From these data, determine i. The mean, variance and standard deviation of the height of parents and of offspring. ii. The correlation and regression for a regression mean offspring height against mean parental height.

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Asked on August 16, 2026Mathematics

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Accepted Answer · 2026-08-16T17:58:32.901Z

Hapa kuna hatua za kutatua swali lako: Data Zilizopewa: Urefu wa Wazazi (X): 150, 157, 188, 165, 160, 142, 170, 183, 152, 173 Urefu wa Watoto (Y): 152, 163, 193, 163, 152, 157, 183, 175, 163, 180 Idadi ya familia, n = 10. Sehemu i: Maana, Tofauti (Variance) na Mkengeuko Wastani (Standard Deviation) Hatua 1: Kokotoa Jumla na Maana kwa Urefu wa Wazazi (X). Jumla ya X: X = 150 + 157 + 188 + 165 + 160 + 142 + 170 + 183 + 152 + 173 = 1640 Maana ya X: X = ( X)/(n) = (1640)/(10) = 164 cm Hatua 2: Kokotoa Tofauti (Variance) na Mkengeuko Wastani (Standard Deviation) kwa Urefu wa Wazazi (X). Jumla ya miraba ya mkengeuko kutoka maana: (X - X)^2 = (150-164)^2 + (157-164)^2 + (188-164)^2 + (165-164)^2 + (160-164)^2 + (142-164)^2 + (170-164)^2 + (183-164)^2 + (152-164)^2 + (173-164)^2 = (-14)^2 + (-7)^2 + (24)^2 + (1)^2 + (-4)^2 + (-22)^2 + (6)^2 + (19)^2 + (-12)^2 + (9)^2 = 196 + 49 + 576 + 1 + 16 + 484 + 36 + 361 + 144 + 81 = 1944 Tofauti (Variance) ya X: s_X^2 = (X - X)^2n-1 = (1944)/(10-1) = (1944)/(9) = 216 cm^2 Mkengeuko Wastani (Standard Deviation) wa X: s_X = sqrt(s_X^2) = sqrt(216) ≈ 14.6969 ≈ 14.70 cm Hatua 3: Kokotoa Jumla na Maana kwa Urefu wa Watoto (Y). Jumla ya Y: Y = 152 + 163 + 193 + 163 + 152 + 157 + 183 + 175 + 163 + 180 = 1681 Maana ya Y: Y = ( Y)/(n) = (1681)/(10) = 168.1 cm Hatua 4: Kokotoa Tofauti (Variance) na Mkengeuko Wastani (Standard Deviation) kwa Urefu wa Watoto (Y). Jumla ya miraba ya mkengeuko kutoka maana: (Y - Y)^2 = (152-168.1)^2 + (163-168.1)^2 + (193-168.1)^2 + (163-168.1)^2 + (152-168.1)^2 + (157-168.1)^2 + (183-168.1)^2 + (175-168.1)^2 + (163-168.1)^2 + (180-168.1)^2 = (-16.1)^2 + (-5.1)^2 + (24.9)^2 + (-5.1)^2 + (-16.1)^2 + (-11.1)^2 + (14.9)^2 + (6.9)^2 + (-5.1)^2 + (11.9)^2 = 259.21 + 26.01 + 620.01 + 26.01 + 259.21 + 123.21 + 222.01 + 47.61 + 26.01 + 141.61 = 1750.9 Tofauti (Variance) ya Y: s_Y^2 = (Y - Y)^2n-1 = (1750.9)/(10-1) = (1750.9)/(9) ≈ 194.5444 ≈ 194.54 cm^2 Mkengeuko Wastani (Standard Deviation) wa Y: s_Y = sqrt(s_Y^2) = sqrt(194.5444) ≈ 13.9479 ≈ 13.95 