Normal curve probability mean

Each week, John drives to his mother's house. The amount of time required for the trip varies and is normally distributed. On about 16% of trips, it takes him more than 54 minutes to reach his mother's house. On about 2.5% of trips, it takes him less than 33 minutes to reach his mother's house. Which is closest to the mean amount of time required to make the trip?

Full disclosure: I'm studying for a test and this is a question from the practice test and I don't know where to start. I have the answer; I want to know how to get there.

• No worries about asking self-study questions. You probably ought to add the self-study tag though and see the associated wiki. Mar 26 '14 at 23:06
• Your question will need to be modified as described at the link given by TooTone Mar 26 '14 at 23:26

When I read this problem I imagine a normal bell curve. The mean trip duration is exactly in the middle. 50% of the trips take less time than the mean, 50% of the trips take longer.

There is some trip duration below the mean, t1, where 40% of the trips take less time than t1. Conversely there's some trip duration above the mean, t2, where 40% of the trips take longer than t2. Draw t1 and t2 on the x-axis below the curve. Do you see that these two trip durations should be equally distant from the mean?

From there try to draw where 33 minutes and 54 minutes should lie on the x-axis, given the percentages in the question. One should be closer to the mean than the other.

Since the question is flagged as self-study, I'll just provide some hints to (hopefully) help you derive the solution. I'll amend/complete my answer based on your progress.

As a starting point, you might want express the information provided in the problem with some mathematical notation. To do so, define a random variable corresponding to the random quantity. What is its distribution? What probabilities are provided in the problem?