The probability that a randomly selected sample from a normally distributed data with mean, μ, and standard deviation, σ, is less than a value x is given by:
[tex]P(X\ \textless \ x)=P\left(z\ \textless \ \frac{x-\mu}{\sigma} \right)[/tex]
Given that the
mean time it takes for workers at a factory to insert a delicate bolt
into an engine is 15 minutes and the standard deviation of time to insert
the bolt is 4.0 minutes and the distribution of time is approximately
Normally distributed.
The approximate probability the bolt will be inserted in 19 minutes or less by a randomly selected worker is given by:
[tex]P(X\ \textless \ 19)=P\left(z\ \textless \ \frac{19-15}{4} \right) \\ \\ =P\left(z\ \textless \ \frac{4}{4} \right)=P(z\ \textless \ 1)=0.8413\approx84\%[/tex]
