The height of 100 maize plants were measured to the nearest centimeter and the results recorded in the table below. (a) complete the table. (b) Calculate 2 d.p. (i) The mean (ii) The standard deviation. (c) Using the above plot an ogive and use it to find the 30th percentile.

Question

Asked on July 21, 2026Mathematics

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.

Accepted Answer · 2026-07-21T19:07:36.448Z

Huu hapa ni utatuzi wa swali lako: a) Kamilisha jedwali Kwanza, tunabainisha kuwa thamani ya d inatokana na fomula d = (x - A)/(c), ambapo A ni wastani uliochukuliwa (assumed mean) na c ni upana wa darasa (class width). Kutoka jedwali, d=0 wakati x=42, hivyo A=42. Upana wa darasa c = 30-25 = 5. Sasa, hebu tukamilishe safu za fd na fd^2: | Height | Frequency (f) | x | d | d² | fd | fd² | CF | |---------|---------------|-----|-----|-----|------------|------------|-----| | 25-29 | 5 | 27 | -3 | 9 | 5 × -3 = -15 | 5 × 9 = 45 | 5 | | 30-34 | 12 | 32 | -2 | 4 | 12 × -2 = -24 | 12 × 4 = 48 | 17 | | 35-39 | 18 | 37 | -1 | 1 | 18 × -1 = -18 | 18 × 1 = 18 | 35 | | 40-44 | 30 | 42 | 0 | 0 | 30 × 0 = 0 | 30 × 0 = 0 | 65 | | 45-49 | 17 | 47 | 1 | 1 | 17 × 1 = 17 | 17 × 1 = 17 | 82 | | 50-54 | 11 | 52 | 2 | 4 | 11 × 2 = 22 | 11 × 4 = 44 | 93 | | 55-59 | 7 | 57 | 3 | 9 | 7 × 3 = 21 | 7 × 9 = 63 | 100 | | Jumla | f = 100 | | | | fd = 3 | fd^2 = 235 | | b) Kokotoa: i) Wastani (Mean) Step 1: Tumia fomula ya wastani kwa kutumia wastani uliochukuliwa. Fomula ni x = A + ( fd)/( f) × c. Kutoka jedwali, A = 42, fd = 3, f = 100, na c = 5. Step 2: Badilisha thamani kwenye fomula na kokotoa. x = 42 + (3)/(100) × 5 x = 42 + (15)/(100) x = 42 + 0.15 x = 42.15 ii) Mkengeuko Wastani (Standard Deviation) Step 1: Tumia fomula ya mkengeuko wastani kwa kutumia wastani uliochukuliwa. Fomula ni s = c sqrt(( fd^2)/( f) - (( fd)/( f))^2). Kutoka jedwali, c = 5, fd^2 = 235, fd = 3, na f = 100. Step 2: Badilisha thamani kwenye fomula. s = 5 sqrt((235)/(100) - ((3)/(100))^2) Step 3: Kokotoa ndani ya mizizi. s = 5 sqrt(2.35 - (0.03)^2) s = 5 sqrt(2.35 - 0.0009) s = 5 sqrt(2.3491) Step 4: Kokotoa mizizi ya mraba na zidisha. s ≈ 5 × 1.532677 s ≈ 7.663385 Step 5: Zungusha jibu kwa sehemu mbili za desimali. s ≈ 7.66 c) Chora ogive na utumie kupata asilimia ya 30 (30th percentile) Ili kuchora ogive (cumulative frequency curve), utahitaji kupanga masafa ya juu ya darasa (upper class boundaries) dhidi ya masafa jumla (cumulative frequencies). 1. Masafa ya Juu ya Darasa na Masafa Jumla: 25-29: 29.5, CF = 5 30-34: 34.5, CF = 17 35-39: 39.5, CF = 35 40-44: 44.5, CF = 65 45-49: 49.5, CF = 82 50-54: 54.5, CF = 93 55-59: 59.5, CF = 100 Anza ogive kwa (24.5, 0). 2. Kuchora Ogive: Chora grafu ambapo mhimili wa x unawakilisha urefu (Height) na mhimili wa y unawakilisha masafa jumla (Cumulative Frequency). Panga pointi zilizoorodheshwa hapo juu na uziunganishe kwa mstari laini. 3. Kupata Asilimia ya 30 (30th Percentile): Jumla ya masafa ni N = 100. Asilimia ya 30 inalingana na nafasi ya 0.30 × N = 0.30 × 100 = 30. Kwenye ogive, tafuta 30 kwenye mhimili wa y (Cumulative Frequency). Chora mstari mlalo kutoka CF=30 hadi uingiliane na ogive. Kutoka sehemu ya makutano, chora mstari wima chini hadi mhimili wa x (Height). Thamani kwenye mhimili wa x ambapo mstari wima unakatiza ndiyo asilimia ya 30. Kwa kutumia fomula ya kukokotoa asilimia (interpolation) kwa darasa la 35-39 (ambapo CF inatoka 17 hadi 35): P_30 = L + ((30)/(100)N - CF_bf_p) × c Ambapo L = 34.5 (mpaka wa chini wa darasa), N = 100, CF_b = 17 (CF kabla ya darasa), f_p = 18 (masafa ya darasa), c = 5 (upana wa darasa). P_30 = 34.5 + ((30 - 17)/(18)) × 5 P_30 = 34.5 + ((13)/(18)) × 5 P_30 = 34.5 + (65)/(18) P_30 = 34.5 + 3.6111... P_30 ≈ 38.11 3 done, 2 left today. You're making progress.

