Question

1. In cases where we do not know the sample distribution and we have sufficiently large n, then we can apply the Central Limit Theorem. That is to say, according to the CLT, x can have any distribution whatsoever, but as the sample size increases, the distribution of   will approach a normal distribution. The CLT formula is

Where n is the sample size (n ≥ 30), μ is the mean of the x distribution, and σ is the standard deviation of the x distribution.

In the early 1970’s a highway tunnel was opened West of Denver, CO on I-70. One concern of transportation and engineering experts is that vehicles maintain an appropriate speed while passing through the tunnel. If there is too much traffic in the tunnel at any one time (i.e., if traffic is moving too slow), then there could be air quality concerns. If traffic is moving too fast, then the possibility exists that safety issues will arise.

We know the x distribution has µ = 13.6 minutes, σ = 2.5 minutes and there are generally 50 vehicles in the tunnel at one time under normal traffic conditions. Experts suggest that if traffic will pass through the tunnel between 12 and 14 minutes, then all is well.

           

1. What is the percent of time that there should be no traffic safety issues while passing through the tunnel? In other words, what is the probability percent of traffic passing through the tunnel between 12 and 14 minutes?

Answer 1

Suppose, random variable denotes time (in minutes) taken by i-th vehicle.

From the given information,

We define,

By Central Limit Theorem (CLT),

(a)

Probability of traffic passing through the tunnel between 12 and 14 minutes is given by

   [Using R-code 'pnorm(1.131371)-pnorm(-4.525483)']

Hence, in 87.10% time there should be no traffic safety issues while passing through the tunnel.