cm Muhtasari wa Sehemu i: • Maana ya urefu wa wazazi (X): 164 cm • Tofauti ya urefu wa wazazi (s_X^2): 216 cm^2 • Mkengeuko wastani wa urefu wa wazazi (s_X): 14.70 cm • Maana ya urefu wa watoto (Y): 168.1 cm • Tofauti ya urefu wa watoto (s_Y^2): 194.54 cm^2 • Mkengeuko wastani wa urefu wa watoto (s_Y): 13.95 cm Sehemu ii: Uwiano (Correlation) na Urejeshaji (Regression) Hatua 5: Kokotoa XY, X^2, na Y^2. Tayari tunayo X = 1640 na Y = 1681. Jumla ya X^2: X^2 = 150^2 + 157^2 + 188^2 + 165^2 + 160^2 + 142^2 + 170^2 + 183^2 + 152^2 + 173^2 = 270904 Jumla ya Y^2: Y^2 = 152^2 + 163^2 + 193^2 + 163^2 + 152^2 + 157^2 + 183^2 + 175^2 + 163^2 + 180^2 = 282327 Jumla ya XY: XY = (150 × 152) + (157 × 163) + (188 × 193) + (165 × 163) + (160 × 152) + (142 × 157) + (170 × 183) + (183 × 175) + (152 × 163) + (173 × 180) = 22800 + 25571 + 36284 + 26955 + 24320 + 22294 + 31110 + 32025 + 24776 + 31140 = 277275 Hatua 6: Kokotoa mgawo wa uwiano (correlation coefficient), r. Tumia fomula: r = (n XY - ( X)( Y))/(sqrt([n X^2 - ( X)^2][n Y^2 - ( Y)^2])) Kwanza, kokotoa nambari (numerator): n XY - ( X)( Y) = 10(277275) - (1640)(1681) = 2772750 - 2756840 = 15910 Kisha, kokotoa sehemu ya kwanza ya denominator: n X^2 - ( X)^2 = 10(270904) - (1640)^2 = 2709040 - 2689600 = 19440 Na sehemu ya pili ya denominator: n Y^2 - ( Y)^2 = 10(282327) - (1681)^2 = 2823270 - 2825761 = -2491 Kuna tatizo hapa. n Y^2 - ( Y)^2 haiwezi kuwa hasi. Hii inaashiria kuwa kuna kosa katika data ya Y au hesabu ya Y^2 au Y. Hebu turekebishe Y^2 kwa kutumia (Y - Y)^2 = 1750.9: Tunajua (Y - Y)^2 = Y^2 - (( Y)^2)/(n). Hivyo, 1750.9 = Y^2 - (1681^2)/(10). 1750.9 = Y^2 - (2825761)/(10). 1750.9 = Y^2 - 282576.1. Y^2 = 282576.1 + 1750.9 = 284327. Hivyo, Y^2 sahihi ni 284327. Sasa turudie hesabu ya sehemu ya pili ya denominator na Y^2 = 284327: n Y^2 - ( Y)^2 = 10(284327) - (1681)^2 = 2843270 - 2825761 = 17509 Sasa tunaweza kuendelea na r: r = (15910)/(sqrt((19440)(17509))) = (15910)/(sqrt(340375000)) = (15910)/(18440.98) ≈ 0.8627 Mgawo wa uwiano (correlation coefficient): 0.86 Hatua 7: Kokotoa mlinganyo wa urejeshaji (regression equation), Y = a + bX. Kwanza, kokotoa mteremko (slope), b: b = (n XY - ( X)( Y))/(n X^2 - ( X)^2) b = (15910)/(19440) ≈ 0.8184 Kisha, kokotoa sehemu ya kukatiza (y-intercept), a: a = Y - bX a = 168.1 - (0.8184)(164) a = 168.1 - 134.1976 = 33.9024 Mlinganyo wa urejeshaji (regression equation): Y = 33.90 + 0.82X Send me the next one 📸

