Solution
The Point That Flattens the Line
Show the problem again
At $x = 1, 2, 3, 4, 5$ the observed values are $y = 9, 4, 3, 5$ and an unknown fifth value $y_5$. For which $y_5$ is the least-squares slope of $y$ on $x$ exactly zero? Give a reduced fraction or decimal.
Worked solution
The slope is $\sum (x_i - \bar x) y_i / \sum (x_i-\bar x)^2$, and the denominator is $10$ with $x_i - \bar x = -2,-1,0,1,2$. The slope vanishes exactly when $-2(9) - 4 + 0 + 5 + 2y_5 = 0$, so $2y_5 = 17$ and $y_5 = 17/2$. Note that $y_3$ does not matter at all: its point sits at the mean of $x$ and has zero leverage on the slope.
Source: Original exercise written for AxiomIQ.