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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Haya Terry, twendelee na swali hili la Hisabati. Jedwali linaonyesha alama zilizopatikana na wanafunzi katika mtihani wa Hisabati. a) Chora curve ya masafa jumla (cumulative frequency curve) na utumie grafu yako kupata: Step 1: Unda jedwali la masafa jumla (cumulative frequency table). Kwanza, tunahitaji mipaka ya juu ya darasa (upper class boundaries) na masafa jumla (cumulative frequency). | Class | Upper Class Boundary | Frequency (f) | Cumulative Frequency (cf) | | :------ | :------------------- | :------------ | :------------------------ | | 20 - 29 | 29.5 | 0 | 0 | | 30 - 39 | 39.5 | 3 | 3 | | 40 - 49 | 49.5 | 17 | 20 | | 50 - 59 | 59.5 | 27 | 47 | | 60 - 69 | 69.5 | 23 | 70 | | 70 - 79 | 79.5 | 8 | 78 | | 80 - 89 | 89.5 | 2 | 80 | Jumla ya wanafunzi (N) ni 80. Step 2: Chora curve ya masafa jumla (ogive). Kwenye x-axis, chora alama (marks) kwa kutumia mipaka ya juu ya darasa (upper class boundaries). Kwenye y-axis, chora masafa jumla (cumulative frequency). Panga pointi zifuatazo: (29.5, 0), (39.5, 3), (49.5, 20), (59.5, 47), (69.5, 70), (79.5, 78), (89.5, 80). Unganisha pointi hizi kwa kutumia mstari laini (smooth curve). (Kumbuka: Kwa kuwa siwezi kuchora grafu moja kwa moja hapa, maelezo hapo juu yanaeleza jinsi mchoro unavyopaswa kuonekana. Majibu yafuatayo yanatokana na usomaji wa kawaida wa grafu kama hiyo.) i) Median Step 1: Tafuta nafasi ya median. Nafasi ya median ni (N)/(2) = (80)/(2) = 40. Step 2: Soma median kutoka kwenye grafu. Kutoka kwenye y-axis, tafuta 40. Chora mstari mlalo kuelekea curve, kisha chora mstari wima kushuka kwenye x-axis. Thamani ya x-axis itakuwa median. Kutokana na grafu iliyochorwa vizuri, median inapaswa kuwa karibu na 56.9 marks. ii) Quartile deviation Step 1: Tafuta nafasi za quartiles za kwanza na ya tatu. Nafasi ya Quartile ya Kwanza (Q_1) ni (N)/(4) = (80)/(4) = 20. Nafasi ya Quartile ya Tatu (Q_3) ni (3N)/(4) = (3 × 80)/(4) = 60. Step 2: Soma Q_1 na Q_3 kutoka kwenye grafu. Kwa Q_1: Kutoka kwenye y-axis, tafuta 20. Chora mstari mlalo kuelekea curve, kisha chora mstari wima kushuka kwenye x-axis. Q_1 inapaswa kuwa karibu na 49.5 marks. Kwa Q_3: Kutoka kwenye y-axis, tafuta 60. Chora mstari mlalo kuelekea curve, kisha chora mstari wima kushuka kwenye x-axis. Q_3 inapaswa kuwa karibu na 65.2 marks. Step 3: Kokotoa quartile deviation. Quartile Deviation (QD) = (Q_3 - Q_1)/(2) QD = (65.2 - 49.5)/(2) QD = (15.7)/(2) QD = 7.85 Quartile deviation ni 7.85. iii) Idadi ya wanafunzi waliofunga zaidi ya 67. Step 1: Tafuta masafa jumla kwa alama 67. Kutoka kwenye x-axis, tafuta 67. Chora mstari wima kuelekea curve, kisha chora mstari mlalo kuelekea y-axis. Masafa jumla kwa alama 67 (cf_67) inapaswa kuwa karibu na 64 wanafunzi. Step 2: Kokotoa idadi ya wanafunzi waliofunga zaidi ya 67. Idadi ya wanafunzi waliofunga zaidi ya 67 = Jumla ya wanafunzi - cf_67 Idadi = 80 - 64 Idadi = 16 wanafunzi Idadi ya wanafunzi waliofunga zaidi ya 67 ni 16 wanafunzi. Last free one today — make it count tomorrow, or type /upgrade for unlimited.