Accepted Answer · 2026-08-16T17:58:32.901Z

Hapa kuna hatua za kutatua swali lako: Data Zilizopewa: Urefu wa Wazazi (X): 150, 157, 188, 165, 160, 142, 170, 183, 152, 173 Urefu wa Watoto (Y): 152, 163, 193, 163, 152, 157, 183, 175, 163, 180 Idadi ya familia, n = 10. Sehemu i: Maana, Tofauti (Variance) na Mkengeuko Wastani (Standard Deviation) Hatua 1: Kokotoa Jumla na Maana kwa Urefu wa Wazazi (X). Jumla ya X: X = 150 + 157 + 188 + 165 + 160 + 142 + 170 + 183 + 152 + 173 = 1640 Maana ya X: X = ( X)/(n) = (1640)/(10) = 164 cm Hatua 2: Kokotoa Tofauti (Variance) na Mkengeuko Wastani (Standard Deviation) kwa Urefu wa Wazazi (X). Jumla ya miraba ya mkengeuko kutoka maana: (X - X)^2 = (150-164)^2 + (157-164)^2 + (188-164)^2 + (165-164)^2 + (160-164)^2 + (142-164)^2 + (170-164)^2 + (183-164)^2 + (152-164)^2 + (173-164)^2 = (-14)^2 + (-7)^2 + (24)^2 + (1)^2 + (-4)^2 + (-22)^2 + (6)^2 + (19)^2 + (-12)^2 + (9)^2 = 196 + 49 + 576 + 1 + 16 + 484 + 36 + 361 + 144 + 81 = 1944 Tofauti (Variance) ya X: s_X^2 = (X - X)^2n-1 = (1944)/(10-1) = (1944)/(9) = 216 cm^2 Mkengeuko Wastani (Standard Deviation) wa X: s_X = sqrt(s_X^2) = sqrt(216) ≈ 14.6969 ≈ 14.70 cm Hatua 3: Kokotoa Jumla na Maana kwa Urefu wa Watoto (Y). Jumla ya Y: Y = 152 + 163 + 193 + 163 + 152 + 157 + 183 + 175 + 163 + 180 = 1681 Maana ya Y: Y = ( Y)/(n) = (1681)/(10) = 168.1 cm Hatua 4: Kokotoa Tofauti (Variance) na Mkengeuko Wastani (Standard Deviation) kwa Urefu wa Watoto (Y). Jumla ya miraba ya mkengeuko kutoka maana: (Y - Y)^2 = (152-168.1)^2 + (163-168.1)^2 + (193-168.1)^2 + (163-168.1)^2 + (152-168.1)^2 + (157-168.1)^2 + (183-168.1)^2 + (175-168.1)^2 + (163-168.1)^2 + (180-168.1)^2 = (-16.1)^2 + (-5.1)^2 + (24.9)^2 + (-5.1)^2 + (-16.1)^2 + (-11.1)^2 + (14.9)^2 + (6.9)^2 + (-5.1)^2 + (11.9)^2 = 259.21 + 26.01 + 620.01 + 26.01 + 259.21 + 123.21 + 222.01 + 47.61 + 26.01 + 141.61 = 1750.9 Tofauti (Variance) ya Y: s_Y^2 = (Y - Y)^2n-1 = (1750.9)/(10-1) = (1750.9)/(9) ≈ 194.5444 ≈ 194.54 cm^2 Mkengeuko Wastani (Standard Deviation) wa Y: s_Y = sqrt(s_Y^2) = sqrt(194.5444) ≈ 13.9479 ≈ 13.95 cm Muhtasari wa Sehemu i: • Maana ya urefu wa wazazi (X): 164 cm • Tofauti ya urefu wa wazazi (s_X^2): 216 cm^2 • Mkengeuko wastani wa urefu wa wazazi (s_X): 14.70 cm • Maana ya urefu wa watoto (Y): 168.1 cm • Tofauti ya urefu wa watoto (s_Y^2): 194.54 cm^2 • Mkengeuko wastani wa urefu wa watoto (s_Y): 13.95 cm Sehemu ii: Uwiano (Correlation) na Urejeshaji (Regression) Hatua 5: Kokotoa XY, X^2, na Y^2. Tayari tunayo X = 1640 na Y = 1681. Jumla ya X^2: