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Linear Regression (p-2) — Maths (commerce) STD 12 Commerce / Arts — Question

Maharashtra BoardEnglish MediumSTD 12 Commerce / ArtsMaths (commerce)Linear Regression (p-2)4 Marks
Question
For a certain bivariate data
  XY
MEN2520
S.D.43
And r = 0.5 estimate y when x = 10 and estimate x when y = 16
✓

Answer

Given, $\bar{x}=25, \bar{y}=20, \sigma_{\mathrm{x}}=4, \sigma_{\mathrm{y}}=3, \mathrm{r}=0.5$
$
\mathrm{b}_{\mathrm{yx}}=\frac{r \sigma_y}{\sigma_y}=0.5 \times \frac{3}{4}=0.375
$
Regression equation of $Y$ on $X$ is
$
\begin{aligned}
& (Y-\bar{y})=b_{y x}(X-\bar{x}) \\
& (Y-20)=0.375(x-25) \\
& Y-20=0.375 x-9.375 \\
& Y=0.375 x+10.625
\end{aligned}
$
When, $x=10$
$
\begin{aligned}
& Y=0.375(10)+10.625 \\
& =3.75+10.625 \\
& =14.375 \\
& \mathrm{~b}_{\mathrm{xy}}=\frac{r \sigma_x}{\sigma_y}=0.5 \times \frac{4}{3}=0.67
\end{aligned}
$
Regression equation of $X$ on $Y$ is
$
\begin{aligned}
& (X-\bar{x})=b_{y x}(Y-\bar{y}) \\
& (X-25)=0.67(y-20) \\
& (X-25)=0.67 y-13.4 \\
& X=0.67 y+11.6 \\
& \text { When, } Y=16 \\
& X=0.67(16)+11.6 \\
& =10.72+11.6 \\
& =22.32
\end{aligned}
$

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