The following table gives the age and weight of school children in a locality age in year 4 5 7 9 10 11 : : : : : : : : : : : : : weight in kg. 20 25 28 30 32 33 a) Find the co efficient of…

a) Coefficient of Correlation between Age and Weight

To find the coefficient of correlation ($r$) between age ($x$) and weight ($y$), we use the formula:

$$r=\frac{n\sum xy-\sum x\sum y}{\sqrt{[n\sum x^2-(\sum x{)}^2][n\sum y^2-(\sum y{)}^2]}}$$

Step 1: Compute the required sums

Given data:

Compute:

• $n=6$
• $\sum x=4+5+7+9+10+11=46$
• $\sum y=20+25+28+30+32+33=168$
• $\sum xy=(4\times 20)+(5\times 25)+(7\times 28)+(9\times 30)+(10\times 32)+(11\times 33)=80+125+196+270+320+363=1354$
• $\sum x^2=4^2+5^2+7^2+9^2+10^2+11^2=16+25+49+81+100+121=402$
• $\sum y^2=20^2+25^2+28^2+30^2+32^2+33^2=400+625+784+900+1024+1089=4822$

Step 2: Substitute into the formula

$$r=\frac{6\times 1354-46\times 168}{\sqrt{[6\times 402-46^2][6\times 4822-168^2]}}$$

Calculate numerator: $$6\times 1354=8124,46\times 168=7728$$ $$\text{Numerator}=8124-7728=396$$

Calculate denominator: $$6\times 402=2412,46^2=2116$$ $$\text{First term}=2412-2116=296$$

$$6\times 4822=28932,168^2=28224$$ $$\text{Second term}=28932-28224=708$$

$$\text{Denominator}=\sqrt{296\times 708}=\sqrt{209968}=458.23$$

Step 3: Compute $r$

$$r=\frac{396}{458.23}\approx 0.864$$

Answer: The coefficient of correlation between age and weight is $0.864$.

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b) Estimate the Weight when Age is 12 Years

To estimate the weight ($y$) when age ($x$) is 12, we use the linear regression equation:

$$y=a+bx$$

where: $$b=\frac{n\sum xy-\sum x\sum y}{n\sum x^2-(\sum x{)}^2}$$ $$a=\bar{y}-b\bar{x}$$

Step 1: Compute $b$

Using values from part (a): $$b=\frac{396}{296}=1.3378$$

Step 2: Compute $a$

$$\bar{x}=\frac{\sum x}{n}=\frac{46}{6}=7.67$$ $$\bar{y}=\frac{\sum y}{n}=\frac{168}{6}=28$$ $$a=28-(1.3378\times 7.67)=28-10.25=17.75$$

Step 3: Form the regression equation

$$y=17.75+1.3378x$$

Step 4: Estimate weight for $x=12$

$$y=17.75+1.3378\times 12=17.75+16.05=33.8$$

Answer: The estimated weight when age is 12 years is 33.8 kg.