Following are the marks in physics and chemistry of six students Marks in Physics (X) 11 12 13 14 15 16 Marks in Chemistry (Y) 23 24 26 22 25 27 a) Find the coefficient of correlation between X and…

a) Coefficient of Correlation between X and Y

The coefficient of correlation $r$ between two variables $X$ and $Y$ is calculated using 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:

• $n=6$
• $\sum X=11+12+13+14+15+16=81$
• $\sum Y=23+24+26+22+25+27=147$
• $\sum XY=(11\times 23)+(12\times 24)+(13\times 26)+(14\times 22)+(15\times 25)+(16\times 27)$ $=253+288+338+308+375+432=1994$
• $\sum X^2=11^2+12^2+13^2+14^2+15^2+16^2$ $=121+144+169+196+225+256=1011$
• $\sum Y^2=23^2+24^2+26^2+22^2+25^2+27^2$ $=529+576+676+484+625+729=3619$

Step 2: Substitute into the formula

$$r=\frac{6\times 1994-(81)(147)}{\sqrt{[6\times 1011-(81{)}^2][6\times 3619-(147{)}^2]}}$$

Calculate numerator: $$6\times 1994-(81)(147)=11964-11907=57$$

Calculate denominator: $$6\times 1011-(81{)}^2=6066-6561=-495(\text{absolute value: }495)$$ $$6\times 3619-(147{)}^2=21714-21609=105$$ $$\sqrt{495\times 105}=\sqrt{52000}\approx 228.035$$

Step 3: Compute $r$

$$r=\frac{57}{228.035}\approx 0.25$$

Answer: The coefficient of correlation between $X$ and $Y$ is $0.25$.

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b) Estimate the marks in Physics when marks in Chemistry is 30

The regression line of $X$ on $Y$ is given by:

$$X=a+bY$$

where: $$b=\frac{n\sum XY-(\sum X)(\sum Y)}{n\sum Y^2-(\sum Y{)}^2}$$ $$a=\bar{X}-b\bar{Y}$$

Step 1: Compute $b$

$$b=\frac{6\times 1994-(81)(147)}{6\times 3619-(147{)}^2}=\frac{57}{105}\approx 0.5429$$

Step 2: Compute $a$

$$\bar{X}=\frac{81}{6}=13.5,\bar{Y}=\frac{147}{6}=24.5$$ $$a=13.5-(0.5429\times 24.5)=13.5-13.28=0.22$$

Step 3: Regression equation

$$X=0.22+0.5429Y$$

Step 4: Estimate $X$ when $Y=30$

$$X=0.22+0.5429\times 30=0.22+16.287=16.507$$

Answer: The estimated marks in Physics when marks in Chemistry is 30 is $16.51$ (rounded to two decimal places).