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.

Jumla ya X:

$$\sum X = 150 + 157 + 188 + 165 + 160 + 142 + 170 + 183 + 152 + 173 = 1640$$

Maana ya X:

$$\bar{X} = \frac{\sum X}{n} = \frac{1640}{10} = 164 \text{ cm}$$

Jumla ya miraba ya mkengeuko kutoka maana:

$$\sum (X - \bar{X})^2 = (150-164)^2 + (157-164)^2 + (188-164)^2 + (165-164)^2 + (160-164)^2 + (142-164)^2 + (170-164)^2 + (183-164)^2 + (152-164)^2 + (173-164)^2$$ $$= (-14)^2 + (-7)^2 + (24)^2 + (1)^2 + (-4)^2 + (-22)^2 + (6)^2 + (19)^2 + (-12)^2 + (9)^2$$ $$= 196 + 49 + 576 + 1 + 16 + 484 + 36 + 361 + 144 + 81 = 1944$$

Tofauti (Variance) ya X:

$$s_X^2 = \frac{\sum (X - \bar{X})^2}{n-1} = \frac{1944}{10-1} = \frac{1944}{9} = 216 \text{ cm}^2$$

Mkengeuko Wastani (Standard Deviation) wa X:

$$s_X = \sqrt{s_X^2} = \sqrt{216} \approx 14.6969 \approx 14.70 \text{ cm}$$

Jumla ya Y:

$$\sum Y = 152 + 163 + 193 + 163 + 152 + 157 + 183 + 175 + 163 + 180 = 1681$$

Maana ya Y:

$$\bar{Y} = \frac{\sum Y}{n} = \frac{1681}{10} = 168.1 \text{ cm}$$

Jumla ya miraba ya mkengeuko kutoka maana:

$$\sum (Y - \bar{Y})^2 = (152-168.1)^2 + (163-168.1)^2 + (193-168.1)^2 + (163-168.1)^2 + (152-168.1)^2 + (157-168.1)^2 + (183-168.1)^2 + (175-168.1)^2 + (163-168.1)^2 + (180-168.1)^2$$ $$= (-16.1)^2 + (-5.1)^2 + (24.9)^2 + (-5.1)^2 + (-16.1)^2 + (-11.1)^2 + (14.9)^2 + (6.9)^2 + (-5.1)^2 + (11.9)^2$$ $$= 259.21 + 26.01 + 620.01 + 26.01 + 259.21 + 123.21 + 222.01 + 47.61 + 26.01 + 141.61 = 1750.9$$

Tofauti (Variance) ya Y:

$$s_Y^2 = \frac{\sum (Y - \bar{Y})^2}{n-1} = \frac{1750.9}{10-1} = \frac{1750.9}{9} \approx 194.5444 \approx 194.54 \text{ cm}^2$$

Mkengeuko Wastani (Standard Deviation) wa Y:

$$s_Y = \sqrt{s_Y^2} = \sqrt{194.5444} \approx 13.9479 \approx 13.95 \text{ cm}$$

Muhtasari wa Sehemu i:

• Maana ya urefu wa wazazi ($\bar{X}$): $\boxed{\text{164 cm}}$
• Tofauti ya urefu wa wazazi ($s_X^2$): $\boxed{\text{216 cm}^2}$
• Mkengeuko wastani wa urefu wa wazazi ($s_X$): $\boxed{\text{14.70 cm}}$
• Maana ya urefu wa watoto ($\bar{Y}$): $\boxed{\text{168.1 cm}}$
• Tofauti ya urefu wa watoto ($s_Y^2$): $\boxed{\text{194.54 cm}^2}$
• Mkengeuko wastani wa urefu wa watoto ($s_Y$): $\boxed{\text{13.95 cm}}$

Sehemu ii: Uwiano (Correlation) na Urejeshaji (Regression)

Tayari tunayo $\sum X = 1640$ na $\sum Y = 1681$.
Jumla ya $X^2$:

$$\sum X^2 = 150^2 + 157^2 + 188^2 + 165^2 + 160^2 + 142^2 + 170^2 + 183^2 + 152^2 + 173^2 = 270904$$

Jumla ya $Y^2$:

$$\sum Y^2 = 152^2 + 163^2 + 193^2 + 163^2 + 152^2 + 157^2 + 183^2 + 175^2 + 163^2 + 180^2 = 282327$$

Jumla ya $XY$:

$$\sum XY = (150 \times 152) + (157 \times 163) + (188 \times 193) + (165 \times 163) + (160 \times 152) + (142 \times 157) + (170 \times 183) + (183 \times 175) + (152 \times 163) + (173 \times 180)$$ $$= 22800 + 25571 + 36284 + 26955 + 24320 + 22294 + 31110 + 32025 + 24776 + 31140 = 277275$$

Tumia fomula:

$$r = \frac{n \sum XY - (\sum X)(\sum Y)}{\sqrt{[n \sum X^2 - (\sum X)^2][n \sum Y^2 - (\sum Y)^2]}}$$

Kwanza, kokotoa nambari (numerator):

$$n \sum XY - (\sum X)(\sum Y) = 10(277275) - (1640)(1681)$$ $$= 2772750 - 2756840 = 15910$$

Kisha, kokotoa sehemu ya kwanza ya denominator:

$$n \sum X^2 - (\sum X)^2 = 10(270904) - (1640)^2$$ $$= 2709040 - 2689600 = 19440$$

Na sehemu ya pili ya denominator:

$$n \sum Y^2 - (\sum Y)^2 = 10(282327) - (1681)^2$$ $$= 2823270 - 2825761 = -2491$$

Kuna tatizo hapa. $n \sum Y^2 - (\sum Y)^2$ haiwezi kuwa hasi. Hii inaashiria kuwa kuna kosa katika data ya Y au hesabu ya $\sum Y^2$ au $\sum Y$.
Hebu turekebishe $\sum Y^2$ kwa kutumia $\sum (Y - \bar{Y})^2 = 1750.9$:
Tunajua $\sum (Y - \bar{Y})^2 = \sum Y^2 - \frac{(\sum Y)^2}{n}$.
Hivyo, $1750.9 = \sum Y^2 - \frac{1681^2}{10}$.
$1750.9 = \sum Y^2 - \frac{2825761}{10}$.
$1750.9 = \sum Y^2 - 282576.1$.
$\sum Y^2 = 282576.1 + 1750.9 = 284327$.
Hivyo, $\sum Y^2$ sahihi ni 284327.

Sasa turudie hesabu ya sehemu ya pili ya denominator na $\sum Y^2 = 284327$:

$$n \sum Y^2 - (\sum Y)^2 = 10(284327) - (1681)^2$$ $$= 2843270 - 2825761 = 17509$$

Sasa tunaweza kuendelea na $r$:

$$r = \frac{15910}{\sqrt{(19440)(17509)}} = \frac{15910}{\sqrt{340375000}} = \frac{15910}{18440.98} \approx 0.8627$$

Mgawo wa uwiano (correlation coefficient): $\boxed{0.86}$

Kwanza, kokotoa mteremko (slope), $b$:

$$b = \frac{n \sum XY - (\sum X)(\sum Y)}{n \sum X^2 - (\sum X)^2}$$ $$b = \frac{15910}{19440} \approx 0.8184$$

Kisha, kokotoa sehemu ya kukatiza (y-intercept), $a$:

$$a = \bar{Y} - b\bar{X}$$ $$a = 168.1 - (0.8184)(164)$$ $$a = 168.1 - 134.1976 = 33.9024$$

Mlinganyo wa urejeshaji (regression equation):

$$\boxed{Y = 33.90 + 0.82X}$$