Question

Add an intersection with a red light times are normally distributed with a mean Time of three minutes and a standard deviation of .25 minutes approximately what percent of my lights last between 2.5 and 3.5 minutes

Answer 1

2.5 and 3.5 are two standard deviations from the mean, since 2.5 = 3 - 2(0.25) and 3.5 = 3+2(0.25).

The empirical (68-95-99.7) rule says that approximately 95% of a normal distribution falls within two standard deviations of the mean.

Answer 2

Required probability =0.95

Explanation:

Let X be the intersection with red light times

X is Normal with mean = 3 and std dev = 0.25

We have to find out the probability of X lying between 2.5 and 3.5 minutes

P(2.5<x<3.5)

Let us convert these to Z score

`2.5<x`implies

`z>(2.5-3)/(0.25) \\z>-2`

Similarly
`x<3.5 -->\\z<(3.5-3)/(0.25) \\=2`

Hence required probability

=P(
`-2<z<2) =2(0.475)\\=0.95`