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 construct the grouped frequency distribution, frequency polygon, and ogive, we first need to organize the given data. The raw data points are: 3, 104, 43, 66, 200, 81, 81, 13, 14, 16, 12, 125, 44, 180, 205, 13, 33, 101, 106, 36, 35, 36, 35, 198, 106, 6, 28, 76, 74, 81, 40, 39, 46, 126, 198, 71, 41, 91, 100, 63, 83, 41, 78, 140, 254, 36, 43, 45, 56, 71. There are N = 50 data points. Step 1: Determine the range, number of classes, and class width. • Minimum value = 3 • Maximum value = 254 • Range = Maximum value - Minimum value = 254 - 3 = 251 Using Sturges' formula for the number of classes (k): k = 1 + 3.322 _10 N k = 1 + 3.322 _10 50 k = 1 + 3.322 × 1.69897 k ≈ 1 + 5.64 = 6.64 We choose k=7 classes. Class width (w) = Rangek = (251)/(7) ≈ 35.85. We round up to a convenient number, w=40. Step 2: Construct the Grouped Frequency Distribution Table. We define the class intervals starting from a value slightly below the minimum (or the minimum itself) and use a class width of 40. | Class Interval | Class Boundaries | Midpoint (x) | Frequency (f) | Cumulative Frequency (cf) | |:---------------|:-----------------|:---------------|:----------------|:----------------------------| | 0-39 | -0.5 - 39.5 | 19.5 | 15 | 15 | | 40-79 | 39.5 - 79.5 | 59.5 | 16 | 31 | | 80-119 | 79.5 - 119.5 | 99.5 | 10 | 41 | | 120-159 | 119.5 - 159.5 | 139.5 | 3 | 44 | | 160-199 | 159.5 - 199.5 | 179.5 | 3 | 47 | | 200-239 | 199.5 - 239.5 | 219.5 | 2 | 49 | | 240-279 | 239.5 - 279.5 | 259.5 | 1 | 50 | | Total | | | 50 | | Step 3: Construct the Frequency Polygon. A frequency polygon is constructed by plotting the midpoint of each class against its frequency and connecting the points with straight lines. To close the polygon, add a class with zero frequency at each end. The points to plot are: • (-20.5, 0) (midpoint of a hypothetical class before 0-39) • (19.5, 15) • (59.5, 16) • (99.5, 10) • (139.5, 3) • (179.5, 3) • (219.5, 2) • (259.5, 1) • (299.5, 0) (midpoint of a hypothetical class after 240-279) Step 4: Construct the Ogive (Cumulative Frequency Polygon). An ogive is constructed by plotting the upper class boundary of each class against its cumulative frequency and connecting the points with straight lines. The points to plot are: • (-0.5, 0) (lower class boundary of the first class with cumulative frequency 0) • (39.5, 15) • (79.5, 31) • (119.5, 41) • (159.5, 44) • (199.5, 47) • (239.5, 49) • (279.5, 50) That's 2 down. 3 left today — send the next one.