Given the data: $X$: 1, 2, 3, 4, 5 and $Y$: 6, 8, 10, 12, 14, find both regression equations and the coefficient of correlation. Comment on your result.

Question

Accepted Answer

**Step 1 — Basic sums.** $n=5$; $\Sigma X=15 \Rightarrow \bar X=3$; $\Sigma Y = 6+8+10+12+14=50 \Rightarrow \bar Y=10$.
$\Sigma XY = 1(6)+2(8)+3(10)+4(12)+5(14) = 6+16+30+48+70 = 170$.
$\Sigma X^{2}=55$; $\Sigma Y^{2}=36+64+100+144+196=540$.

**Step 2 — Regression coefficients.**
$$b_{yx}=\frac{5(170)-(15)(50)}{5(55)-(15)^2}=\frac{850-750}{275-225}=\frac{100}{50}=2$$
$$b_{xy}=\frac{5(170)-(15)(50)}{5(540)-(50)^2}=\frac{850-750}{2700-2500}=\frac{100}{200}=0.5$$

**Step 3 — Correlation.**
$$r=\sqrt{b_{yx}\cdot b_{xy}}=\sqrt{2	imes0.5}=\sqrt{1}=1$$

**Step 4 — Regression equations.**
$Y$ on $X$: $Y-10=2(X-3)=2X-6 \Rightarrow Y=2X+4$
$X$ on $Y$: $X-3=0.5(Y-10)=0.5Y-5 \Rightarrow X=0.5Y-2$; rearranging for $Y$: $Y=2X+4$ — **the identical line.**

**Step 5 — Independent check.** Each observation satisfies $Y=2X+4$ exactly (e.g. $X=3 \Rightarrow Y=2(3)+4=10$ ✓, $X=5 \Rightarrow Y=14$ ✓), confirming a perfect linear relationship with no scatter at all, matching $r=1$.

> [!ANSWER]
> $r=\mathbf{1}$ (perfect positive correlation); both regression lines reduce to the single line $Y=2X+4$, exactly as the theory predicts for $r=\pm1$.

Q.Given the data: $X$ : 1, 2, 3, 4, 5 and $Y$ : 6, 8, 10, 12, 14, find both regression equations and the coefficient of correlation. Comment on your result.

Concept understanding — Relationship between Regression Coefficients and Correlation

Q.Given the data: XXX: 1, 2, 3, 4, 5 and YYY: 6, 8, 10, 12, 14, find both regression equations and the coefficient of correlation. Comment on your result.

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Q.Given the data: $X$ : 1, 2, 3, 4, 5 and $Y$ : 6, 8, 10, 12, 14, find both regression equations and the coefficient of correlation. Comment on your result.

Unlock everything free for 14 days