Speeds of automobiles on a certain stretch of I-85 at 4:00pm are normally distributed with a mean of 65 mph. 20% of cars are traveling at speeds between 55 and 65mph. What percentage of the cars is going faster than 75mph. Explain your reasoning.
EDIT: Once I find the standard deviation, I know how to solve the problem. I'm having the most trouble determining what the standard deviation is. Should I just plug in numbers for σ in the normalcdf command on the graphing calculator or is there are more accurate way to determine the standard deviation?
You could find the standard deviation, but it is not necessary to do so. The normal distribution is symmetrical about its mean $\mu = 65$ mph. If 20% are at speeds 55 to 65mph, then the rest of the 50% traveling below 65mph are the 30% going slower than 55mph.
Then, by symmetry, there are 20% going between 65 and 75mph, and 30% going faster than 75mph.
If you are also asked to find $\sigma,$ then let $X \sim \mathsf{Norm}(\mu = 65, \sigma)$ be the speed of a randomly selected car. You know that
$$P(55 < X < 65) = P\left(\frac{55-65}{\sigma} < Z < \frac{65-65}{\sigma}=0\right) = 0.20,$$
where $Z$ is standard normal. From printed normal tables you can deduce the value of $-10/\sigma$ and thus of $\sigma.$
Here is a plot of the normal distribution of the speeds of the cars. From left to right areas separated by vertical red lines correspond to probabilities .3, .2, .2, .3, respectively.