6. Normal Distribution and Continuous Random Variables

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.

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.

Suppose the scores earned on Professor McArthur’s third statistics exam are normally distributed with mean 64 and standard deviation 8. Professor McArthur wants to curve the exam scores as follows: The top 6% get an A, the next 14% get a B, the middle 60% get a C, the bottom 6% fail, and the rest earn a D. Any student who can determine these cut-offs earns five bonus points. Determine the cut-offs for Professor McArthur.

The ACT and SAT are two college entrance exams. The composite score on the ACT is approximately normally distributed with mean 21.1 and standard deviation 5.1. The composite score on the SAT is approximately normally distributed with mean 1026 and standard deviation 210. Suppose you scored 26 on the ACT and 1240 on the SAT. Which exam did you score better on? Justify your reasoning using the normal model.