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.
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Another one aaronpro — let's solve it. Question One a) Calculate the mean time Step 1: Find the midpoint (x) for each class interval and the total frequency ( f). The class intervals are 8-12, 13-17, 18-22, 23-27, 28-32, 33-37. The frequencies (f) are 5, 3, 6, 7, 5, 4. Midpoints (x): For 8-12: x = (8+12)/(2) = 10 For 13-17: x = (13+17)/(2) = 15 For 18-22: x = (18+22)/(2) = 20 For 23-27: x = (23+27)/(2) = 25 For 28-32: x = (28+32)/(2) = 30 For 33-37: x = (33+37)/(2) = 35 Total frequency: f = 5 + 3 + 6 + 7 + 5 + 4 = 30 Step 2: Calculate f · x for each interval and find the sum ( f · x). 5 × 10 = 50 3 × 15 = 45 6 × 20 = 120 7 × 25 = 175 5 × 30 = 150 4 × 35 = 140 Sum of f · x: f · x = 50 + 45 + 120 + 175 + 150 + 140 = 680 Step 3: Calculate the mean (x). The formula for the mean of grouped data is x = ( f · x)/( f). x = (680)/(30) = (68)/(3) ≈ 22.666... Rounding to two decimal places: x ≈ 22.67 The mean time is 22.67 minutes. b) Calculate the variance and Standard deviation. Step 1: Calculate x^2 for each midpoint and f · x^2 for each interval. For x=10: x^2 = 10^2 = 100 f · x^2 = 5 × 100 = 500 For x=15: x^2 = 15^2 = 225 f · x^2 = 3 × 225 = 675 For x=20: x^2 = 20^2 = 400 f · x^2 = 6 × 400 = 2400 For x=25: x^2 = 25^2 = 625 f · x^2 = 7 × 625 = 4375 For x=30: x^2 = 30^2 = 900 f · x^2 = 5 × 900 = 4500 For x=35: x^2 = 35^2 = 1225 f · x^2 = 4 × 1225 = 4900 Step 2: Find the sum of f · x^2. f · x^2 = 500 + 675 + 2400 + 4375 + 4500 + 4900 = 17350 Step 3: Calculate the variance. The formula for variance of grouped data is Variance = ( f · x^2)/( f) - (( f · x)/( f))^2. We know f = 30, f · x = 680, and f · x^2 = 17350. Variance = (17350)/(30) - ((680)/(30))^2 Variance = 578.333... - (22.666...)^2 Variance = 578.333... - 513.777... Variance = 64.555... Rounding to two decimal places: Variance ≈ 64.56 The variance is 64.56 minutes^2. Step 4: Calculate the standard deviation. The standard deviation is the square root of the variance. Standard Deviation = sqrt(Variance) = sqrt(64.555...) Standard Deviation ≈ 8.0346... Rounding to two decimal places: Standard Deviation ≈ 8.03 The standard deviation is 8.03 minutes. Question Two Step 1: Identify the given distance and the new time. Distance (D) = 240 km New Time (T) = 1(1)/(2) hours = 1.5 hours Step 2: Calculate the required speed. The formula for speed is Speed = DistanceTime. Speed = 240 km1.5 hours Speed = 160 km/h The speed the aeroplane must use is 160 km/h. Send me the next one 📸