Unformatted text preview: X is calibrated in such a way
that ﬁll weights of containers are approximately normally distributed with mean 8.2 oz and
standard deviation 0.1 oz. The average net weight of a container is advertised as 8.2 oz. A
quality control standard is proposed which would specify that weights of containers must lie
between 8 and 8.3 oz (it is too expensive to overﬁll by more than .1 oz and there might be
trouble with consumer advocates if a container is under-ﬁlled by more than .2 oz). a. If the proposal were adopted, what proportion of containers would lie outside speciﬁcations?
SOLUTION: P(X > 8.3, or X < 8) = 1 − P(8 < X < 8.3) = 1 − P( 8−.8.2 ≤ Z ≤ 8.30−18.2 )
= 1 − [P(Z ≤ 1) − P(Z ≤ −2)] = 1 − (0.8413 − 0.0228) = 1 − 0.8115 = 0.1885.
It has been decided that the quality control standard will be adopted and that at least
86.5% of containers must meet speciﬁcations, and there are two methods of achieving this:
the ﬁrst is to leave the uncertainty of the ﬁlling apparatus unchanged, but to change the
calibration of the apparatus to give a new mean ﬁll weight (and to advertise this new
weight as the average net weight). The second is to improve the precision of the ap...
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