For a bivariate data set, $\bar x=40,\ \bar y=55,\ \sigma_x=5,\ \sigma_y=8$ and $r=0.8$. Obtain the two lines of regression, and estimate (i) $y$ when $x=45$ and (ii) $x$ when $y=60$.

For a bivariate data set, bar x=40,bar y=55,σₓ=5,σy=8 and r=0.8. Obtain the two lines of regression, and estim

Worked Examples · Example 6

\bar x=40,\ \bar y=55,\ \sigma_x=5,\ \sigma_y=8

(i)

y

when

x=45

and

(ii)

x

when

y=60

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Regression coefficients from $r$ and the SDs:

b_{yx}=r\,\frac{\sigma_y}{\sigma_x}=0.8\times\frac{8}{5}=0.8\times1.6=1.28,

b_{xy}=r\,\frac{\sigma_x}{\sigma_y}=0.8\times\frac{5}{8}=0.8\times0.625=0.5.

(Check: $b_{yx}\cdot b_{xy}=1.28\times0.5=0.64=r^2$ , and $\sqrt{0.64}=0.8=r$ — consistent.)

Line of $Y$ on $X$ : $y-\bar y=b_{yx}(x-\bar x)$

y-55=1.28(x-40).

Line of $X$ on $Y$ : $x-\bar x=b_{xy}(y-\bar y)$

x-40=0.5(y-55).

(i) Estimate $y$ when $x=45$ (use $Y$ on $X$ ):

$y=55+1.28(45-40)=55+1.28\times5=55+6.4=61.4.$ …