Question

At an intersection, the red-light times are normally distributed with a mean time of 3 minutes and a standard deviation of 0.25 minutes. Using the empirical rule, approximately what percent of red lights last between 2.5 and 3.5 minutes?

Answer 1

Percent of red lights last between 2.5 and 3.5 minutes is 95.44% .

Explanation:

Step 1: Sketch the curve.

The probability that 2.5<X<3.5 is equal to the blue area under the curve.

Step 2:

Since μ=3 and σ=0.25 we have:

P ( 2.5 < X < 3.5 ) =P ( 2.5−3 < X−μ < 3.5−3 )

⇒ P ( (2.5−3)/0.25 < (X−μ)/σ < (3.5−3)/0.25)

Since, Z = (x−μ)/σ , (2.5−3)/0.25 = −2 and (3.5−3)/0.25 = 2 we have:

P ( 2.5<X<3.5 )=P ( −2<Z<2 )

Step 3: Use the standard normal table to conclude that:

P ( −2<Z<2 )=0.9544

Percent of red lights last between 2.5 and 3.5 minutes is
`0.9544(100) = 95.44`% .