Answer to the prediction taskΒΆ
The best prediction is actually 169.6 cm.
Funny, but the sample average is NOT the best prediction!
Why? Why not larger than 170cm? Why not exactly 170 cm?
I am glad you asked!
It appears that shrinking the prediction toward zero and away from the observed mean is always optimal. By how much it is optimal to shrink, actually, depends on the mean and variance of the data-generating process and on the sample size.
See the mathematical proof in the next section.
Or if you still do not believe me, check out this R code that simulates this inference vs. prediction problem and produces the following graph that visually shows that MSE are indeed minimized not when \(\alpha\) equals one, but when it is smaller.
