Answer
The plant manager should manufacture 57,711 ball bearings to meet the order of 50,000 ball bearings between 4.97 and 5.03 mm.
Work Step by Step
$\mu = 5$, $\sigma = 0.02$
Want to find $P(4.97 < x < 5.03)$
i) Convert 4.97 to a z-score:
z = $\frac{x - \mu}{\sigma}$ = $\frac{4.97 - 5}{0.02}$ = -1.5
ii) Convert 5.03 to a z- score:
z = $\frac{5.03 - 5}{0.02}$ = 1.5
iii) $P(4.97 < x < 5.03) = P(-1.5 < z < 1.5)$
$P(-1.5 < z < 1.5) = P(z < 1.5) - P(z < -1.5) = 0.9332 - 0.0668 = 0.8664$
Therefore $P(4.97 < x < 5.03) = 0.8664$
iv) 0.8664 of the total rods produced will be between 4.97 and 5.03 cm. However, we need 50,000 to be between 4.97 and 5.03 cm. If we let B be the number of ball bearings produced:
0.8664 x B = 50,000
B = $\frac{50,000}{0.8664} = 57, 710.0646$
Thus, to meet the order, the plant manager should produce 57,711 ball bearings in total.