Accepted Answer · 2026-07-21T19:07:36.448Z

Huu hapa ni utatuzi wa swali lako: a) Kamilisha jedwali Kwanza, tunabainisha kuwa thamani ya d inatokana na fomula d = (x - A)/(c), ambapo A ni wastani uliochukuliwa (assumed mean) na c ni upana wa darasa (class width). Kutoka jedwali, d=0 wakati x=42, hivyo A=42. Upana wa darasa c = 30-25 = 5. Sasa, hebu tukamilishe safu za fd na fd^2: | Height | Frequency (f) | x | d | d² | fd | fd² | CF | |---------|---------------|-----|-----|-----|------------|------------|-----| | 25-29 | 5 | 27 | -3 | 9 | 5 × -3 = -15 | 5 × 9 = 45 | 5 | | 30-34 | 12 | 32 | -2 | 4 | 12 × -2 = -24 | 12 × 4 = 48 | 17 | | 35-39 | 18 | 37 | -1 | 1 | 18 × -1 = -18 | 18 × 1 = 18 | 35 | | 40-44 | 30 | 42 | 0 | 0 | 30 × 0 = 0 | 30 × 0 = 0 | 65 | | 45-49 | 17 | 47 | 1 | 1 | 17 × 1 = 17 | 17 × 1 = 17 | 82 | | 50-54 | 11 | 52 | 2 | 4 | 11 × 2 = 22 | 11 × 4 = 44 | 93 | | 55-59 | 7 | 57 | 3 | 9 | 7 × 3 = 21 | 7 × 9 = 63 | 100 | | Jumla | f = 100 | | | | fd = 3 | fd^2 = 235 | | b) Kokotoa: i) Wastani (Mean) Step 1: Tumia fomula ya wastani kwa kutumia wastani uliochukuliwa. Fomula ni x = A + ( fd)/( f) × c. Kutoka jedwali, A = 42, fd = 3, f = 100, na c = 5. Step 2: Badilisha thamani kwenye fomula na kokotoa. x = 42 + (3)/(100) × 5 x = 42 + (15)/(100) x = 42 + 0.15 x = 42.15 ii) Mkengeuko Wastani (Standard Deviation) Step 1: Tumia fomula ya mkengeuko wastani kwa kutumia wastani uliochukuliwa. Fomula ni s = c sqrt(( fd^2)/( f) - (( fd)/( f))^2). Kutoka jedwali, c = 5, fd^2 = 235, fd = 3, na f = 100. Step 2: Badilisha thamani kwenye fomula. s = 5 sqrt((235)/(100) - ((3)/(100))^2) Step 3: Kokotoa ndani ya mizizi. s = 5 sqrt(2.35 - (0.03)^2) s = 5 sqrt(2.35 - 0.0009) s = 5 sqrt(2.3491) Step 4: Kokotoa mizizi ya mraba na zidisha. s ≈ 5 × 1.532677 s ≈ 7.663385 Step 5: Zungusha jibu kwa sehemu mbili za desimali. s ≈ 7.66 c) Chora ogive na utumie kupata asilimia ya 30 (30th percentile) Ili kuchora ogive (cumulative frequency curve), utahitaji kupanga masafa ya juu ya darasa (upper class boundaries) dhidi ya masafa jumla (cumulative frequencies). 1. Masafa ya Juu ya Darasa na Masafa Jumla: 25-29: 29.5, CF = 5 30-34: 34.5, CF = 17 35-39: 39.5, CF = 35 40-44: 44.5, CF = 65 45-49: 49.5, CF = 82 50-54: 54.5, CF = 93 55-59: 59.5, CF = 100 Anza ogive kwa (24.5, 0). 2. Kuchora Ogive: Chora grafu ambapo mhimili wa x unawakilisha urefu (Height) na mhimili wa y unawakilisha masafa jumla (Cumulative Frequency). Panga pointi zilizoorodheshwa hapo juu na uziunganishe kwa mstari laini. 3. Kupata Asilimia ya 30 (30th Percentile): Jumla ya masafa ni N = 100. Asilimia ya 30 inalingana na nafasi ya 0.30 × N = 0.30 × 100 = 30. Kwenye ogive, tafuta 30 kwenye mhimili wa y (Cumulative Frequency). Chora mstari mlalo kutoka CF=30 hadi uingiliane na ogive. Kutoka sehemu ya makutano, chora mstari wima chini hadi mhimili wa x (Height). Thamani kwenye mhimili wa x ambapo mstari wima unakatiza ndiyo asilimia ya 30. Kwa kutumia fomula ya kukokotoa asilimia (interpolation) kwa darasa la 35-39 (ambapo CF inatoka 17 hadi 35): P_30 = L + ((30)/(100)N - CF_bf_p) × c Ambapo L = 34.5 (mpaka wa chini wa darasa), N = 100, CF_b = 17 (CF kabla ya darasa), f_p = 18 (masafa ya darasa), c = 5 (upana wa darasa). P_30 = 34.5 + ((30 - 17)/(18)) × 5 P_30 = 34.5 + ((13)/(18)) × 5 P_30 = 34.5 + (65)/(18) P_30 = 34.5 + 3.6111... P_30 ≈ 38.11 3 done, 2 left today. You're making progress.

The height of 100 maize plants were measured to the nearest centimeter and the results recorded in the table below. (a) complete the table. (b) Calculate 2 d.p. (i) The mean (ii) The standard deviation. (c) Using the above plot an ogive and use it to find the 30th percentile.

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The height of 100 maize plants were measured to the nearest centimeter and the results recorded in the table below. (a) complete the table. (b) Calculate 2 d.p. (i) The mean (ii) The standard deviation. (c) Using the above plot an ogive and use it to find the 30th percentile.

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