1. **Problem Statement:** We have a frequency distribution for weights of 390 materials with frequencies $f_i = [6, 25, 48, 72, 116, 60, 38, 22, 3]$ and class marks $CM_1=112$, $CM_2=117$. We need to construct the full grouped frequency distribution, less than cumulative frequency distribution, find the weight below which 75% of materials lie, and draw histogram, frequency polygon, and less than ogive. 2. **Step 1: Determine class intervals.** Given $CM_1=112$ and $CM_2=117$, the class width $h = 117 - 112 = 5$. Class intervals are centered at class marks, so first class interval is $[112 - 2.5, 112 + 2.5) = [109.5, 114.5)$. Subsequent intervals increase by 5: $[114.5, 119.5)$, $[119.5, 124.5)$, etc. 3. **Step 2: Construct grouped frequency distribution table:** | Class Interval | Class Mark ($CM_i$) | Frequency ($f_i$) | |---------------|---------------------|------------------| | 109.5 - 114.5 | 112 | 6 | | 114.5 - 119.5 | 117 | 25 | | 119.5 - 124.5 | 122 | 48 | | 124.5 - 129.5 | 127 | 72 | | 129.5 - 134.5 | 132 | 116 | | 134.5 - 139.5 | 137 | 60 | | 139.5 - 144.5 | 142 | 38 | | 144.5 - 149.5 | 147 | 22 | | 149.5 - 154.5 | 152 | 3 | 4. **Step 3: Calculate less than cumulative frequency (CF):** Add frequencies cumulatively: $CF_1 = 6$ $CF_2 = 6 + 25 = 31$ $CF_3 = 31 + 48 = 79$ $CF_4 = 79 + 72 = 151$ $CF_5 = 151 + 116 = 267$ $CF_6 = 267 + 60 = 327$ $CF_7 = 327 + 38 = 365$ $CF_8 = 365 + 22 = 387$ $CF_9 = 387 + 3 = 390$ 5. **Step 4: Find weight below which 75% of materials lie.** 75% of 390 is $0.75 \times 390 = 292.5$. Locate class where CF just exceeds 292.5: $CF_5 = 267 < 292.5 < CF_6 = 327$. So, the 75th percentile lies in class interval $[134.5, 139.5)$. Use linear interpolation formula: $$L = 134.5, \quad CF_{prev} = 267, \quad f = 60, \quad h = 5$$ $$P = L + \left(\frac{292.5 - 267}{60}\right) \times 5 = 134.5 + \left(\frac{25.5}{60}\right) \times 5 = 134.5 + 2.125 = 136.625$$ So, 75% of materials weigh less than approximately $136.63$. 6. **Step 5: Drawing histogram, frequency polygon, and less than ogive:** - Histogram: Plot class intervals on x-axis and frequencies on y-axis as bars. - Frequency polygon: Plot points at class marks with frequencies and connect with straight lines. - Less than ogive: Plot class upper boundaries vs cumulative frequencies and connect points with a smooth curve. Since plotting is not requested explicitly, no graph data is included. Final answers: - Grouped frequency distribution and cumulative frequencies as above. - 75th percentile weight approximately $136.63$.