Mean \( \mu = 6.0 \, \text{cm} \)
Standard deviation \( \sigma = 0.2 \, \text{cm} \)
The allowable error in length is \( \pm 0.3 \, \text{cm} \)
Range \( [6.0 - 0.3, 6.0 + 0.3] = [5.7, 6.3] \, \text{cm} \)
Lower bound: \( z_1 = \frac{5.7 - 6.0}{0.2} = -1.5 \)
Upper bound: \( z_2 = \frac{6.3 - 6.0}{0.2} = 1.5 \)
\( P(Z \leq 1.5) = 0.9332 \)
\( P(Z \leq -1.5) = 0.0668 \)
The probability that the length lies within \( [5.7, 6.3] \),
\[P(5.7 \leq X \leq 6.3) = P(Z \leq 1.5) - P(Z \leq -1.5) = 0.9332 - 0.0668 = 0.8664\]
The probability that the length lies outside \( [5.7, 6.3] \),
\[P(\text{Defective}) = 1 - P(5.7 \leq X \leq 6.3) = 1 - 0.8664 = 0.1336\]
Percentage of defectives = 0.1336 * 100 = 13.36%
Probability of defectives to be 5%, the probability of the length lying within \( [5.7, 6.3] \) should be 95%.
Lower bound: \( z_1 = \frac{5.7 - 6.0}{\sigma'} = \frac{-0.3}{\sigma'} \)
Upper bound: \( z_2 = \frac{6.3 - 6.0}{\sigma'} = \frac{0.3}{\sigma'} \)
\( z \)-value corresponding to 95% probability is approximately \( 1.96 \).
\[\frac{0.3}{\sigma'} = 1.96\]
\[\sigma' = \frac{0.3}{1.96} \approx 0.1531 \, \text{cm}\]
