speed greater than 50 km/h is the product of these probabilities: P(both > 50) = (30)/(200) × (29)/(199) P(both > 50) = (870)/(39800) P(both > 50) = (87)/(3980) P(both > 50) ≈ 0.021859... Rounding to 3 significant figures: P(both > 50) ≈ 0.0219 b) iii) a) The upper quartile is 48.3 km/h. (Estimated from interpolation) b) The number of cars with a speed greater than 35 km/h is 159 cars. (Estimated from interpolation) c) The probability that they both have a speed greater than 50 km/h is 0.0219. 3 done, 2 left today. You're making progress.