Question: A bus driver drives the same route multiple times per day. The time it takes to complete one loop of the route follows a normal distribution with a mean of 54 minutes and standard deviation of 3 minutes. What is the probability it takes the driver more than 1 hour to complete one loop? (Use the Empirical Rule.)
My solution: I am thinking here that if one standard deviation is 3 and the 68-95-99.7 rule states that 99.7% of all data falls within 3 standard deviations of the mean, then that would be 54 - (3)(3) for the negative side of the mean and 54 + (3)(3) for the positive side of the mean, so then the 63minutes would exceed the hour mark making the probability would be .997?
Thank you
You have three problems with your answer. The times within three standard deviations of the mean are $45$ to $63$ minutes, which includes times both greater and less than one hour, so that is the wrong number of standard deviations to use. The breakpoint of one hour is mean plus two standard deviations, so you want the chance the random variable is greater than mean plus two standard deviations. The second problem is that you want the time outside the central peak. Your intuition should tell you that the probability should be small the time is outside the peak, while $0.997$ is very close to $1$. The rule says $0.95$ is within two standard deviations of the mean. The third problem is you only want one side of the area outside two standard deviations, so need to divide by $2$.