Question

A sock producer is interested in how uniform the length of socks that are produced at his factory are.
It is known that the sock length follows a normal distribution with a mean of 20 cm and a standard
deviation of 10.5 cm.
. What is the probability that a randomly chosen sock is longer than 25 cm?

Answer 1

By condition,

$a = 20\,\,\sigma = 10.5$

Let's find the probability

$P(x > 25) = \Phi \left( \infty \right) - \Phi \left( {\frac{{25 - a}}{\sigma }} \right) = 0.5 - \Phi \left( {\frac{{25 - 20}}{{10.5}}} \right) = 0.5 - \Phi \left( {0.48} \right) \approx 0.5 - 0.1844 = 0.3156$

Answer: $P(x > 25) \approx 0.3156$