Recover a missing price index and choose the prediction direction

A regression line passes through $(\bar x,\bar y)$; it need not pass through every observation. The seven known housing indices sum to 815.2 and all eight food indices sum to 894.2. Write the missing housing value as $h$. Then

$\begin{aligned}&\frac{894.2}{8}\\ & =54.271\\ &{}+0.48363\frac{815.2+h}{8}.\end{aligned}$

Hence $h=136.0065008\ldots$, which rounds to 136. The supplied coefficients are rounded to five significant figures; this recovery is correspondingly approximate. Substituting only the 2011 food value into the line would incorrectly assume zero residual for that observation. Use the requested recovered integer for the remaining parts.

Plot housing index $x$ horizontally and food index $y$ vertically, retaining all eight pairs and using 136 for 2011. The gradient appears to decrease as housing index rises. This curved pattern makes a single linear model less suitable than the high positive association alone might suggest. The comparison diagram also shows the transformed variables used in (iii).

The known predictor is food index $y$ and the desired response is $\sqrt{x}$. Fit $\sqrt{x}$ on $y$, rather than algebraically reversing the regression of $y$ on $\sqrt{x}$. Retaining full precision gives $\sqrt{x}=a+by$, where

$\begin{aligned}a&=0.8599870849633415,\\b&=0.08963124264600579.\end{aligned}$

The printed rounded equation is $\sqrt{x}=0.0896y+0.860$. Using the retained coefficients,

$\begin{aligned}&x\\ & =(a+134.6b)^2\\ & =167.0388835\ldots\approx167.\end{aligned}$

The input 134.6 exceeds the observed food-index range 100–123.1, so this is extrapolation. The observations cover 2005–2012, while the target is 2016; the relationship may change over time. The positive transformed relationship does not guarantee a reliable future prediction.

Keeping all eight pairings and the recovered 136 gives $r=0.9788756402\ldots$, matching the printed key 0.979. This is correlation between $y$ and $\sqrt{x}$, not between the original $x$ and $y$.

Since $x>0$, $\sqrt{x/100}=\sqrt{x}/10$. The new variables are therefore the old transformed variables multiplied by the positive constants $1/10$ and $1/100$. Positive rescaling changes their units but leaves their correlation unchanged; no recalculation is needed.