6. Normal Distribution and Continuous Random Variables

Explain why P(X < 30) should be reported as < 0.0001 if X is a normal random variable with mean 100 and standard deviation 15.

The arrival times of the bus Alex takes to work follow a normal distribution, with $\mu =12\min$ after the scheduled arrival rime & $\sigma =3\min$. If the bus is scheduled to arrive at Alex's work 10 min before opening, what is the probability that Alex arrives on time (i.e. the bus is less than 10 min late)?

The heights of adult women are approximately normally distributed with a mean of $160\mathrm{cm}$ and a standard deviation of $7\mathrm{cm}$. Find the height $x$ such that $5\%$ of women are shorter than $x$.

Give three interpretations for the area under a normal curve.