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
To find the Interquartile Range (IQR) for the given frequency distribution, we need to calculate the first quartile (Q1) and the third quartile (Q3). First, let's create a table with cumulative frequencies and class boundaries: | Time to travel to work | Frequency (f) | Cumulative Frequency (cf) | Class Boundaries | |------------------------|---------------|---------------------------|------------------| | 1 - 10 | 8 | 8 | 0.5 - 10.5 | | 11 - 20 | 14 | 8 + 14 = 22 | 10.5 - 20.5 | | 21 - 30 | 12 | 22 + 12 = 34 | 20.5 - 30.5 | | 31 - 40 | 9 | 34 + 9 = 43 | 30.5 - 40.5 | | 41 - 50 | 7 | 43 + 7 = 50 | 40.5 - 50.5 | The total number of data points (N) = 50. The class width (c) = 10.5 - 0.5 = 10. The formula for a quartile (Qk) for grouped data is: Qk = L + [((k × N)/4 - cf_b) / f] × c Where: L = lower boundary of the quartile class k = 1 for Q1, 3 for Q3 N = total frequency cf_b = cumulative frequency of the class before the quartile class f = frequency of the quartile class c = class width Step 1: Calculate Q1 (First Quartile) 1. Position of Q1: (1 × N)/4 = (1 × 50)/4 = 12.5th value. 2. Identify Q1 class: The 12.5th value falls in the 11 - 20 class, as its cumulative frequency (22) is the first to exceed 12.5. Lower boundary of Q1 class (L₁) = 10.5 Frequency of Q1 class (f₁) = 14 Cumulative frequency before Q1 class (cf_b₁) = 8 Class width (c) = 10 3. Calculate Q1: Q1 = 10.5 + [(12.5 - 8) / 14] × 10 Q1 = 10.5 + [4.5 / 14] × 10 Q1 = 10.5 + (45 / 14) Q1 = 10.5 + 3.21428... Q1 ≈ 13.71 Step 2: Calculate Q3 (Third Quartile) 1. Position of Q3: (3 × N)/4 = (3 × 50)/4 = 150/4 = 37.5th value. 2. Identify Q3 class: The 37.5th value falls in the 31 - 40 class, as its cumulative frequency (43) is the first to exceed 37.5. Lower boundary of Q3 class (L₃) = 30.5 Frequency of Q3 class (f₃) = 9 Cumulative frequency before Q3 class (cf_b₃) = 34 Class width (c) = 10 3. Calculate Q3: Q3 = 30.5 + [(37.5 - 34) / 9] × 10 Q3 = 30.5 + [3.5 / 9] × 10 Q3 = 30.5 + (35 / 9) Q3 = 30.5 + 3.8888... Q3 ≈ 34.39 Step 3: Calculate the Interquartile Range (IQR) IQR = Q3 - Q1 IQR = 34.3888... - 13.7142... IQR = 20.6746... Rounding to two decimal places: IQR = 20.67