This is the boss. No calculator. Every variance is a perfect square, every z lands on a half, and every correlation squares in your head — the values were chosen so that the arithmetic is never the difficulty. The difficulty is the ORDER.

One audit, and every answer feeds the next: the standard deviation out of the variance, the suspect bottle's z out of that deviation, the fitted line's prediction, the residual out of that prediction, then how much the model explains at all. Write each line down as you go — one early slip rolls the whole way to the verdict.

And the last line is a decision, not a number. That is the point of the whole chapter: a statistic that never ends in a judgement was a calculation, not an analysis. Watch which tool has jurisdiction, too — the plant's rule is written about the mean and the spread, so the regression model, however good or bad, gets no vote on whether the bottle ships. 5★ under par here is the chapter crown. Clock in.