12.9. In packaged weight control, it is not very convenient to obtain directly the weight of the net contents. Instead, we may proceed indirectly by weighing empty containers x, covers y, and total filled packages z. Letting the net weight of contents be w, we thus have w ¼zxy. Now we cannot find sw by use of (12.9), because z, x and y are not independent.

Instead, we can let z ¼xþyþw, in which x, y, and w are likely to be quite independent. Then we use (12.9) for the relation between s2’s. Knowing three of these, we can find sw 2 and sw. However, we can use (12.8) on either form of (12.7)

to find mw. Now, given the actual observed data z: m¼ 189.5, s¼3.5; for y, m¼3.1, s¼1.1 (in pounds) find m and s for w.

Assuming normality estimate the percentage of contents weights below the minimum specification of 165 lb.