For a normally distributed data, with mean, μ, and standard deviation, σ, the probability that a randomly selected data, X, is less than a given value, x, is given by
![P(X\ \textless \ x)=P \left(z\ \textless \ \frac{x-\mu}{\sigma} \right)](https://tex.z-dn.net/?f=P%28X%5C%20%5Ctextless%20%5C%20x%29%3DP%20%5Cleft%28z%5C%20%5Ctextless%20%5C%20%20%5Cfrac%7Bx-%5Cmu%7D%7B%5Csigma%7D%20%5Cright%29)
and the probability that a randomly selected data, X, is greater than a given value, x, is given by
![P(X \ \textgreater \ x)=P \left(z\ \textgreater \ \frac{x-\mu}{\sigma} \right)=1-P \left(z\ \textless \ \frac{x-\mu}{\sigma} \right)](https://tex.z-dn.net/?f=P%28X%20%5C%20%5Ctextgreater%20%5C%20%20x%29%3DP%20%5Cleft%28z%5C%20%5Ctextgreater%20%5C%20%5Cfrac%7Bx-%5Cmu%7D%7B%5Csigma%7D%20%5Cright%29%3D1-P%20%5Cleft%28z%5C%20%5Ctextless%20%5C%20%5Cfrac%7Bx-%5Cmu%7D%7B%5Csigma%7D%20%5Cright%29)
Given that the <span>length of a social media interaction is normally distributed with a mean of 3 minutes and a standard deviation of 0.4 minutes, </span>the probability that an interaction lasts longer than 4 minutes is given by
![P(X\ \textgreater \ 4)=P\left(X\ \textgreater \ \frac{4-3}{0.4} \right) \\ \\ =P(X\ \textgreater \ 2.5)=1-P(X\ \textless \ 2.5)](https://tex.z-dn.net/?f=P%28X%5C%20%5Ctextgreater%20%5C%204%29%3DP%5Cleft%28X%5C%20%5Ctextgreater%20%5C%20%20%5Cfrac%7B4-3%7D%7B0.4%7D%20%5Cright%29%20%5C%5C%20%20%5C%5C%20%3DP%28X%5C%20%5Ctextgreater%20%5C%202.5%29%3D1-P%28X%5C%20%5Ctextless%20%5C%202.5%29)
We use the normal distribution table or calculator to evalute that P(X < 2.5) = 0.99379
Therefore, the probability that <span>an interaction lasts longer than 4 minutes = 1 - 0.99379 = 0.00621
[the last option]</span>