Question

In mountain country, major highways sometimes use tunnels instead of long, winding roads over high passes. However, too many vehicles in a tunnel at the same time can cause a hazardous situation. Traffic engineers are studying a long tunnel in Colorado. If x represents the time for a vehicle to go through the tunnel, it is known that the x distribution has mean 15.5 minutes and standard deviation 4.5 minutes under ordinary traffic conditions.

From a histogram of x values, it was found that the x distribution is mound-shaped with some symmetry about the mean. Engineers have calculated that, on average, vehicles should spend from 14 to 17 minutes in the tunnel. If the time is less than 14 minutes, traffic is moving too fast for safe travel in the tunnel. If the time is more than 17 minutes, there is a problem of bad air quality (too much carbon monoxide and other pollutants).

Under ordinary conditions, there are about 50 vehicles in the tunnel at one time. What is the probability that the mean time for 50 vehicles in the tunnel will be from 14 to 17 minutes? Is there any safety hazard exists for average traffic flow?

Answer 1

By Central limit theorem, the sampling distribution of mean time for 50 vehicles in the tunnel follow Normal distribution with mean = 15.5 minutes and standard deviation = 4.5 / = 0.6363961

Probability that the mean time for 50 vehicles in the tunnel will be from 14 to 17 minutes

= P(14 < < 17)

= P( < 17) - P( < 14)

= P[Z < (17 - 15.5)/ 0.6363961] -  P[Z < (14 - 15.5)/ 0.6363961]

= P[Z < 2.36] - P[Z < -2.36]

= 0.9909 - 0.0091

= 0.9818

Since the probability of mean time for 50 vehicles in the tunnel will be from 14 to 17 minutes is very high, there does not exists safety hazards for average traffic flow.