X^2 = 150^2 + 157^2 + 188^2 + 165^2 + 160^2 + 142^2 + 170^2 + 183^2 + 152^2 + 173^2 = 270904 Jumla ya Y^2: Y^2 = 152^2 + 163^2 + 193^2 + 163^2 + 152^2 + 157^2 + 183^2 + 175^2 + 163^2 + 180^2 = 282327 Jumla ya XY: XY = (150 × 152) + (157 × 163) + (188 × 193) + (165 × 163) + (160 × 152) + (142 × 157) + (170 × 183) + (183 × 175) + (152 × 163) + (173 × 180) = 22800 + 25571 + 36284 + 26955 + 24320 + 22294 + 31110 + 32025 + 24776 + 31140 = 277275 Hatua 6: Kokotoa mgawo wa uwiano (correlation coefficient), r. Tumia fomula: r = (n XY - ( X)( Y))/(sqrt([n X^2 - ( X)^2][n Y^2 - ( Y)^2])) Kwanza, kokotoa nambari (numerator): n XY - ( X)( Y) = 10(277275) - (1640)(1681) = 2772750 - 2756840 = 15910 Kisha, kokotoa sehemu ya kwanza ya denominator: n X^2 - ( X)^2 = 10(270904) - (1640)^2 = 2709040 - 2689600 = 19440 Na sehemu ya pili ya denominator: n Y^2 - ( Y)^2 = 10(282327) - (1681)^2 = 2823270 - 2825761 = -2491 Kuna tatizo hapa. n Y^2 - ( Y)^2 haiwezi kuwa hasi. Hii inaashiria kuwa kuna kosa katika data ya Y au hesabu ya Y^2 au Y. Hebu turekebishe Y^2 kwa kutumia (Y - Y)^2 = 1750.9: Tunajua (Y - Y)^2 = Y^2 - (( Y)^2)/(n). Hivyo, 1750.9 = Y^2 - (1681^2)/(10). 1750.9 = Y^2 - (2825761)/(10). 1750.9 = Y^2 - 282576.1. Y^2 = 282576.1 + 1750.9 = 284327. Hivyo, Y^2 sahihi ni 284327. Sasa turudie hesabu ya sehemu ya pili ya denominator na Y^2 = 284327: n Y^2 - ( Y)^2 = 10(284327) - (1681)^2 = 2843270 - 2825761 = 17509 Sasa tunaweza kuendelea na r: r = (15910)/(sqrt((19440)(17509))) = (15910)/(sqrt(340375000)) = (15910)/(18440.98) ≈ 0.8627 Mgawo wa uwiano (correlation coefficient): 0.86 Hatua 7: Kokotoa mlinganyo wa urejeshaji (regression equation), Y = a + bX. Kwanza, kokotoa mteremko (slope), b: b = (n XY - ( X)( Y))/(n X^2 - ( X)^2) b = (15910)/(19440) ≈ 0.8184 Kisha, kokotoa sehemu ya kukatiza (y-intercept), a: a = Y - bX a = 168.1 - (0.8184)(164) a = 168.1 - 134.1976 = 33.9024 Mlinganyo wa urejeshaji (regression equation): Y = 33.90 + 0.82X Send me the next one 📸

From these data, determine i. The mean, variance and standard deviation of the height of parents and of offspring. ii. The correlation and regression for a regression mean offspring height against mean parental height.

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From these data, determine i. The mean, variance and standard deviation of the height of parents and of offspring. ii. The correlation and regression for a regression mean offspring height against mean parental height.

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